0xDE
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0xDE - LiveJournal.comTue, 01 Sep 2015 04:33:07 GMTLiveJournal / LiveJournal.com110111107784841personalhttp://l-userpic.livejournal.com/32934265/77848410xDE
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100100http://11011110.livejournal.com/315863.htmlTue, 01 Sep 2015 04:33:07 GMTLinkage
http://11011110.livejournal.com/315863.html
<ul><li><a href="http://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/">The Tutte–Coxeter graph</a> and its construction from the outer automorphisms of S<sub>6</sub> (<a href="https://plus.google.com/100003628603413742554/posts/Dog4Lr6HzUD">G+</a>)</li><br /><li><a href="http://blog.christianperone.com/?p=2840&a=2">Google's S2 spatial data structure</a> (<a href="https://plus.google.com/100003628603413742554/posts/4NC1nrdreew">G+</a>)</li><br /><li><a href="https://academia.stackexchange.com/questions/12052/how-many-arxiv-papers-are-uploaded-in-their-final-refereed-versions/12093#12093?newreg=0e423580788e44668fe42d39f6eba2c8">Question: how many arXiv papers are updated to their final journal versions?</a> (Answer: about half of them have non-empty journal-reference metadata; <a href="https://plus.google.com/100003628603413742554/posts/T9eTKXDNs3V">G+</a>)</li><br /><li><a href="https://plus.google.com/101584889282878921052/posts/N6mxUYNDZfX">Keleti's conjecture on the ratio of perimeter to area of a union of unit squares</a> (<a href="https://plus.google.com/100003628603413742554/posts/QGPn36eDNjA">G+</a>)</li><br /><li><a href="http://cstheory.stackexchange.com/q/32332/95">Reference for mixed graph acyclicity testing?</a> (Still not adequately answered, but the bounty has since expired; <a href="https://plus.google.com/100003628603413742554/posts/BPD6mZgWCkQ">G+</a>)</li><br /><li><a href="https://mittheory.wordpress.com/2015/08/15/purifying-spoiled-randomness-with-spoiled-randomness/">Purifying spoiled randomness with spoiled randomness</a> (<a href="https://plus.google.com/100003628603413742554/posts/irBbFtKFsRn">G+</a>)</li><br /><li><a href="https://plus.google.com/117663015413546257905/posts/BcWkaSbhGVR">Six-tetrahedron kaleidocycles don't work</a> (but it takes a computer simulation to demonstrate it, because paper ones do; <a href="https://plus.google.com/100003628603413742554/posts/CZX2trUS5TV">G+</a>)</li><br /><li><a href="http://nielsenhayden.com/makinglight/archives/016262.html">The new single-divisible-vote last-place-elimination scheme for Hugo nominations</a> (or as they call it, e pluribus Hugo; <a href="https://plus.google.com/100003628603413742554/posts/6XDdGCw86VS">G+</a>)</li><br /><li><a href="http://www.smh.com.au/world/russia-orders-wikipedia-page-blocked-over-cannabis-link-20150824-gj6sty.html">Wikipedia blocked in Russia</a> over its refusal to take down information about illegal drugs (since unblocked; <a href="https://plus.google.com/100003628603413742554/posts/RwXkRLHuhGW">G+</a>)</li><br /><li><a href="http://mybiasedcoin.blogspot.com/2015/08/cacm-viewpoints-on-theory-and.html">The two cultures of CS: theory vs experiment</a> (<a href="https://plus.google.com/100003628603413742554/posts/AAjVq5PneKF">G+</a>)</li><br /><li><a href="http://mathoverflow.net/a/215718/440">Six points in general position in 3d can be partitioned into triples determining two linked circles</a> (<a href="https://plus.google.com/100003628603413742554/posts/M2uzpUeznN9">G+</a>)</li><br /><li><a href="http://www.bouletcorp.com/hidden/quantum-pixel/">Boulet imagines what physicists in a pixelated world might think of it</a> (<a href="https://plus.google.com/100003628603413742554/posts/G1UpS5UZY6G">G+</a>)</li><br /><li><a href="http://www.prospectmagazine.co.uk/blogs/philip-ball/why-the-poohsticks-formula-is-wrong">Just because a formula looks mathy doesn't mean it makes any sense</a> (<a href="https://plus.google.com/100003628603413742554/posts/5ew2eu5TKHZ">G+</a>)</li><br /><li><a href="http://recode.net/2015/08/27/google-tells-developers-how-to-get-around-apples-new-security-rules-so-they-can-keep-selling-ads/">Google teaches spammers how to break security to get their ads through</a> (<a href="https://plus.google.com/100003628603413742554/posts/UvfckkmqjBU">G+</a>)</li><br /><li><a href="http://terraoko.com/?p=93652">Hyde Al Dzhazil, Yemeni town built on a rock</a> (<a href="https://plus.google.com/100003628603413742554/posts/hn52AjTJLNH">G+</a>)</li><br /><li><a href="http://westy31.home.xs4all.nl/Golay/GolayCodeAndSymmetry.html">Constructing the Golay error-correcting code from a great dodecahedron</a> (<a href="https://plus.google.com/100003628603413742554/posts/T8AnNLdcdUH">G+</a>)</li></ul>http://11011110.livejournal.com/315863.htmlvotingcorporatizationwikipediagraph theorydata structuressecuritybibliographycirclesgeometrypublic0http://11011110.livejournal.com/315529.htmlSun, 16 Aug 2015 00:17:52 GMTLinkage for the ides of August
http://11011110.livejournal.com/315529.html
<ul><li><a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2015/jul/30/bridges-2015-a-meeting-of-maths-and-art-in-pictures">Art from Bridges 2015</a> (<a href="https://plus.google.com/100003628603413742554/posts/gGrzyjNkipk">G+</a>)</li><br /><li><a href="https://twitter.com/gelada/status/626810879435083776">New convex pentagon tiler</a> (<a href="https://plus.google.com/100003628603413742554/posts/JC8hqyUHuDE">G+</a>; see also <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile">more detailed Guardian article</a>)</li><br /><li><a href="http://scholarlykitchen.sspnet.org/2015/07/16/when-do-citations-reflect-impact/">How both author intent and journal house style affect what a citation means</a> (<a href="https://plus.google.com/100003628603413742554/posts/BUrerbpJvLT">G+</a>)</li><br /><li><a href="http://www.cut-the-knot.org/pythagoras/ModernDaySangaku.shtml">Modern day Sangaku: why are these three circle centers collinear?</a> (<a href="https://plus.google.com/100003628603413742554/posts/K9HpuZudd1e">G+</a>)</li><br /><li><a href="https://plus.google.com/100003628603413742554/posts/D8jYC2fvsyq">Google stands up against extraterritorial censorship in the name of the right to be forgotten</a> (<a href="https://plus.google.com/100003628603413742554/posts/D8jYC2fvsyq">G+</a>)</li><br /><li><a href="http://motherboard.vice.com/read/how-a-victorian-astronomer-fought-the-gender-pay-gap-and-won">Maria Mitchell, Victorian astronomer who fought the pay gap</a> (<a href="https://plus.google.com/100003628603413742554/posts/hbFCzdeZuEN">G+</a>)</li><br /><li><a href="http://mappingignorance.org/2015/07/27/triangulations-are-rigid-you-can-do-better-using-pseudo-triangles/">Rigid structures from pseudo-triangles</a> (<a href="https://plus.google.com/100003628603413742554/posts/HodvVmmPMkP">G+</a>)</li><br /><li><a href="http://www.theatlantic.com/business/archive/2015/08/wikipedia-editors-for-pay/393926/">The fight against promotional editing on Wikipedia's medical articles</a> (<a href="https://plus.google.com/100003628603413742554/posts/QRedXktgTp8">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Dual_graph">Dual graphs</a>, my Wikipedia improvement project for the week (<a href="https://plus.google.com/100003628603413742554/posts/8WAd59thBht">G+</a>)</li><br /><li><a href="http://boingboing.net/2015/08/14/transgenic-mouse-company-pays.html">Bribing researchers to cite your papers</a> (<a href="https://plus.google.com/100003628603413742554/posts/8QhBpmT9x8w">G+</a>)</li><br /><li><a href="https://chrome.google.com/webstore/detail/wikipedia-with-mathjax/fhomhkjcommffnlajeemenejemmegcmi?hl=en-GB">How to make Wikipedia mathematics readable on Chrome</a> (<a href="https://plus.google.com/100003628603413742554/posts/EVf2TtyUXDi">G+</a>)</li></ul>http://11011110.livejournal.com/315529.htmlfeminismfree speechwikipediatilingbibliographycirclesartgeometrypublic0http://11011110.livejournal.com/315337.htmlTue, 11 Aug 2015 06:13:16 GMTWADS photos
http://11011110.livejournal.com/315337.html
<p>I took a few photos on <a href="http://11011110.livejournal.com/315105.html">my recent trip to WADS</a>, of the people at the conference and the scenery on the conference excursion. Below are the ones of individual people:</p>
<div style="clear:both"></div><div align="center"><table border="0" cellpadding="10">
<tr align="center" valign="middle">
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/JorgSack.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/JorgSack-s.jpg" border="2" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/FrankDehne.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/FrankDehne-s.jpg" border="2" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/FaithEllen.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/FaithEllen-s.jpg" border="2" style="border-color:black;" /></a></td>
</tr><tr align="center" valign="middle">
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/DavidKirkpatrick.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/DavidKirkpatrick-s.jpg" border="2" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/JesperNielsen.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/JesperNielsen-s.jpg" border="2" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/YushiUno.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/YushiUno-s.jpg" border="2" style="border-color:black;" /></a></td>
</tr></table><table border="0" cellpadding="10">
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<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/MikeGoodrichAndTsviKopelowitz.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/MikeGoodrichAndTsviKopelowitz-s.jpg" border="2" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/wads15/HaitaoWang.html"><img src="http://www.ics.uci.edu/~eppstein/pix/wads15/HaitaoWang-s.jpg" border="2" style="border-color:black;" /></a></td>
</tr></table></div>
<p><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/wads15/index.html">The rest of the photos</a> )</b></p>http://11011110.livejournal.com/315337.htmlconferencesphotographylandscapepublic0http://11011110.livejournal.com/315105.htmlSun, 09 Aug 2015 03:52:31 GMTReport from WADS
http://11011110.livejournal.com/315105.html
I just returned from Victoria, BC, where the University of Victoria (under the capable local organization of Ulrike Stege) was the host for <a href="http://wads.org">WADS, the Symposium on Algorithms and Data Structures</a>. (The acronym made more sense when it was calling itself a workshop.)<br /><br />Each of the three days of the workshop was scheduled to begin with an invited talk, although one of them had to be rescheduled due to United Airlines cancelling the flight from San Francisco to Victoria that should have carried one of the speakers. Apparently that connection has frequent issues; my taxi driver back to the airport was quite familiar with the phenomenon. My own connections involved Alaska Airlines and Seattle instead, but that didn't prevent three out of four of my flights from being delayed. Fortunately a scheduled two-hour layover saved me from missing the one non-delayed flight. Anyway, the three speakers were Bodo Manthey on smoothed analysis of local search (based on the principles that smoothed instances are unlikely to have small local steps and are likely to have large optimal solution values relative to any local minimum), Cyrus Shahabi on adding "spatial" to social networks (e.g. to infer social connections from colocation in order to design recommendation systems that put you into a bubble where you and your friends see the same recommendations), and Bernard Chazelle on what happens when you mix Markov processes (each state's new value gets replaced by a convex combination of neighboring states' values) with dynamic graphs (the definition of neighboring changes based on some function of the values).<br /><br />Each day also had one of my own talks, on <a href="http://11011110.livejournal.com/307881.html">parametric closures</a>, <a href="http://11011110.livejournal.com/308431.html">cycle bases</a>, and <a href="http://11011110.livejournal.com/303471.html">contact graphs of circular arcs</a>; I've put my talk slides online <a href="http://www.ics.uci.edu/~eppstein/pubs/Epp-WADS-15-slides.pdf">here</a>, <a href="http://www.ics.uci.edu/~eppstein/pubs/EppMcCPar-WADS-15-slides.pdf">here</a>, and <a href="http://www.ics.uci.edu/~eppstein/pubs/AlaEppKau-WADS-15-slides.pdf">here</a>. And in addition I ended up being asked to chair two of the sessions. So my choice of which contributed talks to see was somewhat constrained, and there were several I would have liked to see but didn't. My recollections below of individual papers and talks are necessarily going to be somewhat haphazard, and I've probably gotten some speaker names and results wrong.<br /><br />The first and last contributed talks that I saw were both by Ke Chan. In the first one, Wednesday morning, Chan presented a paper by Adrian Dumitrescu and Csaba Tóth giving an exponential bound on <a href="http://arxiv.org/abs/1411.1303">the number of convex polygons in a triangulated point set</a>, tight to within a polynomial factor. It involved a cute trick for reducing the problem to convex position: by choosing an origin within one of the triangles, mapping the points radially to the unit circle, and careful retriangulation, all of the convex polygons containing the origin could be placed in bijection with the convex polygons of the new set, preserving the points belonging to each polygon but not which ones were vertices. Ke later presented the last talk of the conference, his own work with Dumitrescu on <a href="http://arxiv.org/abs/1409.3600">median-of-medians selection</a>, in which he conjectured that one of the exercises in CLRS is wrong: if you modify the standard groups-of-5 deterministic selection algorithm to use groups of 3 or 4 instead, the natural recurrence for its runtime solves to O(<i>n</i> log <i>n</i>) instead of linear, but it may not be possible to find an input that is as bad as the recurrence suggests, and slight modifications to the algorithm can be proven to be linear.<br /><br />Also on Wednesday morning, Michael Kerber spoke on work with Sergio Cabello in which they solve motion planning problems by <a href="http://arxiv.org/abs/1502.03690">subdividing the free space of a moving object by a quadtree</a>, whose cells are either completely free, completely obstructed, or mixed (still needing subdivision to determine whether they can be passed through). To test whether the obstructed cells completely separate the start and goal positions, they use a union-find data structure augmented by parity bits that can be used to detect a cycle of obstacles that crosses the start-goal line segment an odd number of times. But I think there are additional interesting data structural questions to study for this problem. In particular, some mixed cells might be completely surrounded by a ring of free cells, making it useless to subdivide them: any path through them could be replaced by a path through the ring. Can we detect these useless cells quickly somehow?<br /><br />One of the Wenesday afternoon talks was by Fahad Panolan on the <a href="http://people.scs.carleton.ca/~wads/WADS2015-papers/paper_85.pdf">parameterized complexity of matroid girth</a>. The girth of a matroid is the size of its smallest circuit (or equivalently of its smallest dependent set); for graphic matroids it's the same as the graph girth, for transversal matroids it's the size of the smallest Hall set (a subset of one side of a bipartite graph that cannot be the set of endpoints of a matching), for binary matroids it's the size of the smallest "even set" (apparently a notorious open problem in parameterized complexity) and for linear matroids of low rank it includes the problem of detecting whether a point set is in general position or not. The paper presented strong complexity-based evidence that the problem is not FPT for linear matroids when parameterized by rank or solution size, but is FPT when parameterized by a combination of rank and field size. It would be interesting to clarify whether field size can be replaced by field characteristic here, or whether the lower bounds for large fields work even when those fields have low characteristic.<br /><br />If WADS had a best presentation award (it doesn't), my vote would have gone to Bill Bird, who spoke Thursday morning on his work with Wendy Myrvold on algorithms to generate <a href="http://people.scs.carleton.ca/~wads/WADS2015-papers/paper_49.pdf">all non-isomorphic colorings of graphs</a> and for determining the distinguishing numbers of graphs (the minimum number of colors needed in a coloring that destroys all the symmetries of the graph). His algorithms aren't necessarily faster than previously-known ones, but they use significantly less memory, something that can be even more of a bottleneck than runtime when things are exponential. Unfortunately seeing this one meant that I missed the parallel talk by Chang and Yen, that among other things used Courcelle's theorem to prove <a href="http://people.scs.carleton.ca/~wads/WADS2015-papers/paper_43.pdf">fixed parameter tractability of finding polygonal contact representations</a> of planar graphs parameterized by polygon complexity and graph treewidth.<br /><br />There was only one Thursday afternoon session, making time for the conference excursion to Butchart Gardens followed by a nice beachside dinner. I had a choice between parameterized graph algorithms versus data structures, and ended up choosing the algorithms. The first paper of the session, by Li, Wang, Chen, and Cao, was on <a href="http://arxiv.org/abs/1412.8296">finding a spanning tree with at least <i>k</i> internal nodes</a>. It was known that, if the given graph contains an independent set whose neighborhood has at most half its size, then the problem can be reduced to a smaller one, and previous approaches involved finding a depth-first spanning tree and (if it's not itself good enough but the graph is still large) using its leaves as the independent set, but that leads to a kernel of size 3<i>k</i>. Instead, the new paper shows that using 6-opt local moves on the tree to reduce its number of leaves leads to a 2<i>k</i>-size kernel and an O(4<sup><i>k</i></sup>)-time parameterized algorithm.<br /><br />Two other talks on Thursday, one in the morning and one in the afternoon, both considered puzzles in which you are trying to get from one point in a large state space to another by a sequence of local moves. The morning's paper, by a group of Japanese authors with David Kirkpatrick, involved <a href="http://people.scs.carleton.ca/~wads/WADS2015-papers/paper_102.pdf">getting from one coloring of a graph to another</a> by swapping the colors of adjacent vertices, using as few swaps as possible. It's NP-hard even for cubic planar bipartite graphs, but when there are only two colors it can be solved optimally by finding a minimum-weight collection of paths connecting pairs of opposite-colored mismatched vertices. In the afternoon, Amer Mouawad spoke about <a href="http://arxiv.org/abs/1502.04803">getting from one independent set to another by a sequence of 2-opt moves</a> (removing one vertex and adding another). Without even trying to minimize the number of moves, it's hard for graphs of bounded treewidth, pathwidth, bandwidth, etc., although it can be solved polynomially for bounded tree-depth. Mouawad and his co-authors showed that nevertheless it's fixed-parameter tractable, parameterized by the independent set size, for a much broader class of sparse graphs including the graphs of bounded degeneracy and the nowhere-dense graphs.<br /><br />There was a brief business meeting during the conference dinner. The WADS conference series has been organized for the last 26 years by Frank Dehne and Jörg Sack, but at the meeting Jörg announced that Frank would be stepping down, with Faith Ellen taking his place. I don't think the location for the next WADS (in two years) has been decided yet; in any case it wasn't announced.<br /><br />Friday we could all sleep in because of the delayed invited talk, except for a couple of unfortunate attendees who left the dinner early and didn't hear the announcement about the changed schedule. Yannik Stein spoke first, about his work with Korman, Mulzer, van Renssen, Roeloffzen, and Seiferth on <a href="http://page.mi.fu-berlin.de/yannikstein/publications/triang_tradeoff.pdf">time-space tradeoffs for Voronoi diagrams</a>. If you have only constant working memory, but linear read-only input memory and write-only output memory, you can still construct a Voronoi diagram in quadratic time. Stein's method trades off this bound with the usual sequential time bound for Voronoi diagrams by computing a good sample of the input, using it to divide the problem into many independent constant-space subproblems, and running all the subproblems in parallel so that when they need to scan the input they can all do so in a single batch.<br /><br />In the same session, Amir Nayyeri spoke about an interesting definition of the distance between two points in the plane (or some higher dimensional space) given some sample of points on a manifold, that is supposed to capture the property that <a href="http://arxiv.org/abs/1502.08048">paths through the manifold are cheaper than paths across space</a>. To define the length of a curve, one integrates the local feature size (distance to the nearest sample) along the curve; if you follow a polygonal path through sample points the result is approximately the sum of squares of edge lengths, but the definition works more generally. With only a single sample point, the optimal curve between any two points can be found by viewing the points on the plane through the two points and the sample as complex numbers (with the sample at zero), squaring them all, and then either connecting the two given points by a line segment (if they started within 90 degrees of each other) or by a two-segment path through the origin (otherwise). For larger samples they form a Voronoi diagram, place gateway points along the boundaries of the cells, and use the single-sample method within each cell, giving a polynomial time approximation scheme for the optimal curve.<br /><br />Finally, in the afternoon session, Thatchaphol Saranurak spoke about <a href="http://arxiv.org/abs/1506.08319">the dynamic optimality conjecture</a>. There are now two candidates for a self-adjusting online binary search tree data structure that might have a constant competitive ratio against all offline strategies for adjusting a binary search tree to be fast on an input sequence: splay trees and another data structure based on a greedy solution to a geometric reformulation of the problem (augmenting a point set to prevent any two points of the augmented set from determining an empty rectangle). If either of these methods is competitive, it must in particular be competitive against a deque, which can access items at the start and end of the sorted list only in constant time per operation, because it's not hard to simulate a deque by a binary search tree. However, only an inverse-Ackermann competitive ratio was known for splay trees. This talk presented an exponential-of-inverse-Ackermann competitive ratio for the greedy strategy, based on showing that the augmented point set avoids a certain pattern in the shape of a W. But more than this specific result, I think it's also interesting for handling insertion and deletion operations directly; past work in this area instead assumed a static set of keys.<br /><br />I also have some photographs of attendees that I took during the conference; I haven't yet finished processing them but will put up another post here once I do.<a name='cutid1-end'></a>http://11011110.livejournal.com/315105.htmlcomputational geometrygraph algorithmsconferencestalksalgorithmsdata structurespaperspublic0http://11011110.livejournal.com/314841.htmlWed, 05 Aug 2015 06:11:42 GMTMathML considered harmful
http://11011110.livejournal.com/314841.html
If you've been paying any attention to my blog posts and other online activity, you probably know that I'm a huge fan of Wikipedia. I think it's a great way to communicate theoretical research to a wider audience, a great way to practice writing in a setting that encourages writing for readability, and a great place to publish survey-like material. Since I began editing Wikipedia in 2006, I have made <a href="https://en.wikipedia.org/wiki/Wikipedia:List_of_Wikipedians_by_number_of_edits">over 90000 edits</a> and created <a href="https://en.wikipedia.org/wiki/Wikipedia:List_of_Wikipedians_by_article_count">over 700 new articles</a> (not counting redirects etc), most of them on mathematical subjects. I've also regularly been using collections of Wikipedia readings as textbooks in some of my classes for which there is no published text that matches the material I want to cover. I've encouraged others to contribute their expertise and will take the opportunity to do so again: Edit Wikipedia! Contribute your knowledge to the broader world!<br /><br />But if you've read many Wikipedia article on mathematical subjects, you'll know that they can have a few issues. The content may sometimes be amateurish and topics may be missing, but those can be fixed with some effort. (Edit Wikipedia! Contribute your knowledge!) Another, more stubborn issue concerns the formatting of its mathematical equations.<br /><br /><br /><b>What's wrong with Wikipedia's equation formatting</b><br /><br />In Wikipedia's default view (what you get if you don't create an account, log in, and change your appearance preferences), what you see for many equations is a bitmap graphics image of the equation. It's nicely typeset by TeX or something that mimics TeX, but the fonts don't match the text of the articles (especially in formulas that include pieces text themselves rather than just mathematical symbols or variables), the font sizes are generally too big, the text alignment is wrong, they look pixelated rather than matching the sharpness of the other text, you can't select or copy text from the middle of them, they are stuck in \displaystyle even for inline formulas (leading to ugly irregularities in line spacing), and they don't change color to indicate the existence of a link the way real text does.<br /><br />In 2006, that was a lot better than the mathematics formatting that you could do anywhere else on the web. On Wikipedia, you could edit formulas using familiar TeX markup and they would automatically appear. Everywhere else, you had the choice of editing html manually (and being limited to what could be formatted in html), or writing in other document-processing systems such as LaTeX and either publishing pdf files or using tools like <a href="https://en.wikipedia.org/wiki/LaTeX2HTML">LaTeX2HTML</a> to batch-produce HTML files with bare-bones formatting and bitmap image formulas.<br /><br />Later, some other web sites began springing up providing the ability to do the same thing as Wikipedia: enter TeX markup and get a bitmap image that you could include in your html files, with the same problems of font and alignment and pixelization. I probably still have several of these among my old blog posts, because LiveJournal has still never even caught up to the 2006 Wikipedia's ability to format math.<br /><br />And then along came <a href="https://www.mathjax.org/">MathJax</a>.<br /><br /><br /><b>MathJax and Wikipedia</b><br /><br />Introduced in 2009 after a joint project of the AMS, SIAM, and others, MathJax provided code to do the work of turning mathematical formulas into html markup so that you don't have to. Their slogan is "it just works", and it's true. Add one line of code to your web pages:<pre><script type="text/javascript" src="//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></pre>and then start formatting math using \(...\) for inline formulas and \[...\] for displayed equations. (If you want $...$ the setup is a few more lines.)<br /><br />MathJax has some minor issues, involving slow formatting and reflowing already-displayed text, and it's kind of a hack, but it really is easy and it really does work well, producing equations that are formatted as pieces of text, looking as sharp as any other piece of text, using fonts that match the surrounding text, and scaling properly with the surrounding text. MathJax quickly became the de facto standard for mathematics formatting on the web and has become widely used by sites including MathSciNet, MathOverflow, arXiv, the commercial journal publishers, etc.: almost everyone who publishes mathematics on the web.<br /><br />MathJax has newer imitators, as well. A good example is the Khan Academy's <a href="https://khan.github.io/KaTeX/">KaTeX</a>. Unlike many instances of MathJax, KaTeX runs on the server rather than the browser, making it feel faster and preventing some of the reflow glitches that affect MathJax. Unfortunately, it is also more limited than MathJax in the types of formulas that it can handle. When it works it works well but it's not a complete solution.<br /><br />The exception to all of this mathematics formatting wonderfulness has been Wikipedia. What does Wikipedia use as its default method of displaying mathematics in 2015, six years after MathJax became available? The same bitmap images it was using in 2006. It did add a MathJax user option (for logged in users only) <a href="http://11011110.livejournal.com/252718.html">in 2012</a>, but only grudgingly, and now <a href="https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_Signpost/2015-07-22/Technology_report">it's taken even that much away again</a>.<br /><br />If you want MathJax in Wikipedia, you'll have to hack it together yourself with custom Javascript, and of course that will do nothing for how most people see the articles you edit. And what is the reason for all of this foot-dragging, and the reason for this long post? <a href="https://en.wikipedia.org/wiki/MathML">MathML</a>, a failed non-solution to mathematics rendering that the Wikimedia developers are still trying to promote.<br /><br /><br /><b>Editors vs developers</b><br /><br />Some background on Wikipedia versus Wikimedia is probably appropriate here. It's confusing because the names are so similar, but they're different things. Wikipedia is a big multi-language encyclopedia, whose volunteer contributors are usually called editors. It has a big body of text, lots of rules about how to edit, and various pieces of bureaucracy to enforce those rules, but the bureaucrats are also volunteers. Mediawiki is the software on which Wikipedia runs, and Wikimedia is the nonprofit organization that develops and runs that software. It has actual computer farms and actual paid employees, some of whom are software developers, but it also uses volunteer software developers to augment the efforts of the paid ones. The Wikipedia editors and Wikimedia developers are not the same people, and have often had major disagreements on how best to improve the Wikimedia software. Even when they share common goals, such as expanding and diversifying the base of editors (Edit Wikipedia! Contribute your knowledge!) they often disagree on how to achieve them. Major flashpoints have included<ul><li>The <a href="https://en.wikipedia.org/wiki/Wikipedia:VisualEditor">Visual Editor</a>, a GUI intended by the developers to replace text editing of wiki-markup,</li><li><a href="https://en.wikipedia.org/wiki/Wikipedia:Flow">Flow</a>, a threaded discussion system intended to replace the article talk pages used by editors to discuss controversial changes to articles, compare alternative versions of article text, or request changes that other editors might be better able to make, and</li><li><a href="https://en.wikipedia.org/wiki/Wikipedia:Pending_changes">Pending changes</a>, a system for preventing new editors' changes from becoming visible until they are vetted by more experienced editors.</li></ul>Compared to these controversies, mathematics formatting and MathML are small potatoes: a small part of the encyclopedia, important to only a small fraction of editors and developers, but very important to that small number of people.<br /><br /><br /><b>Standards-based semantic markup</b><br /><br />Two issues that developers care much more about than editors and readers are semantic content markup and standards-based web design. From long before the dawn of the web, computerized document processing systems have tended to use one of three strategies for describing a piece of text and its formatting:<ul><li>One is to specify precisely what each mark in the output document should look like and where it should be placed; this geometry-based approach is used, for instance, in the pdf files commonly used to encode and transmit formatted scientific journal articles.</li><li>A second is to specify the intended human meaning ("semantics") of a piece of text, letting the software that receives the document figure out how that meaning should be translated into formatting. In LaTeX when we specify that something is a theorem (\begin{theorem} ... \end{theorem}) or in HTML when we specify that something is a top-level header (<h1> ... </h1>) or paragraph (<p> ... </p>) we are using semantic markup.</li><li>And a third is to mix the text with a sequence of ad-hoc commands that interact with a specific formatting engine and tell it what to do, but with no clearly defined geometry or semantics (like, for instance, plain TeX).</li></ul>In the early days of HTML, it used a mix of semantic and ad-hoc markup, leading to chaotic results such as web pages that were carefully coded to look nice on one browser displaying badly or not at all on others. Then, through the mid-to-late 1990s, there was a push to standardize both the coding of HTML markup and the formatting of the results by browsers, leading to the current system under which the basic document markup is purely <a href="https://en.wikipedia.org/wiki/Semantic_HTML">semantic HTML</a> (describing paragraphs or other divisions in a piece of text) but with an associated stylesheet that describes in a precise way how that semantics should be translated into formatting. As a result, one can now design a web page and be confident that it behaves in a predictable way on all modern browsers, subject only to unavoidable variations such as screen width.<br /><br />The same push for standards-based markup that led to the cleanup in HTML also led to many other new XML-based markup languages, and the hope that these languages could be used to represent pieces of HTML-based documents that required specialized markups. One of the more successful languages of this type has been <a href="https://en.wikipedia.org/wiki/Scalable_Vector_Graphics">SVG</a>, a language for storing scalable images built from simple geometric objects like lines and circles. Unlike previous vector graphics formats such as postscript and pdf, SVG images could be mixed with text in the middle of an HTML document, and unlike previous inline bitmap image formats such as GIF, PNG, and JPG they could be scaled to arbitrarily large sizes without loss of quality and did not require a separate image file download. SVG files are widely used today; for instance, they are the standard format for vector graphics on Wikipedia, and the LiveJournal service on which I keep my blog uses them as part of the login banner that you see on the top of many of its pages.<br /><br />But the success of this project has led the Wikipedia developers to push the same ideas into other areas where they fit less well, and in particular into mathematics formatting using another of the XML languages designed at this time, MathML.<br /><br /><br /><b>What is MathML?</b><br /><br />MathML is intended to provide a structured way to describe mathematical expressions, but from the start there were ambiguities in what this meant, that caused the language to be schizophrenic, really forming two languages in one and mirroring the split between presentation and markup described above. One of the two languages, called <a href="http://www.w3.org/TR/MathML2/chapter4.html">content markup</a> in the MathML standard, is intended to describe the underlying mathematical content of an expression: what it means in terms of calculating numbers from other numbers. For instance, the expression <i>x</i> + <i>y</i>/<i>z</i> means that you should divide <i>y</i> by <i>z</i> and then add <i>x</i> to the result. The markup used to represent this expression involves an "apply" token that represents the application of a function to its arguments; here there are two functions, addition and division, each with two arguments, which are either variables or the results of other function applications. So we get a piece of markup that looks like<pre><semantics>
<apply>
<plus/>
<ci> x </ci>
<apply>
<divide/>
<ci> y </ci>
<ci> z </ci>
</apply>
</apply>
</semantics></pre>In contrast, the other language within MathML, <a href="http://www.w3.org/TR/MathML2/chapter3.html">presentation markup</a>, is intended to describe the visual appearance of a mathematical expression. As in TeX, expressions are formed from subexpressions that are either concatenated together horizontally (as in the example expression <i>x</i> + <i>y</i>/<i>z</i> which has the five symbols <i>x</i>, "+", <i>y</i>, "/", <i>z</i> in a row), vertically (as in the other way of writing the same expression using a vertical fraction instead of a slash), or sometimes in more complex grids. These concatenations of symbols can be nested within each other, and the MathML specification recommends that this nesting should follow the semantics of the expression (even in presentation markup), so the same expression written using presentation markup would look like<pre><mrow>
<mi> x </mi>
<mo> + </mo>
<mrow>
<mi> y </mi>
<mo> / </mo>
<mi> z </mi>
<mrow>
</mrow></pre>(don't ask me why this isn't surrounded by a presentation tag; I copied this expression directly from the MathML manual).<br /><br /><br /><b>How to edit mathematics</b><br /><br />Needless to say, none of this is how mathematicians actually write mathematics using LaTeX, the standard document formatting system for mathematical publications. Instead, the same expression would be written in LaTeX as<pre>$x+y/z$</pre>or<pre>\[ x+y/z \]</pre>Here, just as in the MathML examples, the spaces are unimportant. The $...$ or \[...\] delimiters mark off mathematics from other text. Clearly, LaTeX is much more concise, and much more closely resembles the expression it describes. For those reasons it is also much easier to read and write by humans than either form of MathML. Fortunately, I don't think anyone has proposed making Wikipedia editors type in MathML formulas directly. Instead, for actually writing formulas in Wikipedia articles or elsewhere, there are two plausible options:<ul><li>Keep using LaTeX-like textual markup, and somehow translate it into MathML, or</li><li>Force everyone to use a GUI (see above for the huge wars between editors and developers already caused by this idea, before we even get to the mathematical part) and tack on a GUI editor for mathematics expressions.</li></ul>Fortunately, so far, option (1) seems to be prevailing on Wikipedia.<br /><br />So, as we've seen, MathML isn't a human-usable method of editing formulas. It's just too cumbersome compared to the alternatives. Instead, the best option for editing formulas is the one we've already been using, a more streamlined LaTeX-based markup format. But having a streamlined editing language isn't unique to mathematics. After all, the bulk of Wikipedia articles are written in a streamlined wiki-markup format, designed to be easier for humans to edit than pure html. For instance, in wiki-markup, paragraph breaks are indicated very simply by leaving a blank line between the paragraphs, just like LaTeX and quite unlike the way they are indicated in HTML. The Wikimedia engine takes care of translating this markup into proper HTML, and for the most part the translation goes very smoothly. Can't we do the same for mathematics markup, editing LaTeX but then translating it into proper MathML? That way, we would have convenient markup for editors, clean semantics for the server-browser communication channel, and nice readable formulas for readers, right? This, or something like this, seems to be the fervent hope of the Wikimedia developers. This hope is everything they have been working towards, and it is the reason they have so badly neglected any other way of rendering mathematics. But does it work? Can it work?<br /><br />To answer these questions, it is necessary to distinguish again between content (semantic) MathML and presentation MathML. Unless we're careful to distinguish these two languages, it's too easy to argue for or against the use of MathML by picking and choosing the advantages or disadvantages of one or the other style of markup, considering content MathML when it suits the argument and switching to presentation MathML when that choice would make a stronger argument. So let's treat them separately, beginning with content MathML.<br /><br /><br /><b>Why content MathML doesn't work</b><br /><br />To begin with, content MathML was never intended to be used for editing and then publishing text documents. That's what presentation MathML is for. Content MathML is aimed more at applications that need the ability to manipulate and evaluate mathematical functions: for instance, in symbolic algebra systems, or in the cells of a spreadsheet. But content MathML does have some tempting attractions for document publishing nevertheless. Foremost among them, I think, is that by representing the meaning of a mathematical expression, rather than its appearance, it would allow expressions to be presented to users of Wikipedia in other ways. For instance, maybe a blind person could have the formula described to them audially rather than visually? Additionally, if a document describes the meaning of an expression, it might be possible to allow readers to plug in example values and see what it does to them.<br /><br />However, this mode of MathML comes with severe drawbacks that, I think, make it unusable for Wikipedia. One of these is that the problem of inferring meaning from a mathematical expression is too difficult and requires too much context. If I write the expression <i>xRy</i>, what do I mean? It could be that I wish to multiply three real numbers, symbolized by the three letters. It could be instead that the lower-case letters are vectors and the upper case letter is a matrix; again, I might want to multiply them, but the multiplication function for these objects is a different function than the one for real numbers. It could be that I am writing about context-free grammars, that the lower case letters are terminal symbols of a grammar and the upper case letter is a non-terminal symbol, and that writing them next to each other represents string concatenation. It could be that the upper case letter represents a binary relation, and the expression represents a truth value, true when <i>x</i> and <i>y</i> have the given relation to each other and false when it doesn't. Or it could even be that (as used to be popular among some types of logician and for all I know could still be) this expression is using the Polish notation for function applications, that the first two letters are unary functions, and that the expression describes the application of these functions to the value represented by the third letter. Each of these possibilities would have a different expression in content MathML (and a different sequence of words as spoken rather than written mathematics), but the LaTeX markup just doesn't include the information needed to tell which is intended. Unless we develop a true artificial intelligence that can read and understand the surrounding text, we can't transform LaTeX into content MathML correctly, and we've already seen that the other options for directly editing content MathML are too cumbersome.<br /><br />Even if we could edit content MathML, we'd have a second problem: converting it back into a visual representation for the bulk of the Wikipedia users, the ones who read the articles and formulas. But this problem is also highly ambiguous. If an expression involves division, should it be presented with a slash or a vertical fraction? If it involves function application, should it be presented with the standard mathematical notation of a function name followed by a parenthesized argument list, with the Polish notation of the logicians, or maybe (for binary functions) as an infix operator? If it involves multiplication, should it be presented with a diagonal cross multiplication symbol, a centered dot multiplication symbol, or by just placing the arguments next to each other with no symbol between them? These are editorial decisions. They need to be left to the human editors of wikipedia, and represented in the markup that they edit. But they don't translate into content MathML, because it's the wrong kind of MathML for representing these distinctions.<br /><br />With no way to accurately translate wiki markup to content MathML and no way to translate content MathML back into a visual formula in a way that respects the wishes of the editors, it is a non-solution.<br /><br /><br /><b>Why presentation MathML doesn't work</b><br /><br />That leaves presentation MathML, and there the difficulties are more about practicality than about theoretical ambiguities. Can LaTeX markup be translated to well formatted presentation MathML? Sure, that conversion already exists and works reasonably well for simple formulas. When editors get too tricky with their LaTeX markup (for instance <a href="http://tex.stackexchange.com/questions/174118/not-independent-sign-in-latex">using negative spacing to cobble together a "not independent of" slashed double-up-tack symbol from smaller pieces</a>) the quality of the presentation MathML will suffer, but it can presumably still represent the same combination of pieces and produce roughly the same visual appearance.<br /><br />Does presentation MathML produce a semantically clean server-browser communications channel? No, the semantics are in the other kind of MathML, but this issue is unimportant to anyone but the developers.<br /><br />Can presentation MathML be used to produce a spoken rather than visual presentation of a formula? No, because the meaning that would allow a good spoken translation has been lost, and using speech to describe the visual layout of a mathematical formula would be worse than just spelling out the original streamlined markup. Sight-impaired users would need to avoid the translation to presentation MathML, and instead just run the LaTeX markup directly through their text-to-speech systems, but that option already exists. So (unlike semantic markup) presentation markup is no help in accessibility, but (also unlike semantic markup) it can actually be made to work.<br /><br />Yes, but can presentation MathML be made to actually work well? Here's where it falls down badly. It is neither necessary nor sufficient for good presentation of mathematical formulas.<br /><br />It is not necessary, even if one wants a clean server-browser communications channel without browser-side hacks, because all it's doing is describing the relative position of visual marks on a screen, which is a function that HTML already provides. Systems such as MathJax and KaTeX have already shown that HTML alone is perfectly capable of describing the visual presentation of mathematical formulas.<br /><br />And it is not sufficient, because formatting complex mathematical expressions is just too much of a niche market compared to formatting HTML text and SVG graphics that it's not worth the time of the big browser developers to put much effort into doing it well. Some major browsers do a credible job of formatting presentation MathML. Some claim to be able to format presentation MathML but do it very badly. And some have never supported it or (like Chrome, the one I use most often) have retracted their support for MathML. I don't see this situation as likely to change, because it's not just a matter of programming it once and forgetting about it; any large chunk of code in a browser requires continued development, development costs money, and the cost-benefit ratio for MathML to the browser developers is too low to make its continued development worthwhile.<br /><br />So, when you format mathematics using presentation MathML, you can't be guaranteed that your readers can see good results, and you can't be guaranteed that they will see any results at all. Instead, you have to detect the failure to format MathML and have a fallback formating method that can be used whenever MathML doesn't work. But that fallback is actually the common denominator for your users, the one that limits the quality of your presentation to them. For Wikipedia, so far, that fallback has been bitmap images, with all their problems. The plan for the foreseeable future is to eventually change it to be SVG images, with almost all of the same problems: they scale better than bitmaps, and require less browser-server communication than bitmaps, but they have all of the other problems that bitmaps have already revealed. Any developer effort put into improving MathML support is developer time not spent on making the fallback work, nor on choosing a better fallback.<br /><br /><br /><b>Blame MathML</b><br /><br />It would be one thing if we just had ugly but readable formulas. But because Wikipedia's built-in mathematical formatting has been so bad for so long, with so little hope of improvement, Wikipedia editors have been drawn to several alternative solutions for Wikipedia, that integrate mathematical formulas better into the surrounding text at the expense of formatting them worse as mathematics. Some articles use Wikipedia's native formatting capabilities (such as changes to slanted rather than upright fonts) together with HTML coding (for subscripts and superscripts); others use more complicated templates that change fonts to make mathematical formulas look like LaTeX math. These have their own problems; the wiki-formatting solution uses sans-serif fonts, not a good idea for math (where it is important to be able to distinguish upright lower-case L's, the number one, and vertical bars from each other), the font-switching used by the templates doesn't work on all platforms (notably, not on the Android Wikipedia app), and neither method can handle complex formulas. So most articles that use these workarounds also use Wikipedia's math, leading to different appearances for the same variable names. And they have no accessibility features. Despite all that, most editors consider these methods to be better than the alternative, Wikipedia's actual math formatting capabilities. Even if Wikipedia eventually does switch to better math formatting, it will take a long time and a lot of editing work to get the articles to catch up.<br /><br />So that's where we are with Wikipedia, mathematics, and MathML. Content MathML can't work, but has distracted the developers with the false hope of clean semantic markup and accessibility. Presentation MathML can't be edited, doesn't have clean semantics, isn't accessible, and doesn't actually work for most users. In the meantime everyone is stuck seeing image files for formulas, despite all their drawbacks and despite the existence of better solutions. The developers have put in very little effort even to reduce the more easily-fixed of the problems with image files, nor to find a better solution, because they have their sights fixed on MathML and won't work on anything else. Logged-in editors are able to set a preference for Wikipedia to translate math into MathML for their own browsing, allowing editors who use the right browsers to get formatting that's as good as, but no better than, MathJax, but this doesn't help the rest of us and it doesn't help the people who just want to read Wikipedia. Editors have lost hope of ever having good mathematics formatting and have switched to half-baked workarounds with their own problems. In effect, by distracting the developers and preventing them from improving the default mathematics formatting on Wikipedia, the existence of MathML has made Wikipedia both harder to edit (because editors need to deal with all the workarounds, whose markup is much worse than LaTeX) and harder to read (compared with other mathematics sites now using MathJax). That seems a big price to pay for very little benefit.<br /><br />So, by all means, edit Wikipedia! Contribute your knowledge to the broader world! Just don't expect the mathematical formulas to be as pretty as you might expect anywhere else on the web. And blame MathML for the problem, rather than thinking of it as any kind of solution.<br /><br /><b>( <a href="https://plus.google.com/u/0/100003628603413742554/posts/XEfbfT6C1M6">See also more discussion of this post on Google+</a> )</b><a name='cutid1-end'></a>http://11011110.livejournal.com/314841.htmlwikipediamathematicspublic7http://11011110.livejournal.com/314526.htmlMon, 03 Aug 2015 05:39:22 GMTZio Ziegler in Irvine
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Street art? In Irvine? Apparently, now, the answer is yes. Local clothing manufacturer <a href="http://www.tillys.com/">Tillys</a> somehow <a href="http://www.ocregister.com/articles/ziegler-670632-wall-work.html">persuaded the planning commission</a> to allow them to commission <a href="https://en.wikipedia.org/wiki/Zio_Ziegler">Zio Ziegler</a> (new Wikipedia article) to decorate one of their warehouses, right next to interstate 405, where approximately 240,000 daily drivers will see it.<br /><br />You can't exactly stop on the freeway to take photos, but I found enough other more accessible vantage points to get a few shots:<br /><br /><div align="center"><a href="http://www.ics.uci.edu/~eppstein/pix/tillys/6.html"><img src="http://www.ics.uci.edu/~eppstein/pix/tillys/6-m.jpg" border="2" style="border-color:black;" /></a></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/tillys/index.html">The rest of the photos</a> )</b>http://11011110.livejournal.com/314526.htmlgraffitiphotographypublic0http://11011110.livejournal.com/314209.htmlSat, 01 Aug 2015 04:39:27 GMTLinkage
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<ul><li><a href="https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Lorenz_cipher">W. T. Tutte's WWII cryptography work</a> (<a href="https://plus.google.com/100003628603413742554/posts/fWEQSPQqdkT">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=SL0HZ5wNrV8">Cube into rhombic dodecahedron dissection video</a> (<a href="https://plus.google.com/100003628603413742554/posts/Zx77brWvJJ4">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2015/07/the-sandy-beach-architecture-of-calvin-seibert/">Calvin Seibert's geometric sand castles</a> (<a href="https://plus.google.com/100003628603413742554/posts/5SdMxzh5Twh">G+</a>)</li><br /><li><a href="http://www.mathopt.org/?nav=fulkerson_2015">Paco Santos' Hirsch counterexample wins him the Fulkerson Prize</a> (<a href="https://plus.google.com/100003628603413742554/posts/df4on188Hzm">G+</a>)</li><br /><li><a href="http://www.islamicgeometric.com/#!gallery-one/c9im">Tessellating calligraphy</a> (<a href="https://plus.google.com/100003628603413742554/posts/i31x9zARNGL">G+</a>)</li><br /><li><a href="https://plus.google.com/+Aperiodical/posts/CjApiKYBdYV">MacTutor History of Mathematics website honored by the London Mathematical Society</a> (<a href="https://plus.google.com/100003628603413742554/posts/Zm7crptZp93">G+</a>)</li><br /><li><a href="https://cameroncounts.wordpress.com/2015/07/20/good-news-on-metrics/">The backlash against using numerology to measure research quality gains strength</a> (<a href="https://plus.google.com/100003628603413742554/posts/Umcr98gDDQL">G+</a>)</li><br /><li><a href="http://www.scientificamerican.com/article/developing-brains-fold-like-crumpled-paper-to-get-their-convolutions1/">Turns out that folded brain surfaces and crumpled paper balls obey the same scaling laws relating uncrumpled area, crumpled size, and surface thickness</a> (<a href="https://plus.google.com/100003628603413742554/posts/A1zf9HvCeWZ">G+</a>)</li><br /><li><a href="http://chronicle.com/article/The-Myth-That-Academic-Science/231413/">The <i>Chronicle of Higher Education</i> debunks recent claims that anti-women discrimination in academia is a thing of the past</a> (<a href="https://plus.google.com/100003628603413742554/posts/V5n1sKYeamm">G+</a>)</li><br /><li><a href="http://blogs.oregonstate.edu/glencora/2015/07/20/1181/">Academics deserve weekends too</a> (<a href="https://plus.google.com/100003628603413742554/posts/7pB9NgJoxoD">G+</a>)</li><br /><li><a href="http://techcrunch.com/2015/07/27/google-weans-itself-off-of-google/">Why it's a good idea to keep reposting these link roundups here</a> rather than trusting G+ to stay alive (<a href="https://plus.google.com/100003628603413742554/posts/AqaLfYKv5u5">G+</a>)</li><br /><li><a href="http://www.wired.com/2015/07/finally-know-graphene-good-origami/">Graphene kirigami nano-scale springs</a> (<a href="https://plus.google.com/100003628603413742554/posts/S1zxTr6aiqw">G+</a>)</li><br /><li><a href="http://www.itechpost.com/articles/15436/20150729/german-security-watchdogs-want-facebook-to-allow-use-of-aliases.htm">Germans to Facebook: allow pseudonyms!</a> (<a href="https://plus.google.com/100003628603413742554/posts/EMhRavdZytG">G+</a>)</li><br /><li><a href="http://blogs.ams.org/visualinsight/2015/08/01/heawood-graph/">Visualizing the Heawood graph</a> (using Wikipedia illustrations by me and others; <a href="https://plus.google.com/100003628603413742554/posts/TXPW42qtRPe">G+</a>)</li></ul>http://11011110.livejournal.com/314209.htmldissectiontoolsgraph drawingcryptographyanonymityscienceartacademiapublic0http://11011110.livejournal.com/314016.htmlTue, 28 Jul 2015 06:55:47 GMTOrange County Fair
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My daughter is doing a summer internship at the visual arts competition of the Orange County Fair: she helped bring in the submitted pictures, hung them, and assisted the judges, and now while the fair is on she gets to stand around telling people not to bring drinks into the exhibit. All very educational...<br /><br />Anyway, this weekend we went to the fair ourselves, had Sara show us her favorite photos in the exhibit (as a staff member she got to assign three staff award ribbons to the entries), visited the ice sculptures, tasted some of the wines (my favorite: the Gold Hill 2011 El Dorado Zinfandel), ate greasy food on a stick (we did not try the deep-fried koolaid nor the caviar-encrusted twinkies), watched the pig races, etc. And of course, took plenty of photos.<br /><br />The one below is of the stiltwalking leader of a steampunk band that passed us at one point during our visit:<br /><br /><div align="center"><a href="http://www.ics.uci.edu/~eppstein/pix/ocfair15/SteampunkParade2.html"><img src="http://www.ics.uci.edu/~eppstein/pix/ocfair15/SteampunkParade2-m.jpg" border="2" style="border-color:black;" /></a></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/ocfair15/index.html">The rest of the photos</a> )</b>http://11011110.livejournal.com/314016.htmlphotographypublic0http://11011110.livejournal.com/313774.htmlSat, 25 Jul 2015 07:24:51 GMTWhen AVL trees are perfect
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One of my students, Will Devanny, is teaching a summer-session offering of our lower-division undergraduate data structures class. (My university forbids graduate students from being the instructor of record for classes during the regular term, despite allowing equally-qualified non-student lecturers, but encourages students to teach during the summer; don't ask me why.) Anyway, he asked his students the following question: Suppose you are given as input a sorted sequence of numbers. Describe an efficient algorithm for constructing a perfectly balanced binary search tree with these numbers as keys. (Here perfectly balanced means that the depths of the leaves of the tree are all within one of each other.)<br /><br />There are many different ways of answering the question correctly, of varying degrees of trickiness. If I were given the question, I'd be tempted to give an answer that uses the following function to determine the index of the parent of the item at position x (with zero-based indexing):<pre>def parent(x):
y = (x+1 &~ x)
z = x & (y+y)
return (x | y) &~ z</pre>(And then I'd get marked down for a solution that has the optimal depth but is not balanced...) Instead, since the students had just learned about <a href="https://en.wikipedia.org/wiki/AVL_tree">AVL trees</a>, some of them thought that AVL trees must be part of this problem's solution. They gave as their answer: insert the numbers into an AVL tree in sorted order.<br /><br />You might think that this is a wrong answer, because AVL trees don't have to be perfectly balanced. And probably the students giving this answer misunderstood the question, or the behavior of AVL trees, or both. But it turns out to be correct! If you insert items into an AVL tree in sorted order, without doing any deletions, you will always get a perfectly balanced tree. More precisely, as can be shown by induction, the right spine of the tree will always have a sequence of complete trees dangling from it, so that the leaves within each of these trees are all at the same level and the leaves of any two of these trees are all at levels that are within one of each other. Here are the first few steps of the process:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/IncrementalAVL.png"></div><br /><br />The next steps, inserting 10 and 11, do the same thing to the right subtree that the insertions of 6 and 7 did to the whole tree. But when we insert 12, we get a right subtree that looks like the whole tree for 8, too deep to balance the left subtree, so we rebalance the root making the left subtree be a complete binary tree one level deeper, etc.<br /><br />One potential flaw of the AVL tree solution is that inserting <i>n</i> points into an AVL tree normally takes time <i>O</i>(<i>n</i> log <i>n</i>), while other solutions take linear time. But because of the special insertion order, as long as you maintain a pointer to the last node in the tree and do the insertion bottom-up instead of top-down, the total time can still be made to be linear.<br /><br />Morals of the story: check your student's crazy answers instead of just assuming that they must be wrong. For that matter check your own answers instead of just assuming that they must be right. And remember this cute fact about AVL trees as possibly forming the basis of a homework or exam question in a more advanced class, or at least as a spoiler answer to watch out for when you ask Will's question.<a name='cutid1-end'></a>http://11011110.livejournal.com/313774.htmlalgorithmsdata structurespublic2http://11011110.livejournal.com/313473.htmlWed, 15 Jul 2015 20:33:58 GMTLinkage
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<ul><li><a href="http://www.dataisnature.com/?p=2183">Thomas Sopwith’s wooden models of the earth's geological strata</a> (<a href="https://plus.google.com/100003628603413742554/posts/5yfGMPiTtK8">G+</a>)</li><br /><li><a href="http://theadvocate.com/news/12794452-123/national-organization-defends-lsu-professor">LSU president F.King Alexander fires tenured professor for saying "F.King" too many times in front of the students</a> (<a href="https://plus.google.com/100003628603413742554/posts/SH7mBZChBLD">G+</a>)</li><br /><li><a href="http://www.pe.com/articles/course-772388-baron-college.html">College administration backs faculty against student demands for trigger warnings on a course about graphic novels</a> (<a href="https://plus.google.com/100003628603413742554/posts/SAwAvp1pTP2">G+</a>)</li><br /><li><a href="http://hyrodium.tumblr.com/post/123270340099/the-reason-why-involute-gears-turn-smoothly-fig">A graphical explanation of how involute gears work and why they turn so smoothly</a> (<a href="https://plus.google.com/100003628603413742554/posts/WFAMuvY5kzP">G+</a>)</li><br /><li><a href="http://boingboing.net/2015/07/07/kickstarting-custom-cellular-a.html">Cellular automaton scarf kickstarter</a> now with only one week left (<a href="https://plus.google.com/100003628603413742554/posts/MZJCpLZkEoh">G+</a>)</li><br /><li><a href="https://unlockingresearch.blog.lib.cam.ac.uk/?p=192">Dutch universities boycott Elsevier</a> and <a href="http://www.buzzfeed.com/janebradley/as-assad-butchered-his-people-this-london-firm-helped-his-ba">Elsevier's support of Syria and Iran</a> (<a href="https://plus.google.com/100003628603413742554/posts/YineWoD67zb">G+</a>)</li><br /><li><a href="http://boingboing.net/2015/07/09/pirate-meps-copyright-reform.html">Freedom of panorama in Europe safe again for now</a> (<a href="https://plus.google.com/100003628603413742554/posts/PtFjaN4u3BH">G+</a>)</li><br /><li><a href="http://www.academiaobscura.com/academic-nursery-rhymes/">Academic nursery rhymes</a> (<a href="https://plus.google.com/100003628603413742554/posts/eNxyC8sKsCP">G+</a>)</li><br /><li><a href="https://plus.google.com/+ChandlerCarruth/posts/BaJayWnkiWw">Why and how you should turn off auto-running plugins now</a> (<a href="https://plus.google.com/100003628603413742554/posts/AKTznAR4zoB">G+</a>)</li><br /><li><a href="http://www.dataisnature.com/?p=2191">Mark A Reynolds – Intersecting the Void by Intervals</a> (constructivist art involving overlaid tilted grids and curves; <a href="https://plus.google.com/100003628603413742554/posts/VLNvUiHWVUu">G+</a>)</li><br /><li><a href="https://medium.com/matter/the-web-we-have-to-save-2eb1fe15a426">What we lost when we shifted from personal web sites and blogs to mass content aggregation systems</a> (and why I continue to maintain collections of links such as this one somewhere a little more permanent and searchable than my <a href="https://plus.google.com/100003628603413742554/posts/FDPx4btLzuA">G+</a> account)</li><br /><li><a href="https://plus.google.com/+DavidRoberts/posts/gKGSrfxVcMR">Grete Hermann's early work in the formulation of concrete complexity bounds for algorithms</a> (<a href="https://plus.google.com/100003628603413742554/posts/PQ8G5v9gSrz">G+</a>)</li></ul>http://11011110.livejournal.com/313473.htmltoolscellular automataacademiaartalgorithmsphotographypublic0http://11011110.livejournal.com/313272.htmlWed, 08 Jul 2015 04:38:49 GMTWhy you can't fold a paper bag
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<p>You're probably familiar with folding paper bags. You know, the small ones for lunches or the large ones for groceries? They have a triangular folding pattern on their sides that lets them either fold flat or open up into a brick shape. Five sides of the brick are covered by the paper of the bag, and the sixth top side makes an opening that you can put things into.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/BrownPaperBags.jpg" width="600" height="400"><br>
<font size="1">Image from <a href="http://bougieblackgirl.com/we-are-all-black-but-you-as-a-light-skin-person-benefit-from-light-skin-privilege/">this post on racism and the paper bag test</a></font></p>
<p>But did you know that you can't actually open these bags or fold them up again purely by folding? To change them from their folded to their unfolded state, or vice versa, you also have to bend and twist the paper in the regions between the folds. Try it with a real bag: you should feel a distinct snap in the paper as you unfold it. And although you can start to fold it up again without the same snap, at least to the point where the top closes up (lke the upper middle bag in the image above), this partially folded state involves subtly twisting the sides of the bag, so that the creases down the middles of the sides are opened out near the bottom of the bag and folded up near the top of the bag. In fact, although both the folded and unfolded states keep the paper flat wherever it is not creased, none of the intermediate states between folded and unfolded can be realized with flat paper. So if you tried to make one of these bags from something more rigid like sheets of glass, connected by hinges at the creases, it wouldn't work.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/GlassCranes.jpg" width="480" height="480"><br>
<font size="1"><a href="https://www.etsy.com/listing/209462085/stained-glass-origami-sadakos-peace">Stained glass cranes by Suncatchercreations on Etsy</a></font></p>
<p>I have a small part in a new preprint analyzing this phenomenon, "<a href="http://arxiv.org/abs/1507.01644">Rigid Origami Vertices: Conditions and Forcing Sets</a>" (with Abel, Cantarella, Demaine, Hull, Ku, Lang, and Tachi, arXiv:1507.01644). Even paper bags are too complicated, so we simplify the situation by looking at folding patterns formed from a flat sheet of paper by a set of creases that all meet at a single vertex. When does such a pattern have a continuous folding motion that starts from its flat state and uses all the creases while avoiding any bends between the creases? It turns out to be necessary and sufficient for the pattern to contain what we call a "bird's foot". This is a configuration that has folds of both types (mountain and valley), such that for at least one of these two types of folds, the convex hull of the folds has the vertex in its interior. There are four different ways of doing this: there can be three mountain folds separated by angles that are all less than <i>π</i> together with a single valley fold, four mountain folds on two crossed lines together with a single valley fold, or the same things with mountains and valleys swapped. Extra folds are ok, as long as one of these patterns is in there somewhere.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/BirdsFeet.png"></p>
<p>So how does our analysis apply to the paper bag folding pattern? Paper bags have three types of vertex:</p>
<ul>
<li>There are four vertices on the corners of the bag, with three mountain folds (the three edges of the bag that meet at the corner) and one valley fold (part of the triangle of creases that allows the bag to fold flat). Our theory doesn't apply to these vertices, because they can't be spread out to a flat sheet of paper. But they're not a problem, because they can be continuously folded in a range of states between their flat-folded and unfolded states.</li>
<li>There are two vertices in the middle of the two folded sides of the bag, forming the apexes of the triangles of creases. They have three valley folds (connecting the vertex to two corners of the bag and the top midpont of the side) and one horizontal mountain fold. This forms a bird's foot, so again it's ok.</li>
<li>The remaining two vertices lie on the edges of the bag. At these vertices, four folds meet at right angles. Two of the folds are mountain folds along the edge of the bag, and the other two are horizontal creases, one a mountain and the other a valley. This is not a bird's foot, because there are only three mountain folds and two of them are separated by an angle of <i>π</i>, while a bird's foot should have angles less than that. It has a continuous motion from its flat state to its folded state, but not one that happens all at once. Instead, you have to first fold the two parallel mountain folds (the ones along the edge of the bag) keeping the other two folds completely unfolded. Then, once the two mountain folds have been folded completely flat, you can start foldng the other two folds. But this two-stage motion is incompatible with the motion of the other vertices of the bag, and this is what prevents the bag from folding without bending or twisting.</li>
</ul>
<p>Is the inability to fold a paper bag a flaw in its design? I don't think so. First, obviously it is still possible to open or close the bag, because paper is not rigid. And second, I think the fact that a little snap is needed to open or close the bag may actually give the bag some strength: it stays fixed in its closed or open states easily, without flopping between the two states.</p>
See also two earlier papers by Ballcom, Demaine, and others, more specifically about paper bag folding, in the <a href="http://erikdemaine.org/papers/PaperBag_CGW2004/">2004 Fall Workshop</a> and in <a href="http://erikdemaine.org/papers/PaperBag_OSME2006/">Origami<sup>4</sup></a>.<a name='cutid1-end'></a>http://11011110.livejournal.com/313272.htmlorigamipaperspublic0http://11011110.livejournal.com/312903.htmlTue, 07 Jul 2015 04:16:15 GMTFast farthest-first traversal
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There are a lot of ways of ordering points so that nearby points are likely to be nearby in the ordering; space-filling curves are one good example. But what if you want nearby points to be far apart in the ordering? In this case, a good choice is the farthest-first traversal (also known as greedy permutation), the subject of my latest preprint (with Sariel Har-Peled and Tasos Sidiropoulos, <a href="http://arxiv.org/abs/1507.01555">arXiv:1507.01555</a>). This is an ordering of the given points determined by a greedy algorithm, in which each successive point is chosen to be as far as possible from all previously-chosen points.<br /><br />As far as I can tell, this permutation was first introduced as the ordering used for the "farthest insertion" heuristic for the traveling salesman problem, by Rosenkrantz, Stearns, and Lewis (SICOMP 1977), but it is better known from the work of Gonzalez on clustering (TCS 1985). One nice property of this permutation is that every prefix of it gives a well-spaced sample of the whole point set. So, as Gonzales showed, taking the first k points of this permutation as cluster centers, and assigning all remaining points to the nearest center, gives a good approximation to the k-center problem of partitioning the points into k clusters while minimizing the maximum cluster radius. You can compute the permutation only once, and then use the same permutation for clusterings with different numbers of clusterings, rather than having to recompute the whole clustering when k changes.<br /><br />But the same properties are useful for many other problems, and have caused this same ordering to have been re-used and re-discovered for many other applications. The introduction of the preprint surveys a few of these, including choosing a small number of representative colors from the colors used by a given bitmap image in order to quantize the image (say when compressing down from png-16 to png-8); choosing a small number of representative points from the free space of a robot in order to build a compact road map for the free space; and selecting a well-spaced subset of points in the plane at a variable range of densities to use for dithering different grayscale levels into black-and-white dot patterns.<br /><br />Our paper follows up a previous work by Sariel and M. Mendel (SICCOMP 2006) which defines an approximate version of the farthest-point traversal. Instead of choosing each point to be as far as possible from the set of previous points, we relax this constraint and allow points that are within a (1 + <i>ε</i>) factor of the farthest possible distance from the set of previous points. The farthest-point traversal can be computed in quadratic time, but Har-Peled and Mendel showed that for low-dimensional Euclidean spaces (or other spaces with similar properties) a near-linear time approximation is possible. The new contribution of our preprint is to show a similar near-linear time bound for metric spaces defined by the distances in sparse graphs, based on a randomized-incremental version of Dijkstra's algorithm. We also use locality-sensitive hashing techniques to find approximate greedy permutations in high-diensional Euclidean spaces in subquadratic time, and range-searching data structures to find exact greedy permutations in low-treewidth graphs in subquadratic time.<br /><br />The tongue-twister title of this post was one that we had used for an earlier draft of the paper, but my co-authors made me take it out. Boo!<a name='cutid1-end'></a>http://11011110.livejournal.com/312903.htmlcomputational geometrypaperspublic0http://11011110.livejournal.com/312741.htmlSun, 05 Jul 2015 05:57:16 GMTJuly 4th parade
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I'm in Mendocino once more for their July 4th parade. As usual it was an irreverent mixture of fire trucks, local businesses and politicians, theatre troupes, hippies, and protesters in a beautiful oceanside setting. Lots of fun, and I took <a href="http://www.ics.uci.edu/~eppstein/pix/j4p15/index.html">lots of photos</a>. Here's just one shot, of the start of the parade, where all the local fire departments (mostly volunteers) bring their trucks and set off their sirens.<br /><br /><div align="center"><a href="http://www.ics.uci.edu/~eppstein/pix/j4p15/TheParadeBegins.html"><img src="http://www.ics.uci.edu/~eppstein/pix/j4p15/TheParadeBegins-m.jpg" border="2" style="border-color:black;" /></a></div><br /><br />The fog bank visible in the background is not a good omen for the success of tonight's fireworks show...http://11011110.livejournal.com/312741.htmlmendocinophotographypublic0http://11011110.livejournal.com/312432.htmlWed, 01 Jul 2015 18:20:17 GMTLinkage for the end of June
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I know, it's already July; I forgot that June only has 30 days.<br /><ul><li><a href="http://puzzlepicnic.com/puzzle?4694">Spiral galaxies</a>, my latest puzzle addiction (<a href="https://plus.google.com/100003628603413742554/posts/RzTcvx57F5y">G+</a>)</li><br /><li><a href="http://gizmodo.com/why-mathematicians-are-hoarding-this-special-type-of-ja-1711008881">An endangered species of mathematical chalk</a> (<a href="https://plus.google.com/100003628603413742554/posts/awYsAnAsK6E">G+</a>)</li><br /><li><a href="http://www.bach-bogen.de/blog/thecelloupgrade/zwischen-e-und-f">Mathematical art/music project in Stuttgart</a> (<a href="https://plus.google.com/100003628603413742554/posts/Dr9k2SXEpQt">G+</a>)</li><br /><li><a href="http://www.slate.com/blogs/the_eye/2015/06/15/joris_laarman_mx3d_the_pedestrian_bridge_will_be_3_d_printed_over_an_amsterdam.html">3d-printed fractal bridge in Amsterdam</a> (<a href="https://plus.google.com/100003628603413742554/posts/LYC2vThcozm">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_Signpost/2015-06-17/In_focus">Freedom to take and use photographs in public places in Europe endangered by newly proposed EU law</a> (<a href="https://plus.google.com/100003628603413742554/posts/G6YQijJjDns">G+</a>)</li><br /><li><a href="http://thecreatorsproject.vice.com/blog/a-paper-origami-sculpture-that-shrinks-from-your-touch">Touch-sensitive kinetic origami</a> (<a href="https://plus.google.com/100003628603413742554/posts/8kQXkYYaEsH">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=2g3sdzgSABM">Video about 3d immersions of the Klein bottle</a> (<a href="https://plus.google.com/100003628603413742554/posts/Lzj2SN95Jee">G+</a>)</li><br /><li><a href="http://retractionwatch.com/2015/06/25/one-publisher-appears-to-have-retracted-thousands-of-meeting-abstracts-yes-thousands/">IEEE clears away some of its junk publications</a> (<a href="https://plus.google.com/100003628603413742554/posts/c8fAPuTJzv6">G+</a>)</li><br /><li><a href="https://www.msri.org/system/cms/files/132/files/original/Lander-Case_for_Research.pdf">Why curiosity-driven basic research is important (and should continue to get government funding),</a> by mathematician and biologist Eric Lander (<a href="https://plus.google.com/100003628603413742554/posts/Hc4Ab2nSRwY">G+</a>)</li><br /><li><a href="http://www.improbable.com/2015/06/27/preference-peculiarities-curves-good-or-angles-bad/">Curves good, or angles bad?</a> (<a href="https://plus.google.com/100003628603413742554/posts/Lj78vLR6FKq">G+</a>)</li></ul>http://11011110.livejournal.com/312432.htmltopologyfree speechmusicscienceartorigamiphotographypublic0http://11011110.livejournal.com/312298.htmlWed, 01 Jul 2015 01:11:29 GMTNew preprint on track layouts
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Although it only hints at the connection, one way of interpreting my latest preprint is about higher-dimensional graph drawings. The paper is "<a href="http://arxiv.org/abs/1506.09145">Track Layouts, Layered Path Decompositions, and Leveled Planarity</a>" (with Bannister, Devanny, Dujmović, and Wood, arXiv:1506.09145).<br /><br />The track layouts of the title can be interpreted geometrically as being embeddings of the vertices of a graph on the positive coordinate axes of <i>d</i>-dimensional space, such that each edge forms a curve lying in the quarter-plane between two axes and no two edges cross. For instance, for three tracks, you get a drawing on the three rays and three quarter-planes of an orthant of three-dimensional space, and if you look at that orthant from a point of view somewhere on its symmetry axis, you get a picture looking something like the right side of this figure:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/3-track-spiral.png"></div><br /><br />The left side of the figure shows a different style of graph drawing, a leveled planar drawing in which the vertices are arranged in rows and each edge connects two consecutive rows; the blue shading shows how any leveled planar drawing can be spiraled around between the three rays of the orthant to produce a 3-track drawing. Not every 3-track drawing arises in this way: for instance, you can easily find a 3-track drawing of a triangle, while every leveled planar drawing is bipartite. But it turns out that every bipartite 3-track drawing is also leveled planar. Although this is a nice and non-obvious equivalence between two seemingly different drawing styles, it's a bit unfortunate, because testing whether a graph has a leveled planar drawing was known to be NP-complete and therefore the same is true for 3-track drawing (answering a question posed by Dujmović, Pór, and Wood in 2004).<br /><br />Despite this hardness result, there are several natural graph classes that always have 3-track drawings, including outerplanar graphs, Halin graphs, and squaregraphs. Since outerplanar graphs have treewidth two and Halin graphs have treewidth three, you might think that the series-parallel graphs (also treewidth two but more general than outerplanar) would be sandwiched between them and also have 3-track drawings, but that turns out not to be true. Here's a counterexample:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/ApexTree.png"></div><br /><br />The apex vertex connected to everything else would force the rest of the graph to live on the remaining two tracks, but the only two-track graphs are caterpillars, trees in which all vertices are within distance one of a central path. Since the tree formed from this graph by removing the apex is not a caterpillar, the graph itself does not have a 3-track drawing.<br /><br />This paper also introduces the notion of layered pathwidth, but I don't want to take much credit for that part because it came from some earlier not-yet-published work by Dujmović, Wood, and others. The definition is a bit too technical to repeat here (read the paper), but the leveled planar graphs turn out to be exactly the graphs of layered pathwidth one. So the hardness of testing leveled planarity also shows that testing layered pathwidth is hard. The apex-binary tree example above has bounded track-number (at most four) but unbounded layered pathwidth (for sufficiently large binary trees) showing that track-number and layered pathwidth are distinct concepts. But we think that the graphs of track number three (even the non-bipartite ones) should have bounded layered pathwidth, although we haven't yet been able to prove that conjecture.<a name='cutid1-end'></a>http://11011110.livejournal.com/312298.htmlgraph drawingpaperspublic0http://11011110.livejournal.com/312061.htmlTue, 30 Jun 2015 01:55:34 GMTThe white village of Thorn
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Here's another town in the Netherlands that I visited just before Computational Geometry Week: <a href="https://en.wikipedia.org/wiki/Thorn,_Netherlands">Thorn</a>, also known as "the white village". The story goes that when Napoleon took over the Netherlands, he instituted a building tax based on how many windows each building had. So the villagers bricked up many of their windows and then, to make the change less obvious, whitewashed the buildings. The buildings are still painted white and give the place a distinctive look.<br /><br />It's a small town, so not something that would likely fill a whole day of sightseeing, but very pretty. Behind the church we found an art gallery where an older man had put on a show of his art about trains, including paintings, prints based on old engineering drawings, and a giant model of the bridge over the river Kwai; it was the first day of the show and we were the first to visit.<br /><br /><div align="center"><a href="http://www.ics.uci.edu/~eppstein/pix/thorn/7.html"><img src="http://www.ics.uci.edu/~eppstein/pix/thorn/7-m.jpg" border="2" style="border-color:black;" /></a></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/thorn/index.html">The rest of the photos</a> )</b>http://11011110.livejournal.com/312061.htmlarchitecturenetherlandsphotographypublic1http://11011110.livejournal.com/311631.htmlSun, 28 Jun 2015 21:51:46 GMTDelft
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I arrived a day early in the Netherlands for Computational Geometry Week, to allow me longer to get used to the nine-hour time change. One of the things I did with the extra time was to visit Delft, one of many pretty Dutch canal cities, which turned out to be holding a fun flea market that day as well as some sort of children's marching band competition. I didn't take any photos of those, but I did get some from a tour of the Royal Delft Museum and Factory there. Royal Delft is probably best known for its ornamental blue-and-white painted plates, but I was more interested in their architectural ceramics:<br /><br /><div align="center"><a href="http://www.ics.uci.edu/~eppstein/pix/delft/Columnade.html"><img src="http://www.ics.uci.edu/~eppstein/pix/delft/Columnade-m.jpg" border="2" style="border-color:black;" /></a></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/delft/index.html">The rest of the photos</a> )</b>http://11011110.livejournal.com/311631.htmlarchitecturenetherlandsphotographypublic0http://11011110.livejournal.com/311383.htmlSat, 27 Jun 2015 22:33:59 GMTReport from Geometry Week
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I just returned from visiting Eindhoven, the Netherlands, for <a href="http://www.win.tue.nl/SoCG2015/">Computational Geometry Week</a>, including the 31st International Symposium on Computational Geometry, the 4th Annual Minisymposium on Computational Topology, the Workshop on Geometric Networks, the Workshop on Stochastic Geometry and Random Generation, the Workshop on Geometric Intersection Graphs, the Young Researchers Forum, and the CG Week Multimedia Exposition, almost all of which I attended pieces of (it was not possible to attend everything because SoCG had two parallel sessions and the workshops were run in parallel to each other).<br /><br />After a welcoming reception the evening before at campus café De Zwarte Doos ("the black box", but I was warned not to search Google for that phrase because some other company is using it for other purposes) the conference itself began Monday morning (June 22) with the best-paper talk from SoCG, "<a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.1">Combinatorial discrepancy for boxes via the γ<sub>2</sub> norm</a>", by Jirka Matoušek (posthumously) and Aleksandar Nikolov. The question they study is, given a set of <i>n</i> points in <i>d</i>-dimensional Euclidean space, how to label the points with +1 and –1 so that the sum of the labels in each axis-aligned box is as small as possible. They provide a lower bound for how big these sums might be, of the form log <i>n</i><sup><i>d</i> + O(1)</sup>, nearly tight and roughly squaring the previous bound. The method is simple and elegant and was presented very clearly by Nikolov; it consists of showing that the discrepancy (the quantity in question) is nearly the same as the γ<sub>2</sub> norm of an associated matrix (the product of the maximum row norm and column norm of two matrices that multiply to the given one, chosen to have small rows and columns respectively), that the points can be assumed to form a grid, that the matrix for points in a grid is a Kronecker product of lower-dimensional matrices of the same type, that the one-dimensional matrix (an all-one lower triangular matrix) has logarithic γ<sub>2</sub> norm, and that the γ<sub>2</sub> norm is multiplicative with respect to the Kronecker product.<br /><br />Next, we saw the three videos and one demo of the Multimedia Exposition. My favorite was "Tilt: the video", on the hardness of puzzles in which you tilt a panel with multiple moving balls and fixed obstacles in order to try to move the balls into given configurations:<br /><br /><div align="center"><lj-embed id="58" /></div><br /><br />Another highlight from Monday was Jeff Erickson's talk at the computational topology workshop. Listed in the program as "????", it turned out to be about <a href="http://jeffe.cs.illinois.edu/pubs/talks/prehistory.pdf">the pre-history of computational topology and computational geometry</a>, in which Jeff informed us that most of the basic concepts and algorithms in Chapter 1 of O'Rourke's computational geometry book have been known for centuries. In particular winding numbers and turning numbers can be traced to Thomas Bradwardine in 1320. Algorithms for computing the signed areas of curves and polygons (by summing over trapezoids or triangles determined by pieces of their boundaries) come from Albrecht Meister in 1770, as do some pretty strong hints about the Whitney–Graustein theorem that turning number is a complete topological invariant for regular homotopy. The algorithm for testing whether a point is inside or outside of a polygon by shooting a ray for the point and counting how many times it crosses the boundary comes from Gauss, who also asked which chord diagrams represent the crossing sequences of immersed circles. Even later, Max Dehn gave the first full proof of the Jordan curve theorem for polygons by proving the existence of triangulations, proving the existence of ears in the triangulations (often called Meisters' ear theorem, but this is a different Meisters, later than both Dehn and Meister), and doing the obvious induction.<br /><br />Tuesday started with a talk by Mikkel Abrahamsen on <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.198">finding bitangents of polygons</a>. The method is very simple, and uses only two pointers into each polygon. Two of these pointers point to vertices from each polygon that define a candidate bitangent. The other two pointers walk around the two polygons in tandem, checking whether everything else is on the correct side of the bitangent. If not, we get a new bitangent to try and we reset the two walks. The technical advantage of this over previous methods (involving computing convex hulls of convex polygons) is that it uses only constant space instead of linear space, but it's the sort of thing I was very happy to see at SoCG and find very difficult to imagine getting into a more general theoretical conference like FOCS and STOC. Next, in the same session, Luis Barba presented what turned out to be the winner of the best student presentation award, a talk on <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.209">finding geodesic centers of polygons</a>, points such that the whole polygon can be reached by paths within the polygon that are as short as possible.<br /><br />Also on Tuesday was the first of two invited talks, by Ben Green on his work with Terry Tao on <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.405">the number of ordinary lines</a>. If you have a finite set of points in the plane, not necessarily in general position but not all on a line, then there will be at least one "ordinary line" containing exactly two points; this is the <a href="https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem">Sylvester–Gallai theorem</a>. But there will generally be more than one ordinary line; for instance if all but one point is on a line then the number of ordinary lines is still large. What Green and Tao showed is that (for very large point sets) the number of ordinary lines is at least <i>n</i>/2 when <i>n</i> (the number of points) is even, and 3<i>n</i>/4 when it is odd, matching known upper bounds. The method involves taking the projective dual line arrangement, using Euler's formula to show that when the number of ordinary lines (simple crossings in the dual) is small then most crossings involve three lines and most faces in the line arrangement are triangles, and using these properties to show that this implies that the points can be covered by a small number of cubic curves. Then there's a big case analysis involving the various different possible cubic curves; the one that gives the fewest ordinary lines turns out to be a conic plus a line. Towards the end of the talk he reviewed several related conjectures including one that he credited to <a href="http://users.monash.edu.au/~davidwo/papers/KPW-VisCol-DCG05.pdf">Kára, Pór, and Wood</a> on whether every large set of points contains either a large collinear subset or a large pairwise-visible subset; there are some hints that algebraic methods can also be used for this but it looks trickier than for counting ordinary lines.<br /><br />Unfortunately the Young Researchers' Forum papers don't seem to be individually linked from the conference web site, but a <a href="http://www.computational-geometry.org/YRF/cgyrf2015.pdf">book of all two-page abstracts</a> is available as a single large pdf file. Two of the Tuesday afternoon talks caught my attention. Andrew Winslow spoke about <a href="http://arxiv.org/abs/1504.07883">tiling the plane with translates of a polyomino</a>. It was known that this is possible if and only if the boundary of the polyopmino can be partitioned into six pieces such that opposite pieces of boundary fit precisely together. Winslow translates this into a string problem in which one describes the polyomino by a string over four characters describing the four directions a boundary segment can be oriented. Then the problem becomes one of finding a cyclic rotation of this string, and a partition of it into six substrings, such that opposite substrings are reverse complements. Some combinatorics on words involving "admissible factors" reduces the problem to something that (like many string problems) can be solved using suffix trees. And Anika Rounds spoke in the same session, showing some hardness results on realizability of linkages (systems of rigid bodies connected by pins). Unlike some of the other work on linkages I've reported on here, the bodies are not allowed to overlap, but this constraint makes the realizability problem strongly NP-hard.<br /><br />Also of note Tuesday was the afternoon snack of <a href="https://en.wikipedia.org/wiki/Bossche_bol">Bossche bollen</a>, a Dutch specialty in the form of chocolate-covered whipped cream bombs.<br /><br />One of the Wednesday morning talks had an odd title: "<a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.436">Low-quality dimension reduction</a>". What this turns out to mean is that one has a high-dimensional nearest neighbor query problem and one translates it into a lower-dimensional <i>k</i>-nearest-neighbor problem. The result is an approximate nearest neighbor data structure with truly linear space and sublinear query time, with the query time exponent depending only on the approximation quality and no exponential dependence on dimension. Another of the Wednesday morning talks, by Hsien-Chih Chang, concerned <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.689">a generalization of Voronoi diagrams to points with vector weights</a>, where one wants to find all points whose vector of weights, augmented with one more coordinate for the distance to a query point, is not dominated by any other point. When the weight coordinates are independent random variables this turns out to have near-linear complexity and near-constant output size per query.<br /><br />After an enthusiastic presentation by Benjamin Burton at the Wednesday afternoon business meeting, we decided to go to Brisbane, Australia, in 2017. The 2016 location has already been set for Boston, co-located with STOC. Other business meeting topics included reports from the various program chairs, a diiscussion of the possibility of setting up a non-profit organization to run the conference (about which we'll probably see more online later this year), a change to the steering committee elections to institute staggered terms, and a report from Jack Snoeyink of the NSF on some new initiatives for international cooperation. The rest of the day was occupied by the excursion (a boat ride and walking tour in nearby Den Bosch) and dinner (in the Orangerie of Den Bosch, a decommissioned Gothic church). At the dinner, Pankaj Agarwal and Vera Sacristán presented moving rememberances of Jirka Matoušek and Ferran Hurtado, respectively, both of whom were important figures in the computational geometry community and both of whom died this past year.<br /><br />In the first section of Thursday morning, the well-synchronized timing of the parallel sessions came in handy. Daniel Dadush described <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.704">deterministic algorithms for estimating the volume of a convex body</a> by using a symmetric subset of the body to construct a lattice whose set of points within a slightly-expanded verson of the body can be easily listed and approximate its volume well. The problem has a randomized approximation scheme and deterministic algorithms must be at least exponential-time, but unlike previous ones this one is only single-exponential. The other parallel session included a nice <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.754">APX-hardness reduction for <i>k</i>-means clustering</a>, from vertex covers n trangle-free graphs. The dmension is high (each vertex becomes a dimension and each edge becomes a point, the sum of two basis vectors) but this can be reduced using the Johnson–Lindenstrauss lemma. And back to the first session, Timothy Chan greatly simplified an old algorithm of Bernard Chazelle for <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.733">constructing the intersection of two convex polyhedra in linear time</a>.<br /><br />In the next session we saw what I thought was one of the better student presentations, but one that was ineligible for the best-presentation award because it was scheduled after the voting closed. Its presenter, Arie Bos, is a retiree who has gone back to school, and he told me he thought it would be unfair to compete for the prize because he has so much experience making presentations in his past career. The subject was generalized Hilbert curves, fractal curves that can be formed by taking a Hamiltonian tour of a hypercube and then repeatedly replacing each vertex of the path by a shrunken copy of the same tour. The result of the paper is a new method for generating these tours in such a way that subtours are particularly compact: the bounding box of every subtour is at least 1/4 full, regardless of dimension.<br /><br />Susanne Albers presented the second of two invited talks, on data-dependent algorithm analysis. Her thesis was that, although worst-case analysis has been effective at encouraging the development of efficient algorithms and at distinguishing efficient from inefficient ones, it can be too pessimistic and in some cases unable to distinguish which of two competing algorithms is the better one in practice. The first half of the talk concerned randomized data models. The fully random model describes the input to e.g. quicksort but is not realistic enough for most problems. Better randomized models include smoothed analysis, the planted subgraph model (in which one is supposed to fnd a non-random solution part of an input hidden within a larger random part), and the randomized incremental model seen frequently in computational geometry in which a worst-case set of geometric objects is randomly ordered. The second half of the talk concerned some of Albers' own recent work, on deterministic online algorithms, focusing on modeling the locality of referencing in online input sequences. For online caching, rather than previous work using access graphs, she instead looks at the vector of distances between requests of the same types, where the distance is measured by the number of distinct other types between two requests of one type. By restricting the definition of the competitive ratio to input sequences with the same vector, she shows that the classical LRU strategy is much more tightly competitive than could be shown by the classical input-length based analysis, and that it is significantly better than other choices. She also performs a similar analysis for list updates, but using a dfferent data model in which one looks at runs and long runs in the subsequence of inputs having two particular data values.<br /><br />After a lunch featuring Hollandse Nieuwe herring (eaten raw in small pieces on toothpicks rather than the traditional method of lowering a whole herring down one's throat like a bird) I spent the remainder of the conference at the Workshop on Geometric Intersection Graphs. There's an <a href="http://cgweek15.tcs.uj.edu.pl/problems.pdf">open problem list</a> from the workshop online; eventually it is supposed to include problems from the open problem session but currently it seems to be only a pre-provided list of problems. I contributed one on the complexity of coloring <a href="http://en.wikipedia.org/wiki/Circle_graph">circle graphs</a>. A 1992 paper of Walter Unger claimed a proof that this is NP-complete for 4-coloring and polynomial for 3-coloring, but I don't believe the 3-coloring part: it was presented too sketchily and there is no full journal version. So I think it should still be considered open whether 3-coloring of circle graphs is easy or hard. Both the 3-coloring and 4-coloring questions are also interesting for triangle-free circle graphs, which are always 5-colorable.<br /><br />Overall, I found it to be a well-run, entertaining, interesting, and informative conference. I would have liked to see more application papers both in the main symposium and the satellites, but that's been a perennial issue since the start of SoCG. The new open access publisher and formatting seems to be working well, and I'm looking forward to next year in Boston.<a name='cutid1-end'></a>http://11011110.livejournal.com/311383.htmlcomputational geometryunsolvednetherlandsconferencestalkspaperspublic2http://11011110.livejournal.com/311043.htmlWed, 17 Jun 2015 03:30:07 GMTTwo new papers
http://11011110.livejournal.com/311043.html
Somehow I seem to have two new papers online that I haven't mentioned here before.<br /><br />First, among the many newly-online papers of the newly-open-source (yay!) <a href="http://drops.dagstuhl.de/portals/extern/index.php?semnr=15005">Proceedings of the 31st International Symposium on Computational Geometry</a>, I have one with Drago Bokal and Sergio Cabello, "<a href="http://drops.dagstuhl.de/opus/volltexte/2015/5113/pdf/30.pdf">Finding All Maximal Subsequences with Hereditary Properties</a>". Despite the abstract name, this is really about a concrete problem: given trajectory data (a sequence of points describing the motion of someone or something), answer questions about the shape of different parts of the trajectory. For instance, in the path below, one part is pretty much straight, a second part is nearly stationary, and the third part is moving in one general direction but not by a straight line. We want to be able to figure that out.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/WindowedFootprints.png"></div><br /><br />More technically, we set up a data structure that can query whether any subsequence of the trajectory has one of these three properties, formalized as having low convex hull area, having low diameter, or having a direction with respect to which it is monotone. The data structure itself is very simple — just store for each starting point of a subsequence the farthest ending point that gives a yes answer — but the harder part is building the data structure by finding all of these farthest ending points, efficiently. That's what the "maximal subsequences" in the title means, and where I'll leave you here, to read the paper if you want to find more.<a name='cutid1-end'></a><br /><br />Second, over on the arXiv, I have a new preprint "<a href="http://arxiv.org/abs/1506.04380">Genus, Treewidth, and Local Crossing Number</a>", arXiv:1506.04380, with Vida Dujmović and David Wood. Here local crossing number means the maximum number of crossings per edge, related to but not the same as the global crossing number (proportional to average crossings per edge). For instance, a <a href="https://en.wikipedia.org/wiki/1-planar_graph">1-planar graph</a> is a graph with local crossing number one. It was known that, like planar graphs, the graphs of low crossing number obey a <a href="https://en.wikipedia.org/wiki/Planar_separator_theorem">separator theorem</a> (they can be recursively partitioned into small pieces with few vertices appearing on the boundary between the pieces) but the functional dependence of the separator size on the local crossing number wasn't known before. Now it is, even when we consider embeddings on surfaces of high genus instead of the plane.<br /><br />A second result in the same paper involves finding low-crossing-number embeddings of arbitrary graphs. It was known that any graph with <i>m</i> edges can be embedded onto a surface of your favorite genus <i>g</i> in such a way that the average crossings per edge is nearly (within a polylog factor of) <i>m</i>/<i>g</i>, as good as one could hope to get. We strengthen this to get the same bounds for local crossing number instead of global crossing number. Like the previous result for global crossing number, the proof uses a method of Leighton and Rao for routing paths on expanders, but with some additional machinery: a load-balancing preprocessing phase that helps avoid problems with irregularities in the degree distribution of the graph.<a name='cutid2-end'></a>http://11011110.livejournal.com/311043.htmltopologygraph drawingdata structurespaperspublic0http://11011110.livejournal.com/310818.htmlTue, 16 Jun 2015 04:44:37 GMTLinkage
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<ul><li><a href="https://www.youtube.com/watch?v=UxcJfaoK5xg">Marble machine with 11000 marbles</a> (video; <a href="https://plus.google.com/100003628603413742554/posts/L35u5geW4J4">G+</a>)</li><br /><li><a href="http://www.theonion.com/graphic/pros-and-cons-standardized-testing-50388">Pros and cons of standardized testing</a>, only slightly exaggerated (<a href="https://plus.google.com/100003628603413742554/posts/UjvJ7c344wp">G+</a>)</li><br /><li><a href="http://mitpress.mit.edu/sites/default/files/titles/content/alife14/978-0-262-32621-6-ch084.pdf">Conservation of genki</a> in <a href="https://en.wikipedia.org/wiki/Critters_%28block_cellular_automaton%29">Critters</a> (<a href="https://plus.google.com/100003628603413742554/posts/8YQgpBt4y2E">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2015/06/bruce-shapiros-mesmerizing-kinetic-sand-drawing-machines/">Bruce Shapiro's sand drawing machines</a> (<a href="https://plus.google.com/100003628603413742554/posts/JbpKBkAsRyz">G+</a>)</li><br /><li><a href="http://scholarlykitchen.sspnet.org/2014/08/21/the-mystery-of-a-partial-impact-factor/">The mystery of a partial impact factor</a> or, yet another reason not to take these numbers seriously (<a href="https://plus.google.com/100003628603413742554/posts/GFejYnoCGMs">G+</a>)</li><br /><li><a href="http://america.aljazeera.com/opinions/2015/6/killing-tenure-is-academias-point-of-no-return.html">Scott Walker looks to kill tenure in Wisconsin</a> (<a href="https://plus.google.com/100003628603413742554/posts/PsDgwAXUHnC">G+</a>)</li><br /><li><a href="http://www.mathematicalgemstones.com/gemstones/can-you-prove-it/">Can you prove it?</a> A cute theorem about a coincidence of line segments connecting centers of tangent circles (<a href="https://plus.google.com/100003628603413742554/posts/5LQ8r4Rsmpx">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=0hlvhQZIOQw">Numberphile video on Ford circles and mediants</a> (<a href="https://plus.google.com/100003628603413742554/posts/ibyCaNmey3s">G+</a>)</li><br /><li><a href="http://www.theguardian.com/education/2015/jun/11/nobel-laureate-sir-tim-hunt-resigns-trouble-with-girls-comments">Even a Nobel prize won't save you if you say something sexist enough</a> (<a href="https://plus.google.com/100003628603413742554/posts/6UrfUrfcFij">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=bXn4_JkVFVo">Video of a cat playing a theremin</a> (<a href="https://plus.google.com/100003628603413742554/posts/2Qxotz8H6V4">G+</a>)</li><br /><li><a href="http://aperiodical.com/2015/06/math-stack-a-really-pretty-deck-of-cards-with-maths-on/">Playing cards decorated with mathematics</a> (<a href="https://plus.google.com/100003628603413742554/posts/NmgoQ8HUSwm">G+</a>)</li><br /><li><a href="http://www.wired.com/2015/05/attack-geosciences-congress/">Congress's attack on the geosciences</a> (<a href="https://plus.google.com/100003628603413742554/posts/6EgHn2tCCjq">G+</a>)</li><br /><li><a href="http://www.researchgate.net/publication/275966051_Manifesto_of_Editorial_Independence_of_Editors_of_Frontiers_Medical_Journals">Manifesto of editorial independence</a> <a href="http://scim.ag/OAconflict">gets editors sacked from for-profit open source publisher</a> (<a href="https://plus.google.com/100003628603413742554/posts/hnUtXSC86BR">G+</a>)</li><br /><li><a href="https://ideas.repec.org/top/top.person.all.html">Economists rank themselves</a> (<a href="https://plus.google.com/100003628603413742554/posts/cEjEdjxutYA">G+</a>)</li><br /><li><a href="http://drops.dagstuhl.de/portals/extern/index.php?semnr=15005">The new open-source SoCG proceedings are here</a> (<a href="https://plus.google.com/100003628603413742554/posts/QS2eafsTDkm">G+</a>)</li></ul>http://11011110.livejournal.com/310818.htmlfeminismcomputational geometrycellular automataconferencescirclesartacademiapaperspoliticspublic0http://11011110.livejournal.com/310630.htmlMon, 08 Jun 2015 01:11:58 GMTMetric dimension for subdivided graphs
http://11011110.livejournal.com/310630.html
I have another new preprint out this evening: "<a href="http://arxiv.org/abs/1506.01749">Metric Dimension Parameterized by Max Leaf Number</a>", arXiv:1506.01749. The <a href="https://en.wikipedia.org/wiki/Metric_dimension_(graph_theory)#Properties">metric dimension</a> of a graph is the minimum number of vertices you need to choose as landmarks so that all other vertices are uniquely determined by their distances to the landmarks.<br /><br />The result in the new paper is small: it says that if you form big graphs from smaller ones by subdividing their edges into paths, you can solve the problem in a time that depends exponentially on the size of the small graph, but only linearly on the number of added subdivision vertices. So the result seems to apply to only a very restricted class of graphs, but as I argue in the paper there are some natural real-world graphs that have the structure of subdivided smaller graphs: the graphs of public transportation systems tend to have this form, because they consist of long lines or tracks with many stops on them. For instance, here's a graph of the Toronto subway system, from Paulshannon <a href="https://commons.wikimedia.org/wiki/File:TTCsubwayRTmap-2007.svg">on Wikimedia commons</a>:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/TTCsubwayRTmap-2007.svg.png"></div><br /><br />It's also known how to compute the metric dimension efficiently on trees. So putting these two results together, it seems at least plausible that there should be an efficient algorithm on the graphs formed by subdividing smaller graphs and gluing trees onto them. That is, I would like a fixed-parameter tractable algorithm parameterized by the <a href="https://en.wikipedia.org/wiki/Circuit_rank">cyclomatic number</a> (or slightly better the almost-tree number), rather than by the max leaf number. But despite some effort I wasn't able to get that.<a name='cutid1-end'></a>http://11011110.livejournal.com/310630.htmlgraph algorithmspaperspublic0http://11011110.livejournal.com/310518.htmlMon, 01 Jun 2015 01:59:04 GMTLinkage
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<ul><li><a href="http://nielsenhayden.com/makinglight/archives/016246.html">An interesting new voting system arises from the ashes of a broken science-fiction award process</a> (<a href="https://plus.google.com/100003628603413742554/posts/g2JPvVap3B7">G+</a>)</li><br /><li><a href="https://vimeo.com/2719929">Jorg Meyer, scientific glassblower</a>, <a href="https://en.wikipedia.org/wiki/Jorg_Meyer">whale rider, and falconer</a> (<a href="https://plus.google.com/100003628603413742554/posts/CvqaSqTA4DY">G+</a>)</li><br /><li><a href="https://plus.google.com/+JoergFliege/posts/VWFHj2vYBGG">Oklahoma oilman fails to get a university researcher fired for publishing about ties between fracking and earthquakes</a> (<a href="https://plus.google.com/100003628603413742554/posts/Jxc8nNTnCYs">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1502.07597">Women mathematicians in France in the mid-twentieth century</a> (<a href="https://plus.google.com/100003628603413742554/posts/44e4NEu3k2t">G+</a>, with links to Wikipedia articles on the same people)</li><br /><li><a href="http://www.sciencemag.org/content/348/6234/479">What happens when you mix open-records laws, state university professors' email histories, and the anti-science movement</a> (<a href="https://plus.google.com/100003628603413742554/posts/PWBvQDiDJwn">G+</a>)</li><br /><li><a href="http://blogs.oregonstate.edu/glencora/2015/05/21/with-great-privilege-comes-great-responsibility/">Silent Glen gets tenure</a> (<a href="https://plus.google.com/100003628603413742554/posts/j2bDTNRYzdu">G+</a>)</li><br /><li><a href="http://thrilling-tales.webomator.com/derange-o-lab/pulp-o-mizer/pulp-o-mizer.html">Pulp magazine cover generator</a> (<a href="https://plus.google.com/100003628603413742554/posts/3mQrBK8UzVh">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Reuleaux_triangle">Reuleaux triangles</a> (<a href="https://plus.google.com/100003628603413742554/posts/Ls2pYDMkCSy">G+</a>)</li><br /><li><a href="https://plus.google.com/+PeterSuber/posts/Bn6K3ZCGaM4">Do hybrid open access journals double-dip?</a> Answer: sometimes they even triple-dip (<a href="https://plus.google.com/100003628603413742554/posts/9cR9iH9yKxg">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1505.06508">Pattern-avoiding permutation counting functions are nastier than we thought</a> (<a href="https://plus.google.com/100003628603413742554/posts/NdueUTsAC6b">G+</a>)</li><br /><li><a href="http://jeff560.tripod.com/mathsym.html">Earliest uses of mathematical symbols</a> (<a href="https://plus.google.com/100003628603413742554/posts/5vuYmEcpZZA">G+</a>)</li><br /><li><a href="http://www.lindengledhill.com/">Crystal microphotographs by Linden Gledhill</a> (<a href="https://plus.google.com/100003628603413742554/posts/9rdNzj4hYjj">G+</a>)</li><br /><li><a href="http://www.nature.com/news/sleeping-beauty-papers-slumber-for-decades-1.17615">Sleeping beauty papers</a> with many citations after a long sleep (<a href="https://plus.google.com/100003628603413742554/posts/gof4gKPz77V">G+</a>)</li><br /><li><a href="http://boingboing.net/2015/05/29/miniature-origami-robot-self-f.html">Tiny self-folding robot</a> (<a href="https://plus.google.com/100003628603413742554/posts/3xUV12qot32">G+</a>)</li><br /><li><a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2602683">Extending freedom of panorama in Europe</a> (<a href="https://plus.google.com/100003628603413742554/posts/6y4qBN9zUGj">G+</a>)</li></ul>http://11011110.livejournal.com/310518.htmlfeminismvotingpermutation patternswikipediageometryacademiauciorigamipublic0http://11011110.livejournal.com/310028.htmlWed, 20 May 2015 07:37:43 GMTGraham on Erdős on Egyptian fractions
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In a recent paper Ron Graham <a href="http://www.math.ucsd.edu/~ronspubs/13_03_Egyptian.pdf">surveys the work of Paul Erdős on Egyptian fractions</a>. Did you know that Erdős' second paper was on the subject? I didn't. It proved that the sum of a harmonic progression can never form an Egyptian fraction representation of an integer (there is always at least one prime that appears in only one term). Graham himself is also a fan, having studied Egyptian fractions in his Ph.D. thesis.<br /><br />Another of Erdős' papers surveyed by Graham is also somewhat related to the subject of my recent blog posts on sequences of highly composite numbers. This paper (famous for formulating the Erdős–Straus 4/n = 1/x + 1/y + 1/z conjecture) included another conjecture that every rational number x/y (between 0 and 1) has an Egyptian fraction representation with O(log log y) terms. However, the best bound known so far is larger, O(sqrt log y).<br /><br />For any number z, let D(z) be the smallest number with the property that every positive integer less than z can be expressed as a sum of at most D(z) divisors of z (not necessarily distinct). Then a stronger version of Erdős' conjecture (for which the same bounds are known) is that, for every y, there exists a number z larger than y (but not too much larger) with D(z) = O(log log z). With such a z, you can split x/y into floor(xz/y)/z + remainder/yz and then use the sum-of-divisors property of z to split each of these two terms into a small number of unit fractions.<br /><br />Computing D(z) for small values of z is not particularly hard, using a dynamic programming algorithm for the subset sum problem. So, based on the guess that the highly composite numbers would have small values of D(z), I tried looking for the biggest highly composite number with each value. In this way I found that D(24) = 3; D(180) = 4; D(5040) = 5; and D(1081080) = 6. That is, every positive integer less than 1081080 can be represented as a sum of at most six divisors of 1081080, and some require exactly six. Based on this, every x/y with y at most 1081080 can be represented as at most a 12-term Egyptian fraction.<br /><br />Each number in the sequence 2, 6, 24, 180, 5040, 1081080, ... is within a small factor of the 1.6 power of the previous number; another way of saying the same thing is that the numbers in this sequence obey an approximate multiplicative Fibonacci recurrence in which each number is approximately the product of the previous two. The next number in the sequence might still be within reach of calculation, using a faster programming language than my Python implementation. If that 1.6-power pattern could be shown to continue forever, then Erdős' log-log conjecture would be true.<a name='cutid1-end'></a>http://11011110.livejournal.com/310028.htmlegyptian fractionsnumber theorypublic0http://11011110.livejournal.com/309894.htmlSat, 16 May 2015 05:36:58 GMTMid-May linkage
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<ul><li><a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2560572">An economic analysis of public domain photos on Wikipedia</a> shows that "massive social harm was done by the most recent copyright term extension that has prevented millions of works from falling into the public domain since 1998" (<a href="https://plus.google.com/100003628603413742554/posts/7DaUpswgdY2">G+</a>)</li><br /><li><a href="http://blogs.ams.org/visualinsight/2015/05/01/twin-dodecahedra/">An infinite tree of regular dodecahedra</a> sharing a cube of vertices between each neighboring pair (<a href="https://plus.google.com/100003628603413742554/posts/UpKo5xNmZn9">G+</a>)</li><br /><li><a href="https://igorpak.wordpress.com/2015/05/02/you-should-watch-combinatorics-videos/">Combinatorics videos</a> collected by Igor Pak (<a href="https://plus.google.com/100003628603413742554/posts/KuGUzGoTqZw">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=a3QqKBWHarA">The inspiration for some of Man Ray's art in a collection of mathematical models</a> (<a href="https://plus.google.com/100003628603413742554/posts/UhfwavcvYdo">G+</a>)</li><br /><li><a href="https://plus.google.com/+FrancoisDorais/posts/E3Yh9YwQTQN">Did you know you could get bibtex directly from a doi?</a> (<a href="https://plus.google.com/100003628603413742554/posts/Qbx6xEERaup">G+</a>)</li><br /><li><a href="http://blogs.plos.org/everyone/2015/05/01/plos-one-update-peer-review-investigation/">Journal editor canned for using sexist referee report</a> (<a href="https://plus.google.com/100003628603413742554/posts/jmxuXn5GZ1W">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=ploETyBDM7I">Trilingual powers of two</a> in a video on street-vendor cookie-making (<a href="https://plus.google.com/100003628603413742554/posts/5HQMVbkkJ9S">G+</a>)</li><br /><li><a href="http://www.metafilter.com/149171/The-International-Journal-of-Proof-of-Concept-or-Get-The-Fuck-Out">Winner, best name of an actual publication</a> (hacker zine PoC||GTFO; <a href="https://plus.google.com/100003628603413742554/posts/SgfjCrpmKPP">G+</a>)</li><br /><li><a href="https://www.chromeexperiments.com/experiment/100000-stars">3d visualization of nearby stars</a> (<a href="https://plus.google.com/100003628603413742554/posts/XQsfYnUMWsE">G+</a>)</li><br /><li><a href="https://doajournals.wordpress.com/2015/05/11/historical-apc-data-from-before-the-april-upgrade">Over 2/3 of listed open access journals charge no author fees</a> (<a href="https://plus.google.com/100003628603413742554/posts/geLxaXBzBge">G+</a>)</li><br /><li><a href="http://www.theguardian.com/technology/2015/may/14/dear-google-open-letter-from-80-academics-on-right-to-be-forgotten">Open letter to Google by 80 academics</a> asking for greater transparency on "right to be forgotten" (<a href="https://plus.google.com/100003628603413742554/posts/WGj2wwQtU1J">G+</a>)</li></ul>http://11011110.livejournal.com/309894.htmlfeminismbibliographycombinatoricswikipediageometryartpublic0http://11011110.livejournal.com/309622.htmlFri, 15 May 2015 20:32:41 GMTParametric knapsacks for number-theoretic sequences
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One of the key principles of <a href="http://11011110.livejournal.com/307881.html">parametric optimization</a> is that, when you are faced with optimizing the nonlinear combination of two linear values (sums of element weights, costs, etc) you should instead look at the set of optima for all possible linear combinations of the same two values. Let's see how this applies to the <a href="http://11011110.livejournal.com/305481.html">number-theoretic knapsack problems</a> I posted about <a href="http://11011110.livejournal.com/309343.html">earlier this week</a>.<br /><br />In the knapsack problem, we are trying to optimize the total profit of a subset of the given elements, subject to the condition that their total size is at most a given threshold. This can be expressed as a nonlinear combination of these two linear values in which the function of profit and size is the identity function on profit when the size is small enough and zero otherwise. This isn't the nice sort of quasiconvex function that parametric methods are best-suited for, but the fractional knapsack problem instead involves a greedy algorithm for maximizing the profit/size ratio, and this sort of ratio is quasiconvex. So in any case, following the parametric approach, let's replace both of these nonlinear combinations by the linear combination profit − λ·size, let the parameter λ vary, and see what solutions we get.<br /><br />For any particular value of λ, the answer is very simple: the optimal solutions are the ones that take all elements for which profit/size > λ (the ones that make a positive contribution to the solution value), and any subset of the elements for which profit/size = λ (the ones whose contribution is zero). The smallest-size optimal solution is the one that takes only the elements for which profit/size > λ. So the set of all smallest-size optimal solutions is almost exactly the same as the set of solutions generated by the greedy algorithm that adds one element at a time in order by the profit/size ratio. To make this algorithm generate exactly the smallest-size optimal solutions, we need to modify it so that when there are ties in profit/size ratio it adds all tied elements at once rather than adding them one at a time. When the set of profit/size values is discrete (as it is in our problems) this set of solutions also has the property that each solution is the unique optimal solution for a nonempty range of parameter values.<br /><br />Now suppose we go back to the number-theoretic sequences that I started with (the highly abundant numbers and the highly composite numbers), expand out the definition of the profit and size functions in the parametric optimization functions profit − λ·size, and eliminate the logs in these functions by exponentiating. Then the sequence of smallest-size optimal solutions for these objective functions are exactly how the colossally abundant numbers and superior highly composite numbers are defined. That is, it is no coincidence that starting with the knapsack-problem formulations of the highly abundant and highly composite numbers, and then applying the greedy algorithm to the resulting knapsack problems, gave these other two sequences: it falls out directly from the parametric analysis above and the definitions of these sequences.<br /><br />However, OEIS states that <a href="http://oeis.org/A073751">the correctness of the generation algorithm</a> for the successive factors of the colossally abundant numbers is still conjectural rather than proven. How can this be, when we have seen above that the greedy algorithm always works for sequences like this? The part that must still be unknown concerns the possibility of ties: is it ever possible for two or more knapsack elements to have the same profit/cost ratio? If so we must take both or all of them at once rather than letting them be chosen one at a time. And this is problematic from the algorithmic point of view because it involves testing complicated expressions involving logarithms for exact equality.<br /><br />Specifically, in the highly abundant number version of the problem, we need to know whether there can exist two prime powers <i>p<sup>i</sup></i> with the same value of the expression log<sub><i>p</i></sub>(<i>p</i><sup><i>i</i> + 1</sup> − 1)/(<i>p</i><sup><i>i</i></sup> − 1). In the highly composite number version of the problem, we need to know whether there can exist two prime powers with the same value of the expression log<sub><i>p</i></sub>(<i>i</i> + 1)/<i>i</i>. In both cases, it seems unlikely, but obviously that's not a proof. More generally, Alaoglu and Erdős conjectured in 1944 (in connection with this problem) that two expressions log<sub><i>p</i></sub><i>q</i> with different prime bases and rational arguments can only be equal if they're both integers, but (although it is known that there can be no three-way ties) this remains unproven.<a name='cutid1-end'></a>http://11011110.livejournal.com/309622.htmlunsolvednumber theoryalgorithmspublic0