urn:lj:livejournal.com:atom1:110111100xDE0xDE0xDE2016-06-25T09:04:32Zurn:lj:livejournal.com:atom1:11011110:330960Open problems from SWAT2016-06-25T09:04:32Z2016-06-25T09:04:32ZI'm now making my way back from Iceland, where I attended the <a href="http://swat16.ru.is">15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)</a>. Rather than providing a run-down of the talks and results I found interesting (you can choose your own from the <a href="http://drops.dagstuhl.de/portals/extern/index.php?semnr=16008">conference proceedngs</a>, this year provided as open-access through <a href="http://drops.dagstuhl.de/opus/institut_lipics.php?fakultaet=04">LIPIcs</a>), I thought it might be more fun to mention a few open problems that people mentioned along the way and that caught my attention.<br /><br />The first invited talk, by Julia Chuzhoy, concerned the relation between grid minor size and treewidth. There exist graphs of treewidth t whose biggest grid minor has side length O(sqrt t/log t), and the results she spoke about give polynomiial lower bounds, but with much smaller exponents (currently around 1/19). As her abstract states, "an important open question is establishing tight bounds" for this problem.<br /><br />From Zachary Friggstad's talk on integrality gaps for directed Steiner trees, we have the situation that (for directed Steiner tree problems with t terminals) it is possible to approximate the solution within a t<sup>ε</sup> factor, for any ε > 0, in polynomial time, or to within a polylog factor in quasipolynomial time. Is a polynomial-time polylog-factor approximation possible?<br /><br />Daniël Paulusma spoke on finding square roots of k-apex graphs; that is, one is given a graph G that could be made planar by the deletion of k vertices, and the goal is to find another graph R on the same vertex set such that two vertices are adjacent in G if and only if their distance in R is at most two. There are a lot of other graph classes for which the problem is known to be polynomial or hard, but he listed as open finding square roots of split graphs or cographs, and finding graph square roots that are planar.<br /><br />Jens Schmidt's talk was motivated by some older work of Mikkel Thorup on recognizing map graphs, the intersection graphs of interior-disjoint simply-connected regions in the plane. These differ from planar graphs because of the four corners phenomenon, where Arizona, Colorado, New Mexico, and Utah all meet at a point (and are all adjacent in the map graph). Thorup gave a polynomial time recognition algorithm but his exponent is huge and his algorithm does not provide a combinatorial certificate of being a map graph (a planar bipartite graph whose half-square is the given graph). Schmidt gave a more satisfactory solution to a special case but would like a better algorithm for the general problem.<br /><br />My own talk concerned <a href="http://11011110.livejournal.com/327681.html">cuckoo filters</a>, a more space-efficient replacement for Bloom filters. A data structure of Pagh et al provides a theoretical but not-yet-practical solution for the same problem (for all false positive rates), but cuckoo filters only work well when the false positive rate is small enough. What about if we want a practically efficient filter that gives a high false positive rate, say 20%? Can we get close to optimal space in this case?<br /><br />Konrad Dabrowski made progress on classifying which sets of forbidden induced subgraphs lead to graph families with bounded clique-width by showing that (diamond,P1+2P2)-free graphs and four other similarly defined classes do have bounded clique-width. But as he reports, there are many remaining open cases.<br /><br />Dana Richards spoke on a problem of comparison-sorting a set of items when only certain pairs of items are allowed to be compared, but his paper has more questions on this problem than answers. Any set of comparisons that you make (a subset of allowed ones) gives you a DAG with an edge for each comparison, oriented from smaller to larger, and you can infer the results of all comparisons in the transitive closure of this DAG. The goal is to either test or infer the results of all allowed comparisons. You can do it in a sublinear number of comparisons when either the set of allowed comparisons is small (by just doing them all) or its complement is small (by Richards' algorithm) but what he would like to prove is a subquadratic (deterministic) bound that holds in all cases.<br /><br />Timothy Chan revisited one of the first kinetic data structures, for closest pairs (and/or nearest neighbors). The static version of the problem can be solved in O(n log n) time in any constant dimension, but previous kinetic structures had a factor of log<sup>dimension</sup> in their time bounds. Chan eliminated this, but at the expense of making the time bound depend on the geometry of the input (the range of values that the closest pair might span) instead of just on the number of points. He asked whether it is possible to eliminate both the dependence on dimension and the dependence on geometry.<br /><br />One of the disadvantages of the move from paper to electronic proceedings (at least in my own experience) is less browsing and more going directly to the papers you already knew you were looking for. And one of the continuing advantages of going to a symposium like this (especially a small and welcoming one like SWAT) is that by listening to talks on subjects you don't already know about, you can broaden your own interests and find out about topics you might not have discovered on your own. But browsing is still possible, even if we no longer are forced to leaf through paper proceedings. So, I'd encourage you to download the full proceedings (from the link at the top left corner of the proceedings link) rather than just the individual papers, and browse them to find many more problems that the papers there have listed as open.<a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:330609Lesley2016-06-20T01:02:53Z2016-06-20T01:03:31ZI've been slow processing this but the arrival of the official shots reminded me: this is from my daughter's graduation a month ago from Lesley (with a B.F.A. in photography).<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/lesley/2-m.jpg" border="2" style="border-color:black;" /></div><br /><br />Sara tells me she likes <a href="http://www.ics.uci.edu/~eppstein/pix/lesley/1.html">the other shot</a> better, though.urn:lj:livejournal.com:atom1:11011110:330247Linkage2016-06-16T05:26:43Z2016-06-16T05:26:43Z<ul><li><a href="http://www.samefacts.com/2016/05/education-2/an-annoying-trend-in-academic-presentations/">Why screenshots of published journal articles make bad talk slides</a> (<a href="https://plus.google.com/100003628603413742554/posts/KAJaVJFkhwp">G+</a>)</li><br /><li><a href="http://rsos.royalsocietypublishing.org/content/3/5/160091">Fibonacci and non-Fibonacci spirals in sunflowers</a> (<a href="https://plus.google.com/100003628603413742554/posts/WUKxEtU9cB6">G+</a>)</li><br /><li><a href="http://bit-player.org/2016/prime-after-prime">More on correlations in the moduli of consecutive primes</a> (<a href="https://plus.google.com/100003628603413742554/posts/hv6jCzodYqE">G+</a>)</li><br /><li><a href="http://www.latimes.com/business/hiltzik/la-fi-hiltzik-university-business-20160602-snap-story.html">Less state funding of universities = more corporate corruption</a> (<a href="https://plus.google.com/100003628603413742554/posts/SXpwqG6uNQP">G+</a>)</li><br /><li><a href="http://chronicle.com/article/After-the-Gold-Rush/236708">My cross-the-street neighbor, caught up in a saga of a MOOC gone bad</a> (<a href="https://plus.google.com/100003628603413742554/posts/Epe6KCqEY1z">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1606.01783">Lonely runner conjecture still not solved</a> (<a href="https://plus.google.com/100003628603413742554/posts/QwoRv1f4YCt">G+</a>)</li><br /><li><a href="https://medium.com/vantage/you-can-tell-europe-to-help-protect-street-photography-6d4b6c4f3442#.2hmpvham2">Europe considers changing its rules on freedom of panorama</a> (i.e. whether that background building prevents you from publishing your photos; <a href="https://plus.google.com/100003628603413742554/posts/CzYi9T3Vsip">G+</a>)</li><br /><li><a href="https://cosmosmagazine.com/mathematics/maths-mystery-solved-after-40-years">5-connected nonplanar graphs have topological <i>K</i><sub>5</sub> minors</a> (<a href="https://plus.google.com/100003628603413742554/posts/cFfEzrdu8rn">G+</a>)</li><br /><li><a href="http://graphdrawing.de/contest2016/contest.html">This year's Graph Drawing contest</a>, plus a corrected issue with only informing insiders of the GD submission rules (<a href="https://plus.google.com/100003628603413742554/posts/SGe6GKuTyy6">G+</a>)</li><br /><li><a href="https://svpow.com/2016/06/09/we-dont-need-oa-in-our-field-everything-is-on-arxiv-nope/">How much is available on arXiv?</a> (<a href="https://plus.google.com/100003628603413742554/posts/Rq8E3pfpfir">G+</a>)</li><br /><li><a href="http://devlinsangle.blogspot.com/2016/06/infinity-and-intuition.html">Uncountable trees with countable height and branching factor</a> (see also discussion on <a href="https://plus.google.com/100003628603413742554/posts/iY73mNFe8Zc">G+</a>)</li><br /><li><a href="http://research.microsoft.com/en-us/um/people/peres/stable/stable.html">Stable marriage for equal area Voronoi-like cells</a> (<a href="https://plus.google.com/100003628603413742554/posts/XgCbvAcPrs5">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=A-lQfVv62V4">Trailer for <i>The Discrete Charm of Geometry</i></a> (<a href="https://plus.google.com/100003628603413742554/posts/2Lo5JYb6LiY">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Widest_path_problem">Widest paths on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/Lc87mt1MJpB">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:330224Robust graph isomorphism and its applications2016-06-02T05:10:21Z2016-06-02T05:10:21ZI think part of the reason graph isomorphism has been such a tricky problem, theoretically, is that in practice it's too easy. Almost all graphs have some small irregularities that can be used as landmarks for identifying all the features of the graph, even when they've been scrambled arbitrarily. Only a small class of highly symmetric graphs pose any actual difficulties, and that's why a deep knowledge of group theory (the study of symmetries) has been so useful in theoretical work on graph isomorphism. It's also why finding counterexamples to crank graph isomorphism algorithms is hard enough that the cranks don't do it themselves and avoid the embarrassment of someone else doing it for them.<br /><br />In many cases, even if you add some noise to a graph, its underlying irregularities will show through, allowing you to recognize it and its individual features. That's the main idea behind my most recent arXiv preprint, "Models and Algorithms for Graph Watermarking" (<a href="https://arxiv.org/abs/1605.09425">arXiv:1605.09425</a>, with Goodrich, Lam, Mamano, Mitzenmacher, and Torres, to appear at <a href="http://manoa.hawaii.edu/isc2016/">ISC 2016</a>).<br /><br />The problem we study is one of watermarking copies of a graph (for instance a large social network) so that if you see one of your copies later you can tell which one it was, by using the graph structure rather than extra information such as vertex labels. To do so, we identify a small number of "landmark" high-degree vertices (generally, the ones with the highest degrees), use the pattern of adjacencies to landmarks to give unique identities to a larger set of (medium-degree) vertices, and flip a small set of randomly selected edges among these vertices. With high probability (when the graph to be watermarked is drawn from a suitable random family) even if an adversary tries to mask the watermark by scrambling the vertices or flipping more edges, the vertex identifications and pattern of flipped edges will be recoverable.<br /><br />Because we're choosing our landmarks in a fairly naive way (by vertex degrees, or as in our implementation with ties broken by neighboring degrees), our algorithms wouldn't work for random regular graphs. But even in such cases, there are other features such as the numbers of triangles they belong to or their distance vectors that often allow some vertices to be distinguished from others. Finding out which features of this type remain robust when noise is added to the graph seems like a promising line of research.urn:lj:livejournal.com:atom1:11011110:329926Linkage2016-06-01T04:07:43Z2016-06-01T04:08:35Z<ul><li><a href="http://voices.norwich.edu/daniel-mcquillan/2016/04/28/a-parity-theorem-for-drawings-of-complete-graphs/">A parity theorem for drawings of complete graphs</a> (<a href="https://plus.google.com/100003628603413742554/posts/DsBQtwX3s7A">G+</a>)</li><br /><li><a href="http://www.nature.com/news/social-sciences-preprint-server-snapped-up-by-publishing-giant-elsevier-1.19932">Social-sciences preprint server SSRN bought by Elsevier</a> (<a href="https://plus.google.com/100003628603413742554/posts/ggHyFHL6Vie">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=1v5Aqo6PaFw">More than one way to draw an ellipse</a> (<a href="https://plus.google.com/100003628603413742554/posts/8ixMAYPig1S">G+</a>)</li><br /><li><a href="https://lamington.wordpress.com/2014/03/04/kleinian-a-tool-for-visualizing-kleinian-groups/">kleinian, a tool for visualizing Kleinian groups</a> (<a href="https://plus.google.com/u/0/100003628603413742554/posts/gjdawsjzDPd">G+</a>)</li><br /><li><a href="http://kottke.org/14/06/cheese-charts">Cheese charts</a>. Really, is it any stranger a thing to call them than pie charts? (<a href="https://plus.google.com/100003628603413742554/posts/2zNSThYNHu9">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Rule_184">Rule 184</a>, three particle systems in one (<a href="https://plus.google.com/100003628603413742554/posts/SzGTuGPvgQJ">G+</a>)</li><br /><li><a href="http://www.slate.com/articles/news_and_politics/cover_story/2016/05/the_thriving_russian_black_market_in_dissertations_and_the_crusaders_fighting.html">Russian black-market dissertations</a> (<a href="https://plus.google.com/100003628603413742554/posts/YrqDr8qofwc">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2016/05/salt-labyrinths-motoi-yamamoto/">Motoi Yamamoto's salt labyrinths</a> (<a href="https://plus.google.com/100003628603413742554/posts/AHXGPjTvy12">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Directed_acyclic_graph">Do you like DAGs?</a> (<a href="https://plus.google.com/100003628603413742554/posts/JjM9KuGc518">G+</a>)<br /><p align="center"><lj-embed id="64" /></p></li><br /><li><a href="http://arstechnica.com/tech-policy/2016/05/google-wins-trial-against-oracle-as-jury-finds-android-is-fair-use/">Android makes fair use of JAVA APIs</a> (<a href="https://plus.google.com/100003628603413742554/posts/12FuG8dinB8">G+</a>)</li><br /><li><a href="https://plus.google.com/+RoiceNelson/posts/hCdQHA9Kgav">The stretched model of the hyperbolic plane</a>, in which most lines are modeled as half-hyperbolas (<a href="https://plus.google.com/100003628603413742554/posts/cA6x9TM62H2">G+</a>)</li><br /><li><a href="http://www.justinobeirne.com/essay/what-happened-to-google-maps">Google maps now more of a route-planner than a usable map</a> (<a href="https://plus.google.com/100003628603413742554/posts/FczAtotGyCj">G+</a>)</li><br /><li><a href="https://plus.google.com/117663015413546257905/posts/dsPiPXsp2QA">Pushing for all funded research to be free to read</a> (<a href="https://plus.google.com/100003628603413742554/posts/hVwsQ8YPHzf">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:329516Too many or too few ways to make change2016-05-30T19:09:48Z2016-05-30T19:21:38Z<p><a href="https://en.wikipedia.org/wiki/Ivan_Pervushin">Ivan Pervushin</a> was a 19th-century Russian amateur mathematician, employed as a cleric, who factored two Fermat numbers and discovered the ninth Mersenne prime 2<sup>61</sup> − 1 = 2305843009213693951. But despite these obvious reasons for fame, his Wikipedia article was in such bad shape (e.g. completely unsourced) that it was recently put up for a deletion discussion, which it survived.</p>
<p>While looking for more sources for this article, I ran across what Google numbers as <a href="https://books.google.com/books?id=zBkpLyUTOjoC&pg=PT140">page 140 of <i>Ripley's Believe It or Not!: In Celebration... A special reissue of the original!</i></a> (Simon and Schuster 2011), which seems to describe a different reason for Pervushin's number 2<sup>61</sup> − 1 to be interesting. It writes (in somewhat hyperbolic language) that this is the number of different ways of making change for $5, using U.S. currency ("cents, nickels, dimes, quarters, halves, and dollars"). Not convinced, I wrote a Python program based on a simple dynamic programming algorithm to compute the number of ways of making change for $5 (also allowing a $5 bill to be used itself, but that only changes the answer by one):</p>
<pre>N = [0]*501
N[0] = 1
for c in [500,100,50,25,10,5,1]:
for i in range(501-c):
N[i+c] += N[i]
print N[500]</pre>
<p>The output was 98412 (Tacoma), much smaller than the number in the book. (The problem is also briefly discussed <a href="https://primes.utm.edu/curios/page.php/2305843009213693951.html">here</a> with essentially the same answer, 98411 if the $5 bill is not counted as a solution.) In retrospect, the small size of this number should not have been a surprise. To make change for <i>n</i> cents using a fixed set of <i>c</i> different coins, the number of possible solutions is asymptotically <i>O</i>(<i>n</i><sup><i>c</i> − 1</sup>): each solution can be represented as a <i>c</i>-dimensional vector of how many coins of each value to use, but the coefficient for the pennies is determined by all the other coefficients. And the leading constant in the <i>O</i>-notation is very small (much smaller than 1) because you can't use anywhere near <i>n</i> of the larger denomination coins. So a polynomial bound with a moderate exponent and a small constant shouldn't generate huge numbers.</p>
<p>But then I thought: maybe they have a different definition of what it means for two ways of making change to differ. It couldn't be that the coins of the same value are distinguishable (different years or mint marks, or different state-by-state obverse designs on the quarters) because then the answer wouldn't be well defined: it would depend on how many different coins of each type you happen to have and not merely on the value of the amount to be changed and the fixed system of coin values. But maybe the order in which I hand you the change matters: if I give you four dollar bills and then four quarters, it's a different way of making change than giving you the quarters first and then the dollars? To test this, I made a small modification to my program to calculate the number of distinct sequences of coins and bills (rather than the number of distinct multisets) that would add to the given amount:</p>
<pre>N = [0]*501
N[0] = 1
for i in range(501):
for c in [500,100,50,25,10,5,1]:
if (i-c >= 0):
N[i] += N[i-c]
print N[500]</pre>
<p>But this time the output was much higher than the published amount: It was 1296142333713114950908964121341365887827473814688245139610006400624. Asymptotically the number of solutions is exponential (because what we are computing is just a linear recurrence) but even with a low base, exponentiating 500 produces big numbers. So this raises a puzzle: can anyone come up with a reasonable definition of the problem for which the Ripley answer is correct?</p><a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:329301Pre-announcement for the SODA call for papers2016-05-26T03:11:36Z2016-05-26T03:11:36ZSODA PC chair Phil Klein asked me to publicize this quick announcement because of delays in getting this information online in a more official way: for next January's <a href="http://siam.org/meetings/da17/">SODA (ACM-SIAM Symposium on Discrete Algorithms) in Barcelona</a>, the submission deadlines will be July 6 (short abstract and paper registration) and July 13 (full submission).<br /><br />For now, you can visit <a href='http://cs.brown.edu/~pnk/#soda17'>http://cs.brown.edu/~pnk/#soda17</a> for the basic information (deadlines, submission site, and program committee).urn:lj:livejournal.com:atom1:11011110:329212Series-parallel duality and read-once functions2016-05-20T02:45:06Z2016-05-20T02:45:06ZIf you draw a <a href="https://en.wikipedia.org/wiki/Series-parallel_graph">series-parallel</a> multigraph in the plane, with an extra edge (the dashed edge below) connecting its two terminals, then its <a href="https://en.wikipedia.org/wiki/Dual_graph">planar dual</a> is also a drawing of the same type, turned sideways. The terminals of the dual are the vertices linked by the dual of the dashed edge. Each series composition in the primal turns into a parallel composition in the dual, and vice versa. Often one talks only about series-parallel graphs, not multigraphs, but for this duality the "multi" part is not easily avoidable: the primal or dual (or both) will have multiple edges between the same two vertices.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/SeriesParallelDual.png"></div><br /><br />The same duality relates terminal-to-terminal paths (or, if you prefer, cycles through the dashed edge) and cuts (minimal subsets of edges whose removal separates the two terminals: a path in one graph is a cut in the dual and vice versa. This works regardless of whether one views the graphs as directed or undirected (the simple paths and minimal cuts are the same). Each pair of a primal and dual path cross exactly once. This one-crossing property is true more generally for <a href="https://en.wikipedia.org/wiki/St-planar_graph"><i>st</i>-planar digraphs</a>, but <i>st</i>-planar graphs don't have the path-cut duality: some of their cuts might not be dual paths (see e.g. the cut formed by the two light green edges below left). On the other hand the path-cut duality works for undirected planar graphs (with two specified terminals and the same extra dashed edge trick to get corresponding terminals in the dual) but they don't generally have the one-crossing property (below right).<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/OneCrossPathCutDual.png"></div><br /><br />Now suppose that some edges might be open (you can go along them as part of a path) and others are closed (blocked off). This gives rise to a Boolean function that takes the state of each edge as input (open = true, closed = false) and produces an output that describes whether there exists an open route from one terminal to the other. The function can be expressed as a Boolean formula in which a series composition becomes an and-operation, and a parallel composition becomes an or-operation. The same function for the dual graph is just the negation of the function for the original graph, and can be obtained by negating all the edge variables and swapping ands for ors. I think this duality of Boolean functions is important for CMOS design but I'm not familiar with the details.<br /><br />The Boolean function that you get this way has a special property: it is <a href="https://en.wikipedia.org/wiki/Read-once_function">read-once</a>, meaning that it has an expression with ands and ors in which each variable appears once (the expression that you get from the graph decomposition). After possibly negating some variables, every read-once function comes from a series-parallel graph in this way, the graph obtained by replacing the ands of a read-once expression by series compositions and the ors of a read-once expression by parallel compositions. If, instead, you express the same function in <a href="https://en.wikipedia.org/wiki/Disjunctive_normal_form">disjunctive normal form</a> you get one prime implicant (conjunction) for each terminal-to-terminal path in the graph. And if you express the same function in <a href="https://en.wikipedia.org/wiki/Conjunctive_normal_form">conjunctive normal form</a> you get one clause (disjunction) for each terminal-to-terminal path in the dual.<br /><br />So the fact that the primal and dual paths of a series-parallel graph have a single crossing expresses, in a graph-theoretic form, a characteristic property of read-once functions (proven in at least three papers, some using these ideas): that their DNF implicants and CNF clauses always intersect in exactly one variable.<a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:328956Fisheye models of Euclidean geometry2016-05-17T23:14:56Z2016-05-18T02:49:46Z<p>You may have heard of the <a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model">Poincaré disk model</a> of the hyperbolic plane, in which the whole hyperbolic plane is mapped to the interior of a Euclidean disk, with hyperbolic lines being transformed into circular arcs that make a right angle with the disk boundary. Or of the <a href="https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model">Beltrami–Klein model</a>, which also maps the hyperbolic plane to the interior of a disk, but with a different map that keeps the lines straight, so that they correspond to the chords of a circle.</p>
<p align="center"><a href="https://commons.wikimedia.org/wiki/File:Uniform_tiling_73-t1_klein.png"><img border="0" src="http://www.ics.uci.edu/~eppstein/0xDE/BeltramiKlein.png"></a></p>
<p>One use for these models is in information visualization, to provide a fisheye view of a drawing in the hyperbolic plane that provides "focus+context": focus on some specific feature in the drawing, and context of all the rest of the drawing, compressed into the outer parts of the disk. Another is as an aid to mathematical intuition: hyperbolic lines may be hard to visualize and understand, but chords of a circle are much more familiar, and this model shows that (in terms of the combinatorial patterns they can perform, if not their distances and angles) they are really the same thing.</p>
<p>But did you know that you can get an analogous fisheye view of the Euclidean plane, by another pair of models that map Euclidean lines to natural families of curves in a disk?</p>
<p>For any function <i>ƒ</i> that maps the ray [0, ∞) one-to-one to an interval [0, <i>r</i>) one can obtain a fisheye view of the Euclidean plane by mapping polar coordinates (<i>θ</i>, <i>r</i>) to (<i>θ</i>, <i>ƒ</i>(<i>r</i>)). But there are a couple of particularly nice choices that can be obtained geometrically, by a three-dimensional projection. Consider the plane as part of a three-dimensional space, with a sphere tangent to the plane at the point you want to focus on, and with the sphere's radius proportional to the size of the feature you want to focus on. Then there are three natural mappings between the sphere and the plane. For any point <i>p</i> on the upper hemisphere of the sphere, there is a unique ray from the center of the sphere through <i>p</i>, and the point where this ray crosses the plane is the <i>central projection</i> of <i>p</i>. Similarly, there is a unique ray from the pole of the sphere (the point opposite its tangency with the plane) through <i>p</i>, and the point where this ray crosses the plane is the <i>polar projection</i> of <i>p</i>. And there is a unique line perpendicular to the plane through <i>p</i>, and the point where it crosses the plane is the <i>perpendicular projection</i> of <i>p</i>. Here they are in cross-section, giving the central, polar, and perpendicular projections between a line and a circle tangent to the line:</p>
<p align="center"><img border="0" src="http://www.ics.uci.edu/~eppstein/0xDE/ThreeProjections.png"></p>
<p>The central projection maps the open upper hemisphere one-to-one to the plane, taking great circular arcs in the hemisphere to lines in the plane. One way to see this is that the lines and arcs are the intersections of planes through the center of the sphere, with the given plane or with the hemisphere respectively. The polar projection maps the whole sphere (except the pole itself) one-to-one to the plane, and takes circles on the sphere to circles or lines in the plane. When restricted to the upper hemisphere, it maps it one-to-one with a disk in the plane, with twice the radius of the sphere. And of course the perpendicular projection also takes the upper hemisphere one-to-one to a disk (with the same radius as the sphere); we saw in <a href="http://11011110.livejournal.com/327966.html">my previous post</a> that it maps great circular arcs to half-ellipses, concentric with the disk and having the same semimajor radius.</p>
<p>Additionally, the polar projection is conformal (it takes angles to equal angles), so it maps circles perpendicular to the equator of the sphere into circles that are perpendicular to the circle bounding the image disk of the hemisphere. And the perpendicular projection takes circles perpendicular to the equator of the sphere into line segments, a chord of the circle bounding the image disk of the hemisphere. So composing the inverse of the polar projection with the perpendicular projection takes the Poincaré model to the Klein model. And composing the inverse of the perpendicular projection with the polar projection takes the Klein model to the Poincaré model.</p>
<p>The two Euclidean disk models are formed in the same way, by composing two of these transformations. Mapping the plane to the hemisphere by an inverse central projection, and then mapping the hemisphere back to the plane by a perpendicular projection, produces a model of the Euclidean plane inside a disk, with Euclidean lines represented as semi-ellipses within the disk (having the disk radius as their semimajor axis):</p>
<p align="center"><img border="0" src="http://www.ics.uci.edu/~eppstein/0xDE/Semielliptical.png"></p>
<p>Mapping the plane to the hemisphere by an inverse central projection, and then mapping the hemisphere back to the plane by a polar projection, produces a model of the Euclidean plane inside a disk, with Euclidean lines represented as circular arcs that cross the boundary of the disk at diametrally opposite points:</p>
<p align="center"><img border="0" src="http://www.ics.uci.edu/~eppstein/0xDE/EuclideanArcModel.png"></p>
<p>In both models, diameter segments also count as degenerate ellipses or arcs, and model Euclidean lines through the origin. One important difference from the hyperbolic case is that neither of these models is conformal: the angles that curves cross in the models aren't the same as the angles that they form in the Euclidean plane. But they can still be used to provide focus+context views of the whole plane. And it follows from the existence of these models that the combinatorial patterns that can be made by lines in the Euclidean plane, by concentric semiellipses with the same semimajor axis, or by circular arcs through diametrally opposite points of a circle, are all the same.</p><a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:328612Linkage2016-05-16T03:27:42Z2016-05-16T03:27:42Z<ul><li><a href="http://arxiv.org/abs/1604.08657">Andrew Suk solves the happy ending problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/T5wa8wGjqhh">G+</a>)</li><br /><li><a href="http://www.sfu.ca/~shermer/CCCG2016/">CCCG is in Vancouver this year</a>; submission deadline is this Tuesday, May 17 (<a href="https://plus.google.com/100003628603413742554/posts/5vi9GNQ4bh4">G+</a>)</li><br /><li><a href="http://www.itsoc.org/news-events/recent-news/sol-golomb-passes-away">Solomon W. Golomb dies</a> (<a href="https://plus.google.com/100003628603413742554/posts/hJmqzQMQJkQ">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2016/05/a-sculptural-geometric-pop-up-book-by-tauba-auerbach/">A sculptural geometric pop-up book by Tauba Auerbach</a> (<a href="https://plus.google.com/100003628603413742554/posts/FyA5z7VKvrR">G+</a>)</li><br /><li><a href="https://blog.vellumatlanta.com/2016/05/04/apple-stole-my-music-no-seriously/">Apple's cloud music storage can wipe your local copies</a> and replace them with ersatz substitutes (<a href="https://plus.google.com/100003628603413742554/posts/hySWEjyD583">G+</a>)</li><br /><li><a href="http://www.metafilter.com/159295/Changing-the-game-at-Harvard">Harvard penalizes members of gender-exclusive organizations</a> (<a href="https://plus.google.com/100003628603413742554/posts/UJmNTJysy7y">G+</a>)</li><br /><li><a href="http://corner.mimuw.edu.pl/?p=811">Matching reviewers to submissions still needs some human attention</a> (<a href="https://plus.google.com/100003628603413742554/posts/4abKgsQRsvY">G+</a>)</li><br /><li><a href="https://twitter.com/dynarski/status/728776167776489472">Removed from a plane for doing mathematics</a> (<a href="https://plus.google.com/100003628603413742554/posts/RyXBm2CEqUp">G+</a>)</li><br /><li><a href="http://chronicle.com/article/No-I-Am-Not-Pregnant/236395">"No, I am not pregnant"</a>: yet another example of everyday sexism in academia, mostly invisible to men (<a href="https://plus.google.com/100003628603413742554/posts/MasdGs8fJe5">G+</a>)</li><br /><li><a href="http://boingboing.net/2016/05/09/australian-government-issues-r.html">Australian government issues report calling for copyright and patent liberalisation</a> (<a href="https://plus.google.com/100003628603413742554/posts/Tr2Be3ttrDm">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Binary_search_algorithm">Binary search</a>, now another Wikipedia good article (<a href="https://plus.google.com/100003628603413742554/posts/1K5RnhuXbBA">G+</a>)</li><br /><li><a href="http://boingboing.net/2016/05/12/researchers-demonstrate-edible.html">Origami robots that can perform surgery on your digestive system after you eat them</a> (<a href="https://plus.google.com/100003628603413742554/posts/9UPp9cgakTb">G+</a>)<br /><p align="center"><lj-embed id="62" /></p></li><br /><li><a href="http://www.slate.com/blogs/bad_astronomy/2016/05/13/eleanor_lutz_created_a_medieval_style_map_of_mars.html">Here Be Robots: A Medieval Map of Mars</a> (<a href="https://plus.google.com/100003628603413742554/posts/bnV6iXcMv9m">G+</a>)</li><br /><li><a href="http://f1000research.com/articles/5-222/v1">New uses for blockchain: ensuring the integrity of medical experiments</a> (<a href="https://plus.google.com/100003628603413742554/posts/cknMhpVyiVZ">G+</a>)</li><br /><li><a href="http://perl.plover.com/yak/Elmo/">An amusing 10-minute talk about NP-completeness and its application to Sesame Street</a> (<a href="http://perl.plover.com/yak/Elmo/">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:328417Linkage2016-05-01T07:14:18Z2016-05-01T17:49:36Z<ul><li><a href="https://en.wikipedia.org/wiki/Book_embedding">Book embedding on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/L6dPgV7aqXd">G+</a>)</li><br /><li><a href="http://www.boredpanda.es/plantas-geometricas">The fractal geometry of plants</a> (<a href="https://plus.google.com/100003628603413742554/posts/gXHW9aki7QZ">G+</a>)</li><br /><li><a href="https://medium.com/@milistjohn/i-am-alex-st-john-s-daughter-and-he-is-wrong-about-women-in-tech-4728545e7c0e#.4szklt59i">A great combination of a success story of an individual woman in tech, a debunking of some nasty misogynist myths, and a collection of helpful resource links for others on the same path</a> (<a href="https://plus.google.com/100003628603413742554/posts/K3Z9Q2cWmRv">G+</a>)</li><br /><li><a href="https://vimeo.com/157620840">Trailer for a documentary on the shifting technology of graphic design from hand layout to computers</a> (<a href="https://plus.google.com/100003628603413742554/posts/SZoAteBLtF7">G+</a>)</li><br /><li><a href="http://www.nature.com/news/peer-review-troubled-from-the-start-1.19763">Complaints about problems with peer review turn out to be far from new</a> (<a href="https://plus.google.com/100003628603413742554/posts/aYq1nTQY2q7">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Curve-shortening_flow">Curve-shortening flow on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/KeABxYM2mDv">G+</a>)</li><br /><li><a href="http://poynder.blogspot.com/2016/04/the-open-access-interviews-sir-timothy.html">The Open Access Interviews: Timothy Gowers</a> (<a href="https://plus.google.com/100003628603413742554/posts/ADFYb5DzBLk">G+</a>)</li><br /><li><a href="http://techcrunch.com/2016/04/26/it-isnt-just-uber-carnegie-mellons-computer-science-dean-on-its-poaching-problem/">CMU’s computer science dean on its poaching problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/PUmAu2BjjkL">G+</a>)</li><br /><li><a href="http://danielwalsh.tumblr.com/post/2173134224/sudo-make-me-a-pseudosphere">From tilted decks of playing cards to hyperbolic geometry</a> (<a href="https://plus.google.com/100003628603413742554/posts/b2fFCynAc3W">G+</a>)</li><br /><li><a href="https://sbseminar.wordpress.com/2016/04/26/springers-copyright-agreement-is-according-to-springer-compatible-with-posting-your-article-on-the-arxiv-under-the-cc-by-0-license/">Springer journal copyrights are compatible with arXiv preprints</a> (<a href="https://plus.google.com/100003628603413742554/posts/QUKWtURCkRX">G+</a>)</li><br /><li><a href="http://twocubes.tumblr.com/post/89441527133/1-2-5">Pinwheel zoom</a> (<a href="https://plus.google.com/100003628603413742554/posts/Va2yewVq731">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:327966The shape of the Kresge Auditorium2016-05-01T02:58:54Z2016-05-01T07:52:45ZThe image below is a study of the geometry of MIT's <a href="https://en.wikipedia.org/wiki/Kresge_Auditorium">Kresge Auditorium</a>.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/Kresge.png"></div><br /><br />I found <a href="http://mathtourist.blogspot.com/2011/12/geometreks-in-boston.html">an article by Ivars Petersen</a> claiming that this building's floor plan is "close to the geometry of a Reuleaux triangle" and I wanted to determine whether that was true. Other sources such as <a href="https://www.technologyreview.com/s/404017/the-amazing-kresge-turns-50/">a 50-year retrospective published by MIT</a> state that the roof of the building has the shape of an eighth of a sphere (a spherical right equilateral triangle); see <a href="http://fab.cba.mit.edu/classes/863.15/section.Harvard/people/Basangwa/week3.html">this link on making a 3d model of the shape</a> for an amusingly-captioned visualization of its construction.<br /><br />So, the floor plan is the projection of an eighth-sphere; what is this shape? The edges of the roof are great circle arcs in 3d, so they project to ellipses in 2d. By my calculation, the aspect ratio of these ellipses is √3:1. To see this, let the sphere be the unit sphere in 3d, with the three corners of the roof at (1,0,0), (0,1,0), and (0,0,1), and project it onto the plane x+y+z=0. Then the semimajor axis of the ellipse is the radius of the sphere, 1, while the semiminor axis is the distance from the origin of the projected midpoint of an arc. The midpoint is √2(1/2,1/2,0), its projection is √2(1/6,1/6,-1/3), and the distance is 1/√3. So I drew three ellipses with that aspect ratio, rotated by a third of a circle around their common center, and to complete the illusion of being three-dimensional (though really it's just a 2d drawing) I added another circle, with radius equal to the semimajor axis of the ellipses. Those are the grey and black parts of the figure. The shape of the auditorium floor plan is the central triangle outlined by black arcs.<br /><br />The red circles in the drawing are centered at the corners of this triangle, and pass through the other two corners. Their intersection forms a Reuleaux triangle, overlaid on the other curved triangle formed by the projected roof. As you can see, the floor plan is not actually a Reuleaux triangle. It differs from Reuleaux in two significant ways: It has slightly less area, and it has elliptical arcs for sides (with variable curvature, bendier near the corners and flatter near the centers of each side) rather than circular arcs. On the other hand, as Petersen stated, it is very close.<br /><br />So, to state the obvious, not all curvy triangles are alike! Another example of this same phenomenon is given by the rotor of the <a href="https://en.wikipedia.org/wiki/Wankel_engine">Wankel rotary engine</a>: also a curved triangle with sharper angles than the Reuleaux, but with another kind of curve for its sides (the <a href="http://demonstrations.wolfram.com/WankelRotaryEngineEpitrochoidalEnvelopes/">envelope of an epitrochoid</a>). I'm pretty sure this envelope is not an ellipse, even though I don't know how to draw it. And the angles are definitely different. So the Wankel would be yet another kind of curved equilateral triangle that differs from the first two.<a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:327681Cuckoo filters and their analysis2016-04-21T03:31:39Z2016-04-21T03:41:43ZDo you have a software project in which you need a fast and space-efficient approximate set data structure, like a <a href="https://en.wikipedia.org/wiki/Bloom_filter">Bloom filter</a>? Then probably what you want is actually a <a href="https://github.com/efficient/cuckoofilter">cuckoo filter</a>, a plug-in replacement for Bloom filters that is faster, more space-efficient, and more versatile (because it allows elements to be deleted as well as inserted).<br /><br />Michael Mitzenmacher has described cuckoo filters in <a href="http://mybiasedcoin.blogspot.com/2014/10/cuckoo-filters.html">an earlier blog post</a> (as well as of course in the <a href="http://www.eecs.harvard.edu/~michaelm/postscripts/cuckoo-conext2014.pdf">published paper about them</a>) but the basic idea is to use a <a href="https://en.wikipedia.org/wiki/Cuckoo_hashing">cuckoo hash table</a> cut down in size by storing only a short fingerprint of each key rather than a whole key-value pair. As in a normal cuckoo hash table, keys (or rather their fingerprints) get moved around to make room for other keys, and that leads to a small complication: when you're moving a fingerprint, you don't know which key it came from, so the location to move it to needs to be computable based only on where it is now and on its value. More specifically, the other location for any fingerprint ends up being the xor of its current location with a hash of its value.<br /><br />Although cuckoo filters have been implemented (see first link) and work well in practice, one drawback is that we didn't know whether they also work well in theory. Conversely, an earlier data structure of Pagh, Pagh, and Rao (<a href="http://www.itu.dk/people/pagh/papers/bloom.pdf">SODA 2005</a>) has all the same advantages of cuckoo filters over Bloom filters, but for it, as far as I am aware, there was no implementation, only theory. In contrast, Bloom filters work both actually and theoretically: there is no major gap between theory and practice.<br /><br />So my new paper, "Cuckoo filter: simplification and analysis" (<a href="http://arxiv.org/abs/1604.06067">arXiv:1604.06067</a>, to appear at SWAT) is aimed at closing this gap, although it does not fully do so. What it does is to show that, if you omit the hash of the fingerprint and instead move each fingerprint to the xor of its current location and its value, then cuckoo filtering works well in theory. This simplification causes the filter to be partitioned into many small sub-filters, which operate independently of each other, with each key being assigned randomly to one of them. The main ideas of the paper are that this assigment of keys to sub-filters is very unlikely to be unbalanced and that, within each sub-filter, the data structure behaves just like a cuckoo hash, without any restriction on which pairs of cells the keys can be mapped to. It uses Chernoff bounds to prove the balancing part (as you do), and then just plugs in the existing analysis of cuckoo hashing for the rest.<br /><br />Of course, it would be better to prove that the actual cuckoo filter works well than this simplification, since I don't think the simplification is likely to be a practical improvement. The graph whose edges connect pairs of cells where a single fingerprint can go has a lot of nice structure and symmetry (usually either a Cayley graph or a disjoint union of hypercubes, compared to a disjoint union of cliques for the simplification), suggesting that spectral graph theory might be helpful, but that's not my forté. Also, my analysis only holds for random hash functions, so it would be good to extend it to realistic methods such as <a href="https://en.wikipedia.org/wiki/Tabulation_hashing">tabulation hashing</a>. Tabulation hashing is known to work for cuckoo hashing, but to get my analysis to work it needs to be extended to blocked cuckoo hashing, a variation that allows multiple keys per cell. Ideally, an analysis of the full cuckoo filter algorithm with a realistic hash function would be best.<br /><br />But you don't need to wait for that analysis to happen to go start using these things in your code: they already work now. We just don't completely understand why they work.<a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:327648Local and inductive properties of graphs2016-04-18T06:23:34Z2016-04-18T06:30:26ZAt LATIN, Allan Borodin brought my attention to a recent paper of his with Yuli Ye, "<a href="http://www.cs.toronto.edu/~bor/Papers/inductive-independence.pdf">Elimination graphs</a>" (TALG 2012), about an idea that combines local properties of graphs with degeneracy.<br /><br />For a graph property P, a graph G is "locally P" if the <a href="https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)">open neighborhood</a> of each vertex v in G (the subgraph of G induced by the neighbors of v) has property P. For instance, a graph is locally cyclic if every vertex's neighbors induce a cycle; these are the graphs of certain well-behaved surface triangulations (every clique of the graph is a simplex in the triangulation). In the hecatohedron below, every neighborhood is a 5-cycle or a 6-cycle.<br /><br /><div align="center"><img src="https://www.ics.uci.edu/~eppstein/junkyard/hecatohedron.gif"></div><br /><br />Local graph properties work particularly well when P is hereditary (that is, when induced subgraphs of graphs with property P also have property P). For then the property of being locally P is also hereditary. And, in this case, being locally P is a generalization of being P. For instance, the <a href="https://en.wikipedia.org/wiki/Claw-free_graph">claw-free graphs</a> are exactly the locally 2-independent graphs (every neighborhood has independence number at most 2); being 2-independent and being claw-free are both hereditary. The graphs of bounded degree are exactly the locally bounded graphs (every neighborhood has constant size).<br /><br />Analogously, Borodin and Ye define an undirected graph to be "inductively P" if its vertices can be ordered into a sequence so that, for each v, the graph induced by the earlier neighbors of v in the sequence has property P. Again, this works particularly well when P is hereditary, because then being inductively P is also hereditary and generalizes P. Additionally, when P is hereditary, the set of orderings in which the vertices can be removed (before reversal) forms an <a href="https://en.wikipedia.org/wiki/Antimatroid">antimatroid</a>, and a valid vertex ordering can be found by a greedy algorithm that repeatedly removes a vertex from the given graph whose neighbors have property P. This algorithm can remove all vertices if and only if the graph is inductively P, in which case the inductive ordering is the reverse of the removal ordering. Therefore, whenever P is hereditary and can be tested in polynomial time, so can the property of being inductively P.<br /><br />With this definition, a graph has bounded <a href="https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)">degeneracy</a> if and only if it is inductively bounded, with the degeneracy equal to the largest earlier-neighborhood size in an ordering that minimizes this size. A graph is <a href="https://en.wikipedia.org/wiki/Chordal_graph">chordal</a> if and only if it is inductively complete (inductively 1-independent), and the inductive ordering in this case is called an elimination ordering. In this way, Borodin and Ye formalize a common generalization of degeneracy orderings and elimination orderings. However, another standard type of vertex ordering, a <a href="https://en.wikipedia.org/wiki/Perfectly_orderable_graph">perfect ordering</a>, can't be defined via inductive hereditary properties, because the orderings are NP-hard to find.<br /><br />The particular property most closely examined by Borodin and Ye is that of being inductively k-independent, which generalizes chordal graphs (k=1), degenerate graphs (k=degeneracy), claw-free graphs (k=2), planar graphs (k=3, beating the degeneracy bound of 5), unit disk graphs (k=3), and disk graphs more generally (order by radius to get k=5). With this property, for instance, a greedy coloring in the inductive order uses at most k times the optimal number of colors (because any coloring of the earlier neighborhood of any vertex uses at most k times the number of colors that would have been used in an optimal coloring). They also show that the inductively k-independent graphs have (k+1)-approximations to their maximum weighted independent set.<br /><br />Borodin and Ye give a time bound of O(n<sup>k+2</sup>) time for recognizing inductively k-independent graphs, or O(n<sup>k+3</sup>) in linear space, when k is a constant. They list improving these time bounds as their most important open problem. Here, some observations already used for recognizing claw-free graphs may help. For instance, each removed vertex can have only O(sqrt m) neighbors (else its neighborhood would be too sparse to be only k-independent), an observation made in the claw-free case by <a href="https://dx.doi.org/10.1016%2FS0020-0190%2800%2900047-8">Kloks, Kratsch & Müller (2000)</a>. And, when seeking a vertex to remove, only O(sqrt m) of them can even have this many neighbors. This immediately improves the linear-space algorithm to O(nm<sup>(k+2)/2</sup>). It seems likely that the fast matrix multiplication methods used to speed up claw-free graph recognition can also be used here to reduce the exponent (as Borodin and Ye already do for some small values of k), but I haven't worked out the details. On the other hand, one can't do much better, because a graph G has independence number at most k if and only if the graph formed from G by adding k+1 independent new vertices, each adjacent to all the original vertices, is inductively k-independent. Therefore, inductive k-independence is no easier than k-independence, and (under the <a href="https://en.wikipedia.org/wiki/Exponential_time_hypothesis">exponential time hypothesis</a>) it requires an exponent linear in k by results of <a href="https://dx.doi.org/10.1016%2Fj.jcss.2006.04.007">Chen et al. (2006)</a>.<a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:327399Photos from LATIN2016-04-17T01:38:06Z2016-04-17T04:45:46ZI recently traveled to Ensenada (my first visit to Mexico, despite its closeness) for <a href="http://latin2016.natix.org/">LATIN 2016</a>. This was our view every morning on our arrival to <a href="https://en.wikipedia.org/wiki/Ensenada_Center_for_Scientific_Research_and_Higher_Education">CICESE</a>, the research center hosting the conference, for the conference breakfast.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/ensenada/CICESE2-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><a href="http://www.ics.uci.edu/~eppstein/pix/ensenada/index.html">The rest of my photos</a> include several from the conference excursion and dinner in the <a href="https://en.wikipedia.org/wiki/Guadalupe,_Baja_California">Valle de Guadalupe</a>, Mexico's wine region.urn:lj:livejournal.com:atom1:11011110:326975Linkage2016-04-16T04:42:36Z2016-04-16T04:46:50ZI continue to find it ironic that the best way I can find to index my posts on Google+, a social network platform launched in 2011 by a major search engine company, and to make it possible for me to search for and find the old posts, is to copy them onto a much older social network platform from 1999 that, except for the Russians, has now mostly fallen into disuse.<br /><ul><li><a href="https://treemapart.wordpress.com/">Ben Schneiderman's treemap art</a> (<a href="https://plus.google.com/100003628603413742554/posts/YLaCJUYxcPe">G+</a>)</li><br /><li><a href="http://evahild.com/">Eva Hild, ceramic artist whose works look like minimal surfaces</a> (<a href="https://plus.google.com/100003628603413742554/posts/ePY22PCkXZt">G+</a>)</li><br /><li><a href="https://plus.google.com/+IsaacCalder/posts/7STiWdYBsiF">Symmetric unfolding of the polytope formed by removing one vertex from a 4-cube</a> (<a href="https://plus.google.com/100003628603413742554/posts/3zGDMzBqsMy">G+</a>)</li><br /><li><a href="http://mentalfloss.com/article/65979/origami-pot-changes-size-plants-grow">Auto-resizing origami flowerpot</a> (<a href="https://plus.google.com/100003628603413742554/posts/NEJHwYjHMz6">G+</a>)</li><br /><li><a href="http://motherboard.vice.com/read/wikipedia-zero-facebook-free-basics-angola-pirates-zero-rating">Is it moral to zero-rate Wikipedia in third-world countries?</a> (<a href="https://plus.google.com/100003628603413742554/posts/3j4bAEJebpn">G+</a>)</li><br /><li><a href="http://crookedtimber.org/2016/04/06/38290/">Some tips to encourage more egalitarian contributions in discussion-based classes</a> (<a href="https://plus.google.com/100003628603413742554/posts/BbcDxgVf7kw">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Linear_probing">Linear probing, now a good article on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/VUSzX6sdXRg">G+</a>)</li><br /><li><a href="http://nautil.us/issue/35/boundaries/why-nature-prefers-hexagons">Hexagonal structures in nature</a> (<a href="https://plus.google.com/100003628603413742554/posts/PAntH5HhQT8">G+</a>)</li><br /><li><a href="http://i.imgur.com/Bbr7s8f.gif">Portrait of the scientist as an infant</a> (shadowplay GIF; <a href="https://plus.google.com/100003628603413742554/posts/DW8HxPbaiRA">G+</a>)</li><br /><li><a href="http://chronicle.com/blogs/ticker/federal-agents-set-up-a-fake-university-to-make-arrests-on-visa-fraud/110079">Fake university exposes visa fraud</a> (<a href="https://plus.google.com/100003628603413742554/posts/cmXdhqjU9HQ">G+</a>; discussion involves the "true" purpose of degree programs)</li><br /><li><a href="https://ilaba.wordpress.com/2016/04/10/arxiv-comments-and-quality-control/">Do open peer-review systems play into sexism?</a> (<a href="https://plus.google.com/100003628603413742554/posts/AmptHg341fE">G+</a>; and if you think they might, be sure to contribute to the <a href="https://cornell.qualtrics.com/SE/?SID=SV_brM4ULfjHGClNJj">arXiv user survey</a> before it closes on April 26)</li><br /><li><a href="https://en.wikipedia.org/wiki/Andrew_M._Gleason">Andrew Gleason</a>, a mathematician who made contributions to Lie group theory, quantum mechanics, Ramsey theory, coding theory, cryptanalysis, and calculus reform, among others (<a href="https://plus.google.com/100003628603413742554/posts/LPCKZSCpSCn">G+</a>)</li><br /><li><a href="http://paleofuture.gizmodo.com/the-untold-story-of-the-teen-hackers-who-transformed-th-1770977586">Teen hackers in the early 1980s</a> (<a href="https://plus.google.com/100003628603413742554/posts/AiHpK7vbUMA">G+</a>)</li><br /><li><a href="http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/">The degree-6 surface with the maximum possible number of nodes (65 of them)</a>. But <a href="https://en.wikipedia.org/wiki/Nodal_surface">for higher degrees the maximum number is still unknown</a> (<a href="https://plus.google.com/100003628603413742554/posts/VNdJ8uemAHK">G+</a>)</li><br /><li><a href="http://katsumihayakawa.com/paperbonsai.html">3d model of an imagined city, made entirely out of paper by Katsumi Hayakawa</a> (<a href="https://plus.google.com/100003628603413742554/posts/XjFpoXLfxCz">G+</a>)</li><br /></ul>urn:lj:livejournal.com:atom1:11011110:326877Linkage2016-04-01T04:45:48Z2016-04-01T04:45:48Z<ul><li><a href="https://www.youtube.com/watch?v=zsjZ2r9Ygzw">John Oliver on the reopening of the cryptowars</a> (<a href="https://plus.google.com/100003628603413742554/posts/Co1SxTyGhZj">G+</a>)</li><br /><li><a href="http://arstechnica.com/security/2016/03/big-name-sites-hit-by-rash-of-malicious-ads-spreading-crypto-ransomware/">Bad ad-screening hygiene exposes thousands to malware</a> so you should use an ad-blocker to be safe (<a href="https://plus.google.com/100003628603413742554/posts/5ai4KkHqDLJ">G+</a>)</li><br /><li><a href="http://www.nature.com/news/turkish-academics-jailed-for-making-terrorism-propaganda-1.19586">Turkey equates petition-signing to terrorism</a>, jails three professors, hounds others from their positions (<a href="https://plus.google.com/100003628603413742554/posts/1E5G9rax5ij">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=IANBoybVApQ">Printable magnets</a> (<a href="https://plus.google.com/100003628603413742554/posts/FPhhAyrygqQ">G+</a>)</li><br /><li><a href="http://www.seas.upenn.edu/~mlazar/voronoi.html">The distribution of cell shapes in Voronoi diagrams of random points in space</a> (<a href="https://plus.google.com/100003628603413742554/posts/ij2ocwZMx4c">G+</a>)</li><br /><li><a href="http://chronicle.com/article/Does-Engineering-Education/235800">The link between engineering education and terrorism</a> (<a href="https://plus.google.com/100003628603413742554/posts/NUNA7jtsjC7">G+</a>)</li><br /><li><a href="https://gilkalai.wordpress.com/2016/03/23/a-breakthrough-by-maryna-viazovska-lead-to-the-long-awaited-solutions-for-the-densest-packing-problem-in-dimensions-8-and-24">Sphere packing solved in dimensions 8 and 24</a> (<a href="https://plus.google.com/100003628603413742554/posts/Drio8d6qp8m">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Split_%28graph_theory%29">Split decomposition of graphs</a> (<a href="https://plus.google.com/100003628603413742554/posts/GkWnDTA6gqY">G+</a>)</li><br /><li><a href="https://news.brown.edu/articles/2016/03/wrinkles">Wrinkles and crumples make graphene better</a> (<a href="https://plus.google.com/100003628603413742554/posts/6JmXKQh84uh">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Integer_sorting">Integer sorting</a> (<a href="https://plus.google.com/100003628603413742554/posts/aU3dsKvb7HW">G+</a>)</li><br /><li><a href="http://themathkid.tumblr.com/post/141688266316/from-proofs-without-words-by-roger-nelsen">Visual proof of the arithmetic-geometric mean inequality</a> (<a href="https://plus.google.com/100003628603413742554/posts/Y4uG8rjcPqd">G+</a>)</li><br /><li><a href="http://www.andrewlipson.com/lego.htm">Lego Escher</a> (<a href="https://plus.google.com/100003628603413742554/posts/iAnkw9Dc1Bs">G+</a>)</li><br /><li><a href="http://www.dailylife.com.au/news-and-views/news-features/top-sydney-university-mathematician-nalini-joshi-laments-gender-discrimination-20160329-gnsywt.html">Nalini Joshi on gender discrimination in Australian mathematics</a> (<a href="https://plus.google.com/100003628603413742554/posts/8xBKv27EjzW">G+</a>)</li><br /><li><a href="http://ringsanity.blogspot.com/2016/03/mathematics-of-ringiana-and-ringsanity.html">Addictive abstract puzzle game based on the theory of Thompson groups</a> (<a href="https://plus.google.com/100003628603413742554/posts/8SDU9X5uvmV">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:326418Random binary heaps, separable permutations, and numbers that multiply to factorials2016-03-20T01:45:49Z2016-03-20T01:45:49Z<p>I recently asked my data structures class the following question: suppose you fill in an <i>n</i>-cell array by a random permutation. What is the probability that the result is a valid binary min-heap?</p>
<p>One way to solve this would be to count heaps, then divide by <i>n!</i>. For instance, for <i>n</i> = 5, there are eight heaps: there are four choices for the right child of the root, and two choices for how to order the two grandchildren, so the probability is 8/120 = 1/15.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/factorial-heaps/Heaps5.png"></p>
<p>Alternatively, you can compute the probability a different way. Each node in the tree must have the minimum value among all its descendants. For a random permutation, the probability that it does is one over the number of descendants (counting the node as a descendant of itself). And the events that two nodes are minimum among their descendants turn out to be independent, so we can just multiply these probabilities together. A five-element heap has nodes whose numbers of descendants are 5, 3, 1, 1, 1, so the probability of getting a heap is one over the product of these numbers, 1/15. This calculation explains why the probability is a unit fraction, something that seems a bit mysterious when you calculate it the other way.</p>
<p>But now we have a nice sequence of integers, the inverses of these probabilities for different choices of <i>n</i>: <a href="http://oeis.org/A132862">1, 2, 3, 8, 15, 36, 63, etc</a>. And when we have a nice sequence of integers, we'd like them to count things. What do these numbers count? One possible answer is that they count a different set of labelings on the same complete binary trees: labelings for which, at each node, all left descendants are smaller than all right descendants. There are 15 such labelings, determined by the choice of any label at the root and any of the three remaining smallest values at its left child:</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/factorial-heaps/AlmostSorted5.png"></p>
<p>More generally, for any <i>n</i>-node tree <i>T</i>, say that a labeling of the nodes of <i>T</i> by the numbers from 1 to <i>n</i> is <i>T</i>-heap-ordered if, for any ancestor-descendant pair (<i>x</i>,<i>y</i>), we have that <i>x</i> < <i>y</i>. And say that a labeling is <i>T</i>-almost-sorted if, for any pair (<i>x</i>,<i>y</i>) that are not in an ancestor-descendant relation, with <i>x</i> to the left of <i>y</i>, we have that <i>x</i> < <i>y</i>. Then the product of the number of <i>T</i>-heap-ordered labelings and the number of <i>T</i>-almost-sorted labelings always equals <i>n</i>!.</p>
<p>We can take this one step farther, from trees to permutations. For any permutation <i>π</i> of the numbers from 1 to <i>n</i>, consider the two-dimensional set of points of the form (<i>i</i>,<i>π</i>(<i>i</i>)), and consider the labelings of these points by the numbers from 1 to <i>n</i>. Define a labeling to be <i>π</i>-upward if, for every two points oriented from southwest to northeast, the southwest point has a smaller label than the northeast point. And define a labeling to be <i>π</i>-downward if, for every two points oriented from northwest to southeast, the northwest point has a smaller label than the northeast point. Do the numbers of <i>π</i>-upward and <i>π</i>-downward labelings have a nice product formula? Not always. For instance, the permutaton 2413 has five <i>π</i>-upward labelings (below) and symmetrically five <i>π</i>-downward labelings. The product of these counts, 25, differs from <i>n</i>! = 24.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/factorial-heaps/2413-upward.png"></p>
<p>However in some sense this is the only possible counterexample. For, when <i>π</i> is a <a href="https://en.wikipedia.org/wiki/Separable_permutation">separable permutation</a> (a permutation that avoids both 2413 and its mirror image 3142 as patterns) then the numbers of <i>π</i>-upward and <i>π</i>-downward labelings always multiply to <i>n</i>!.</p>
<p style="margin-left:2em"><i>Proof sketch:</i> in this case the dot pattern for <i>π</i> can be broken into the dot patterns for two smaller separable permutations, either southwest-northeast of each other (the direct sum of the two smaller permutations) or northwest-southeast (the skew sum). Suppose the sizes of the smaller permutations are <i>k</i> and <i>n</i> − <i>k</i>, and suppose they're combined in a direct sum. Then by induction the product of the numbers of labelings in the two smaller dot patterns is <i>k</i>! and (<i>n</i> − <i>k</i>)!. Two <i>π</i>-upward labelings on the two smaller dot patterns can only be combined in one way to make a <i>π</i>-upward labeling on the whole pattern: every label in the southwest pattern has to be less than every label in the northeast pattern. However, two <i>π</i>-downward labelings on the two smaller dot patterns can be combined in many ways to make a <i>π</i>-downward labeling on the whole dot pattern: we can partition the <i>n</i> labels among the two patterns arbitrarily, and there are Choose(<i>n</i>,<i>k</i>) partitions to choose among. So the number of pairs of an upward and downward label on the whole pattern is the product of the numbers of pairs labelings on the two smaller patterns with the number of ways we can make this combination:
<i>k</i>! × (<i>n</i> − <i>k</i>)! × Choose(<i>n</i>,<i>k</i>) = <i>n</i>!. The case of a skew sum follows by a symmetric argument.</p>
<p>More strongly, this argument shows that any labeling of <i>π</i> can be generated uniquely from the trivial labeling (the one that labels each point by its <i>x</i>-coordinate) by a sequence of riffle shuffles, working bottom up in the recursive structure of <i>π</i> as a skew sum or direct sum of smaller permutations. At each skew sum or direct sum in this decomposition, combining smaller permutations of sizes <i>a</i> and <i>b</i>, we choose one of the Choose(<i>a</i> + <i>b</i>,<i>a</i>) possible riffles and apply it to the labels on the dot patterns of these two smaller permutations. If we do all of the riffles, we get an arbitrary labeling. But if we do only the riffles at skew sums, we get a <i>π</i>-upward labeling, and if we do only the riffles at direct sums, we get a <i>π</i>-downward labeling. So this shows that an arbitrary labeling of <i>π</i> can be decomposed into a unique pair of an upward and downward labeling.</p>
<p>For every tree <i>T</i>, there is a drawing of <i>T</i> as the dot pattern of a separable permutation <i>π</i> such that the <i>π</i>-upward labelings are the same as the <i>T</i>-heap-ordered labelings and the <i>π</i>-downward labelings are the same as the <i>T</i>-almost-sorted labelings. One can construct <i>π</i> as the direct sum of the root of <i>T</i> (represented by a one-element permutation) with the skew sum of the subtrees of <i>T</i> (represented recursively). For instance, the complete binary tree used for 5-element binary heaps is represented in this way by the permutation 13542:</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/factorial-heaps/Tree2Perm.png"></p>
<p>So, the product formula for labelings of trees is a special case of the product formula for separable permutations.</p><a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:326205Holetown2016-03-19T04:07:35Z2016-03-19T04:07:35ZSome scenes near the entrance to the Bellairs Research Institute, in Holetown, Barbados. (You can see the institute's nameplate in the left background to this one). It's actually quite an upscale touristy town, but that wasn't the feeling I was trying to catch in these photos.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/holetown/8-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/holetown/index.html">More photos</a> )</b>urn:lj:livejournal.com:atom1:11011110:326101Linkage2016-03-16T06:02:37Z2016-03-16T06:02:37Z<ul><li><a href="https://www.washingtonpost.com/news/grade-point/wp/2016/02/29/mount-st-marys-university-and-the-dilemma-facing-american-higher-education/">Why aiming to run a university like a business is a worse idea than it sounds like</a> (<a href="https://plus.google.com/100003628603413742554/posts/aLfxrdUxbGs">G+</a>)</li><br /><li><a href="http://discreteanalysisjournal.com/"><i>Discrete Analysis</i> journal</a> publishes its inaugural issue (<a href="https://plus.google.com/100003628603413742554/posts/ceEZuMiYEms">G+</a>)</li><br /><li><a href="https://theconversation.com/new-defence-trade-controls-threaten-academic-freedom-and-the-economy-55310">Australian trade controls threaten academic freedom</a> (<a href="https://plus.google.com/100003628603413742554/posts/Fz92JJhXrp5">G+</a>)</li><br /><li><a href="https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/">Interview with Michael Atiyah</a> with some nice insights into collaborative mathematical research (<a href="https://plus.google.com/100003628603413742554/posts/FT5uFASnG15">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Frankl%E2%80%93R%C3%B6dl_graph">The Frankl–Rödl graph</a> and the challenge it poses to the Unique Games Conjecture (<a href="https://plus.google.com/100003628603413742554/posts/PrYGRkoKSi9">G+</a>)</li><br /><li><a href="http://conwaylife.com/forums/viewtopic.php?f=2&t=2057">Discovery of the Copperhead, a new small c/10 spaceship in Conway's Life</a> (<a href="https://plus.google.com/100003628603413742554/posts/BtcbuZvHR8M">G+</a>)</li><br /><li><a href="http://fivethirtyeight.com/features/a-plagiarism-scandal-is-unfolding-in-the-crossword-world/">Crossword plagiarism</a> (<a href="https://plus.google.com/100003628603413742554/posts/KFdQZBfqM1S">G+</a>)</li><br /><li><a href="http://science.sciencemag.org/content/351/6276/902.full">Academic journal forces pseudonymous authors to reveal their names</a> (<a href="https://plus.google.com/100003628603413742554/posts/Zc78ahxnKxo">G+</a>)</li><br /><li><a href="http://malinchristersson.tumblr.com/post/134524872603/m%C3%B6bius-transformation-of-a-doyle-spiral">Möbius transformation of a Doyle spiral circle packing</a> (<a href="https://plus.google.com/100003628603413742554/posts/aQiRez6Khbm">G+</a>)</li><br /><li><a href="http://blog.computationalcomplexity.org/2016/03/david-johnson-1945-2016.html">David Johnson (1945–2016)</a> (<a href="https://plus.google.com/100003628603413742554/posts/groVYC7wjaS">G+</a>)</li><br /><li><a href="http://blogs.scientificamerican.com/roots-of-unity/how-to-sew-like-a-mathematician/">Using topology to simplify a real-world sewing problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/jgyhN4vn8kG">G+</a>)</li><br /><li><a href="http://www.sfgate.com/education/article/UC-Berkeley-law-dean-resigns-amid-harassment-6882570.php">Yet another academic sexual harrassment case</a> brings down the Berkeley law dean (<a href="https://plus.google.com/100003628603413742554/posts/Jycm648qDPi">G+</a>)</li><br /><li><a href="http://i.imgur.com/I491kLh.jpg">There is no cloud. It's just someone else's computer.</a> (<a href="https://plus.google.com/100003628603413742554/posts/e3ivrz4ryxb">G+</a>)</li><br /><li><a href="https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/">Nonrandomness in the final digits of consecutive primes</a> (<a href="https://plus.google.com/100003628603413742554/posts/aB7T6Z7vvs6">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2016/03/cycloid-drawing-machine/">A fancier spirograph</a> and <a href="http://wheelof.com/sketch/">its online simulator</a> (<a href="https://plus.google.com/100003628603413742554/posts/71pXYjguAjK">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:325782Bellairs2016-03-14T17:35:46Z2016-03-14T17:35:46ZI recently returned from the <a href="http://cglab.ca/~morin/misc/bb2016/">Fourth Annual Workshop on Geometry and Graphs</a> at the Bellairs Research Institute in Barbados. The youngest participant was Günter Rote's daughter:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs16/AdeleErases-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/bellairs16/index.html">More photos of workshop participants</a> )</b>urn:lj:livejournal.com:atom1:11011110:325544Linkage2016-02-29T01:38:05Z2016-02-29T01:58:19Z<ul><li><a href="https://plus.google.com/+JeffErickson/posts/fUzs9WxQKRZ">SoCG accepted paper titles</a> (<a href="https://plus.google.com/100003628603413742554/posts/Sj4a3oWxzG2">G+</a>)</li><br /><li><a href="http://utotherescue.blogspot.com/2016/01/ucop-ordered-spyware-installed-on-uc.html">University of California President Janet Napolitano spies on all campus emails and other internet traffic</a> (<a href="https://plus.google.com/100003628603413742554/posts/PU6ByUUcs6k">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=tDQw21ntR64">Sheep flocking patterns</a> (<a href="https://plus.google.com/100003628603413742554/posts/HpH92r9DFQE">G+</a>)</li><br /><li><a href="http://www.theguardian.com/education/2010/jul/13/perfect-coffee-improbable-research">Using the Thue–Morse sequence to pour coffee</a> (<a href="https://plus.google.com/100003628603413742554/posts/FJ1MxR7opjb">G+</a>)</li><br /><li><a href="http://www2.idsia.ch/cms/fun16/">FUN</a> and <a href="http://optnetsci.cise.ufl.edu/cocoon16/">COCOON</a> submission deadlines extended to early March (<a href="https://plus.google.com/100003628603413742554/posts/NH5CUpK2g2L">G+</a>)</li><br /><li><a href="https://vimeo.com/150929970">Frozen soap bubbles</a> (<a href="https://plus.google.com/100003628603413742554/posts/V8PQu7nCfct">G+</a>)</li><br /><li><a href="http://www.theguardian.com/world/2016/feb/16/prestigious-academic-to-quit-new-zealand-after-autistic-son-refused-residency">Group theorist Dimitri Leemans returns to Europe after NZ denies residency to his son</a> (<a href="https://plus.google.com/100003628603413742554/posts/36DYJmGNq6j">G+</a>)</li><br /><li><a href="http://www.theguardian.com/technology/2016/feb/24/the-fbi-wants-a-backdoor-only-it-can-use-but-wanting-it-doesnt-make-it-possible">Why what the FBI claims to want Apple to do is impossible</a> (<a href="https://plus.google.com/100003628603413742554/posts/D8ZGF7iMx4n">G+</a>)</li><br /><li><a href="http://wpcomics.washingtonpost.com/client/wpc/lio/2016/02/26/">Lio blows geometric soap bubbles</a> (<a href="https://plus.google.com/100003628603413742554/posts/6CTFz53KURE">G+</a>)</li><br /><li><a href="http://mathoverflow.net/questions/38856/jokes-in-the-sense-of-littlewood-examples">Mathematical jokes</a>: theorems and proofs that don't look like they should be valid, but are (<a href="https://plus.google.com/100003628603413742554/posts/HLTuc9jjeRu">G+</a>)</li><br /><li><a href="http://matchstickpuzzles.blogspot.com/2015/12/308-reduce-cubes.html">Matchstick puzzles</a> (<a href="https://plus.google.com/100003628603413742554/posts/edV9vBR4b8i">G+</a>)</li></ul>urn:lj:livejournal.com:atom1:11011110:325238Ordinal numbers as tree-depths of infinite graphs2016-02-25T23:24:31Z2016-02-25T23:24:31ZWhen dealing with finite graphs, it's normal to define (rooted) trees in a graph-theoretic way: a tree is a directed graph with a designated root in which each vertex has a unique walk to the root. The ancestor-descendant relation can be derived from this: an ancestor of <i>x</i> is a vertex that can be reached by a walk from <i>x</i>. However, it's also possible to turn this around, and define trees in terms of their ancestor-descendant relations, with the parent-child relation derived from that. In the turned-around definition, a tree is a partial order with a designated root element that is a predecessor of every element, such that the predecessors of each element are totally ordered. The parent of an element would then be the maximum of its predecessors. So graph-theoretic finite trees and order-theoretic finite trees are really the same things as each other, in a different disguise. But that's no longer true when we go from finite to infinite: we can define trees in either of these two ways, but we get different things.<br /><br />An infinite graph-theoretic tree can be defined in the same way as a finite one: a graph with a designated root in which each vertex has a unique walk to the root. But <a href="https://en.wikipedia.org/wiki/Tree_(set_theory)">the usual order-theoretic definition of a tree</a> takes a little more care: it is a partial order with a designated root element that is a predecessor of every element, as before, but where the predecessors of each element are required to be <a href="https://en.wikipedia.org/wiki/Well-order">well-ordered</a>, not just totally ordered. The ancestor-descendant relation in an infinite graph-theoretic tree gives a valid order-theoretic tree, but one in which the predecessors of each element form a finite total order. Allowing the predecessors to be, instead, any well-ordering gives the order-theoretic trees greater generality over the graph-theoretic ones.<br /><br />This distinction comes up in trying to generalize the notion of <a href="https://en.wikipedia.org/wiki/Tree-depth">tree-depth</a> to infinite graphs. For a finite graph <i>G</i>, the tree-depth is the minimum height of a rooted tree <i>T</i> on the same vertex set such that every edge in <i>G</i> connects an ancestor-descendant pair in <i>T</i>. That is, <i>T</i> should be a depth-first search tree for a supergraph of <i>G</i>. One can try using this definition directly for infinite graphs, with a graph-theoretic infinite tree, but then not all infinite graphs would have a defined tree-depth (for instance, what is the depth of an uncountable clique?). Another quirk of this kind of tree is that, in infinite graphs, depth-first search trees are not the same as spanning trees for which all other edges connect an ancestor-descendant pair. For instance, an uncountable star cannot be explored by depth-first search but clearly is itself such a spanning tree.<br /><br />Instead, I think the right notion of tree-depth, for infinite graphs, uses the order-theoretic notion of a tree. For any graph <i>G</i>, the axiom of choice implies the existence of some order-theoretic trees <i>T</i> such that every edge in <i>G</i> connects a pair of related elements of <i>T</i>. One way of finding such a tree is to choose <i>T</i> to be any well-ordering of <i>G</i>, a tree with only one branch. (A branch of a tree is a maximal totally ordered subset of it.) We can define the tree-depth of <i>G</i> to be the minimum height of a tree <i>T</i> for which each edge of <i>G</i> connects a pair of related elements in <i>T</i>. Here, the height of an element of a tree is the order-type of its strict predecessors and the height of the tree itself is the least ordinal greater than the heights of all its elements. That is, according to this definition, the tree-depth of <i>G</i> should be an <a href="https://en.wikipedia.org/wiki/Ordinal_number">ordinal number</a>. The fact that the ordinals are themselves well-ordered allows the “minimum height” part of the definition of tree-depth above to be well-defined: any set of ordinals, such as the set of heights of valid trees for a given graph <i>G</i>, has a unique smallest element.<br /><br />Every graph with countably many vertices either has bounded tree-depth (a finite number, equivalent to the definition with graph-theoretic trees) or tree-depth <i>ω</i> (the first infinite ordinal number), because we can just use as our tree the set of non-negative integers with their usual ordering (again, a tree with a single branch). However, there are graphs whose tree-depth is a bigger countable ordinal than this. For example, let <i>G</i> = (<i>U</i>,<i>V</i>,<i>E</i>) be a complete bipartite graph where one side of the bipartition <i>U</i> is countable and the other side <i>V</i> is uncountable. We can form a tree <i>T</i> on the vertices of <i>G</i> by choosing any one-to-one correspondence of <i>U</i> with the non-negative integers, and by making each vertex in <i>V</i> have all the vertices in <i>U</i> as its predecessors. This tree has height <i>ω</i> + 1, so <i>G</i> has tree-depth at most <i>ω</i> + 1. But there is no shorter tree for <i>G</i>, because in a shorter tree every vertex would have only finitely many predecessors. If each vertex in <i>U</i> has finitely many predecessors, then there are infinitely many vertices in <i>V</i> that are not predecessors of any vertex in <i>U</i>. Each of these non-predecessor vertices must itself have infinitely many predecessors in a valid tree for <i>G</i>, causing the tree to have height at least <i>ω</i> + 1. So the tree-depth of <i>G</i> is exactly <i>ω</i> + 1.<br /><br />More generally, a transfinite induction shows that every ordinal <i>α</i> is the tree-depth of at least one graph <i>G<sub>α</sub></i>. For, if <i>α</i> is a limit ordinal, we can construct <i>G<sub>α</sub></i> as a limit of the graphs for all smaller ordinals (for instance, by choosing a minimum-height tree for each of these smaller graphs and identifying these trees at their roots). And if <i>α</i> is not a limit ordinal, then it equals <i>β</i> + 1 for some other ordinal <i>β</i>. Choose a cardinal <i>κ</i> bigger than the cardinality of <i>α</i>, a graph <i>G<sub>β</sub></i> with tree-depth <i>β</i>, and a tree <i>T<sub>β</sub></i> realizing the tree-depth of <i>G<sub>β</sub></i>. Then we can construct <i>G<sub>α</sub></i> by adding <i>κ</i> new vertices for each branch of <i>T<sub>β</sub></i>, where each newly added vertex is adjacent to everything in its branch. This graph <i>G<sub>α</sub></i> has an associated tree <i>T<sub>α</sub></i> of height <i>α</i>, in which the predecessor set of each newly added vertex is the branch of <i>T<sub>β</sub></i> that it was added to. An argument like the one for the infinite complete bipartite graph shows that in any tree for <i>G<sub>α</sub></i>, at least one of the new vertices has to be completely above its branch, forcing the tree to have height at least <i>α</i>.<br /><br />So graph-theoretic trees and finite natural numbers are not good enough to define a notion of tree-depth for all infinite graphs, but order-theoretic trees and ordinal numbers suffice. And if you define tree-depth in this way, every ordinal number occurs as the tree-depth of at least one graph.<a name='cutid1-end'></a>urn:lj:livejournal.com:atom1:11011110:325076Black Star Canyon2016-02-17T06:12:42Z2016-02-17T06:12:42ZI spent the president's day holiday hiking in <a href="https://en.wikipedia.org/wiki/Black_Star_Canyon">Black Star Canyon</a>, a wilderness area nearby enough that it's barely outside Irvine and remote enough that in the upper part of the falls trail (where you're basically scrambling up a creek instead of hiking along a trail) there's no sign of civilization to be seen except for the other hikers, airplanes overhead, and occasional painted rocks.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/bsc/11-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><b>( <a href="http://www.ics.uci.edu/~eppstein/pix/bsc/index.html">More photos of Black Star Canyon</a> )</b>urn:lj:livejournal.com:atom1:11011110:324629Linkage2016-02-16T05:36:56Z2016-02-29T01:58:41Z<ul><li><a href="https://www.youtube.com/watch?v=wVH4MS6v23U">Numberphile video on triangle centers</a> (<a href="https://plus.google.com/100003628603413742554/posts/CMYKfvkS1Cu">G+</a>)</li><br /><li><a href="http://www.state.gov/r/pa/prs/ps/2016/01/251577.htm">New US visa requirements for travelers from visa-waiver countries who have visited Iran</a>, may especially effect computational geometers who have been to the Winter School on Computational Geometry (<a href="https://plus.google.com/100003628603413742554/posts/Vwk5VDzYSc6">G+</a>)</li><br /><li><a href="http://vixra.org/abs/1208.0223">Lecture notes from de Bruijn's course on combinatorics</a> have an unlikely host (<a href="https://plus.google.com/u/0/100003628603413742554/posts/fWBTbMvzVwF">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1502.04135">Undecidability of an energy gap in arrays of quantum devices</a> (<a href="https://plus.google.com/100003628603413742554/posts/XgBFEXtru2T">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=IfrJXrzoCwU">Analog rather than digital production of chaotic/fractal images, using video feedback</a> (<a href="https://plus.google.com/100003628603413742554/posts/JEDtopKHJ2S">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Book:Perfect_Graphs">New Wikipedia book on perfect graphs</a> (<a href="https://plus.google.com/100003628603413742554/posts/2uAxv3XytpU">G+</a>)</li><br /><li><a href="https://rjlipton.wordpress.com/2016/02/08/new-nae-members/">Congratulations to the computer scientists newly inducted into the NAE</a> (<a href="https://plus.google.com/100003628603413742554/posts/bqg2GnhwW4m">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=wrsje5It_UU">3d-printing a digital sundial gnomon</a> (<a href="https://plus.google.com/100003628603413742554/posts/Jc784ApdKUD">G+</a>)</li><br /><li><a href="http://fouriestseries.tumblr.com/post/96488907348/cops-and-robbers-and-zombies-and-humans">Zombies vs humans</a>, visualization of a continuous pursuit-evasion model (<a href="https://plus.google.com/100003628603413742554/posts/c7PUrYjNJED">G+</a>)</li><br /><li><a href="http://mathoverflow.net/questions/221676/covering-of-a-surface-of-a-cube-n-times-n-times-n-by-pieces-of-paper-1-times">Rectangle packing on the surface of an integer cube</a> (<a href="https://plus.google.com/100003628603413742554/posts/ckntyJit4YV">G+</a>)</li><br /><li><a href="http://acm-stoc.org/stoc2016/Accepted%20Papers%20-%20STOC%202016.htm">List of accepted papers to STOC 2016</a> (<a href="https://plus.google.com/100003628603413742554/posts/S8Qmn7rRVpL">G+</a>)</li><br /><li><a href="http://www.nola.com/politics/index.ssf/2016/02/board_of_regents_all_universit.html">The Kansas and Louisiana public university systems race to the bottom</a>. But maybe the threat to the football team can act as a parachute? (<a href="https://plus.google.com/100003628603413742554/posts/4KyxAEsm6Z7">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1601.02442">Reversing the curve-shortening flow can send nice curves into very messy singularities</a> (<a href="https://plus.google.com/100003628603413742554/posts/Koq2WTs46bJ">G+</a>)</li><br /><li><a href="http://community.wolfram.com/groups/-/m/t/790393">Ed Pegg's new tetrahedrally-symmetric surface of constant width</a> (<a href="https://plus.google.com/100003628603413742554/posts/E7rWwD7PRUy">G+</a>)</li></ul>