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0xDE - LiveJournal.comWed, 17 Dec 2014 08:17:49 GMTLiveJournal / LiveJournal.com110111107784841personalhttp://l-userpic.livejournal.com/32934265/77848410xDE
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100100http://11011110.livejournal.com/301965.htmlWed, 17 Dec 2014 08:17:49 GMTLinked polytopes and toric grid tessellations
http://11011110.livejournal.com/301965.html
In my recent posting on <a href="http://11011110.livejournal.com/301197.html">four-dimensional polytopes containing linked or knotted cycles of edges</a>, I showed pictures of linked cycles in three examples, the (3,3)-duopyramid, hypercube, and (in the comments) truncated 5-cell. All three of these have some much more special properties: the two linked cycles are induced cycles (there are no edges between two non-consecutive vertices in the same cycle), they include all the vertices in the graph, and their intersection with any two- or three-dimensional face of the polytope forms a connected path.<br /><br />When this happens, we can use it to construct a nice two-dimensional grid representation of the polytope. The set of pairs (<i>x</i>,<i>y</i>) where <i>x</i> is a position on one of the cycles (at a vertex or along an edge) and <i>y</i> is a position on the other cycle form a two-dimensional space, topologically a torus. We can think of this as a grid with wrap-around boundary conditions, where the grid lines correspond to vertex positions on one or the other cycle. The number of grid lines in each dimension is just the length of the cycle. Then, each non-cycle edge of the polytope connects one point from each cycle, so it can be represented as a grid point on this torus. Each two-dimensional face of the polytope has two non-cycle edges, and can be represented as a line segment connecting the corresponding two grid points (perhaps wrapping around from one side of the grid to the other). And when we draw these grid points and line segments, they divide the grid into cells (again, perhaps wrapping around) that turn out to correspond to the 3-dimensional faces of the polytope. So all the features of the polytope that are not part of the two cycles instead show up somewhere on this grid.<br /><br />For instance below, in this two-dimensional grid representation, are the duopyramid (two 3-cycles, so a 3 × 3 grid), hypercube (8 × 8 grid), and truncated 5-cell (10 × 10 grid) again. I've drawn these with the wraparound points halfway along an edge of each cycle in order to avoid placing a grid line on the boundary of the drawing. In the hypercube and truncated 5-cell, the axes are labeled by numberings of the vertices. For the hypercube, the vertices can be numbered by the 16 hexadecimal digits, where two digits are adjacent if they differ in a single bit in their binary representations. The two eight-vertex cycles can be obtained by cycling through the order of which bit changes. For the truncated 5-cell, the vertices can be numbered by ordered pairs of unequal digits from 1 to 5, where the neighbors of each vertex are obtained by changing the second digit or swapping the two digits.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/33dp-q4-t5.png"></div><br /><br />Another way of thinking about this is that, on the three-dimensional surface of a 4d unit sphere, we can draw two linked unit circles, one in the <i>xy</i> plane and the other in the <i>wz</i> plane. The medial axis of these circles (the points on the sphere equally distant from both of them) is a torus, and what we're drawing in this diagram is how a polyhedral version of the same torus slices through the faces of the polytope.<br /><br />You can read off the structure of each 3-dimensional cell in the polytope from the corresponding polygon in the diagram. Recall that these cells are themselves three-dimensional polyhedra whose vertices have been divided into two induced paths. So (just as in the four-dimensional case) we can make a grid from the product of these two paths, represent non-path edges as grid points, and represent two-dimensional faces as line segments connecting grid points. Each two-dimensional face has two non-path sides, and a number of path sides given by the difference in coordinates between the corresponding two grid points. So, the total number of sides of the face is just two plus the Manhattan length of the line segment representing the face. For instance, the unit line segments in the duopyramid diagram represent triangles, and the squares formed by four of these segments represent tetrahedra (four triangles). The segments of Manhattan length two in the hypercube diagram represent quadrilaterals, and the hexagons formed by six of these segments represent cubes (six quadrilaterals). In order to represent a polyhedron in this way, a grid polygon has to have vertical edges at its left and right extremes, and horizontal edges at its top and bottom extremes, because otherwise a vertex at the end of one of the two paths would have only two incident edges, impossible in a polyhedron. For the same reason each intermediate grid line must have a grid point (representing a non-path edge of the polyhedron) on it. We can make a dictionary of small polyhedra that can be decomposed into two induced paths, and their associated grid polygons:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/2pathpoly.png"></div><br /><br />Notice that the grid polygons don't have to be strictly convex: the octahedron has eight grid points, four of which are at the corners of a 2 × 2 square but the other four of which are in the middle of the edges of this square. But in order for a collection of polyhedra to meet up to form the faces of a four-dimensional polytope, each grid point needs at least three line segments connecting to it (each polytope edge has to be surrounded by three or more two-dimensional faces). This can only happen if each grid polygon has at most one slanted side in each of the four corners of its bounding box. So these polygons are convex except for possibly having vertices on their horizontal and vertical sides. There are also some other constraints on their shape; for instance, a hexagon with two diagonal sides within a 2 × 2 square doesn't correspond to a polyhedron, because it forms a shape that is not 3-vertex-connected.<br /><br />Given this dictionary, we can form new patterns by tessellating a rectangular wrap-around grid by these grid polygons, and then ask: does the tessellation represent a 4-dimensional polytope? We have to be a little careful here, because we cannot place horizontal or vertical sides that are subdivided (e.g. in the octahedron) next to similar-looking sides that are not subdivided (e.g. in the cube). There are infinitely many possibilities, some of which give known polytopes, and some of which are unknown to me. For instance, extending the grid of squares shown for the (3,3)-duopyramid to grid of squares in a larger rectangle produces the diagram for another kind of duopyramid.<br /><br />In the drawing below, the left grid tessellation represents a linked-cycle decomposition of the <a href="https://en.wikipedia.org/wiki/Rectified_5-cell">hypersimplex</a> with five tetrahedra and five octahedra (one of the polytopes Gil Kalai asked about in the comments on my previous post). It can be formed from the truncated 5-cell by contracting all of the edges that are not part of tetrahedra; because the cycle edges of the linked cycles of the truncated 5-cell alternate between tetrahedral and non-tetrahedral edges, this contraction preserves the cycle decomposition. The right grid tessellation represents the <a href="https://en.wikipedia.org/wiki/Octahedral_prism">octahedral prism</a>, with eight triangular-prism cells and two octahedral cells. Therefore, both of these polytopes are linked. In both cases I found the infinite tessellation first, found its smallest period of horizontal and vertical translation, and then was able to identify the corresponding polytope with the help of the low number of cells and high symmetry of these tessellations. But I'm confused by the brick wall in the middle. It has the right number of vertices (10) and the right shape of 3-cells (triangular dipyramids) to be the dual hypersimplex, but the number of bricks is wrong: it should be 10 and is instead 12. (A herringbone pattern of bricks will also tessellate nicely, but then the number of vertices in both cycles would be a multiple of four.) It would be nice to have a theorem characterizing which tessellations give polytopes more generally.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/r5c-op-dhc.png"></div><br /><br />One thing is clear: the polytopes that have linked-cycle decompositions are a very special subclass of the 4-polytopes. For instance, for general 4-polytopes, it remains unknown whether they can have fat <a href="https://en.wikipedia.org/wiki/Face_lattice">face lattices</a>. That is, can a polytope with a small number of vertices and 3-cells have a large number of edges and 2-faces? This can't happen in 3d, by a calculation involving Euler's formula, but the same calculation in 4d doesn't rule out this possibility. But in linked-cycle-decomposable polytopes, the number of cycle edges equals the number of vertices. And because the 3-cells are faces of a torus graph, the number of non-cycle edges (vertices of the torus graph) and 2-faces (edges of the torus graph) are bounded by linear functions of the number of 3-cells. In particular, if there are <i>v</i> vertices and <i>c</i> 3-cells, then there can be at most <i>v</i> + 2<i>c</i> edges and at most 3<i>c</i> 2-cells. This bound is tight whenever the torus diagram is simple (has exactly three edges at each vertex), as it is in the hypercube, truncated 5-cell, and hypersimplex cases.<a name='cutid1-end'></a>http://11011110.livejournal.com/301965.htmlgeometrypublic0http://11011110.livejournal.com/301718.htmlWed, 17 Dec 2014 04:46:05 GMTSurvey on k-best enumeration algorithms
http://11011110.livejournal.com/301718.html
When I was asked earlier this year to write <a href="http://dx.doi.org/10.1007/978-3-642-27848-8_733-1">a short survey on <i>k</i>-best enumeration algorithms</a> for the Springer <i>Encyclopedia of Algorithms</i>, I wrote a first draft before checking the formatting requirements. It ended up being approximately five pages of text and seven more pages of references, and I knew I would have to cut some of that. But then I did check the format, and saw that it needed to be much shorter, approximately two pages of text and a dozen references. I don't regret doing it this way; I think having a longer version to cut down helped me to organize the results and figure out which parts were important. But then I thought: why not make the long(er) version available too? I added a few more references, so now it's about six pages of text and ten of references, still closer to an annotated bibliography than an in-depth survey. Here it is: <a href="http://arxiv.org/abs/1412.5075">arXiv:1412.5075</a>.http://11011110.livejournal.com/301718.htmlalgorithmspaperspublic0http://11011110.livejournal.com/301376.htmlTue, 16 Dec 2014 06:25:54 GMTLinkage for mid-December
http://11011110.livejournal.com/301376.html
<ul><li><a href="http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-thing-i-know-about-the-integers/">The number theory behind why you can't have both perfect fifths and perfect octaves on a piano keyboard</a> (with bonus <a href="https://en.wikipedia.org/wiki/Fokker_periodicity_block">lattice quotient music theory</a> link; <a href="https://plus.google.com/100003628603413742554/posts/ZH6ijTsiGDN">G+</a>)</li><br /><li>Sad news of <a href="https://en.wikipedia.org/wiki/Rudolf_Halin">Rudolf Halin</a>'s death (<a href="https://plus.google.com/100003628603413742554/posts/Q53k1pVKEtR">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=R9ogDS-QYT0">Frankenstein vs The Glider Gun</a> video (<a href="https://plus.google.com/100003628603413742554/posts/67p7GUHnumD">G+</a>)</li><br /><li><a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/dec/03/durers-polyhedron-5-theories-that-explain-melencolias-crazy-cube">Günter Ziegler on Dürer's solid</a> (<a href="https://en.wikipedia.org/wiki/Truncated_triangular_trapezohedron">WP</a>; <a href="http://www.metafilter.com/145043/Drers-polyhedron-5-theories-that-explain-Melencolias-crazy-cube">MF</a>; <a href="https://plus.google.com/100003628603413742554/posts/PKLcXds94FK">G+</a>)</li><br /><li><a href="http://www.metafilter.com/144972/Nature-will-make-its-articles-back-to-1869-free-to-share-online">Nature will make its articles back to 1869 free to share online</a>, for certain values of "free" that you might or might not agree with (<a href="https://plus.google.com/100003628603413742554/posts/3Foe25Nxc4o">G+</a>)</li><br /><li><a href="http://polyhedron100.wordpress.com/">Albert Carpenter's polyhedron models</a> (<a href="https://plus.google.com/100003628603413742554/posts/FUkqR1h6WjN">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem">The only complete proof from Fermat</a> and <a href="https://en.wikipedia.org/wiki/Congruum">the gaps in arithmetic progressions of squares</a> (<a href="https://plus.google.com/100003628603413742554/posts/6GmBrCwx1tH">G+</a>)</li><br /><li><a href="http://blog.plover.com/2014/12/01/">Mark-Jason Dominus on how and why he negotiated with his book publishers to be able to keep a free online copy of his Perl book</a> (<a href="https://plus.google.com/100003628603413742554/posts/LsWFCCixZLV">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=rtR63-ecUNo">Video on drawing mushrooms with sound waves</a> (<a href="https://plus.google.com/100003628603413742554/posts/eBSPF2o3PDU">G+</a>)</li><br /><li><a href="http://mashable.com/2014/12/10/senate-wikipedia-torture-report/">Senate staffer tries to scrub "torture" reference from Wikipedia's CIA torture article</a> (<a href="https://plus.google.com/100003628603413742554/posts/PwFE9gJL3DE">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=SL2lYcggGpc">Numberphile video on origami angle trisection</a> (<a href="https://plus.google.com/100003628603413742554/posts/DxdUsDmBjuM">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=ZMByI4s-D-Y">Video on the world's roundest object</a> and why it was made (<a href="https://plus.google.com/100003628603413742554/posts/hyiYUfzEJu3">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1412.2716">How much text re-use is too much?</a> A statistical study of plagiarism on arXiv (<a href="http://www.improbable.com/2014/12/14/lots-and-lots-of-bits-of-copying-in-scientific-literature/">via</a>; <a href="https://plus.google.com/100003628603413742554/posts/PBeiZ56MuVh">G+</a>)</li></ul>http://11011110.livejournal.com/301376.htmlcellular automataopen accesswikipedianumber theorygeometryplagiarismorigamipublic2http://11011110.livejournal.com/301197.htmlSat, 13 Dec 2014 23:53:40 GMTLinks and knots in the graphs of four-dimensional polytopes
http://11011110.livejournal.com/301197.html
<p>The surface of a three-dimensional polyhedron is a two-dimensional space that's topologically equivalent to the sphere. By the Jordan curve theorem, every cycle of edges and vertices in this space cuts the surface into two topological disks. But the surface of a four-dimensional polytope is a three-dimensional space that's topologically equivalent to the hypersphere, or to three-dimensional Euclidean space completed by adding one point at infinity. So, just as in conventional Euclidean space, polygonal chains (such as the cycles of edges and vertices of the polytope) can be nontrivially knotted or linked. If so, this can also be seen in three-dimensions, as a knot or link in the <a href="https://en.wikipedia.org/wiki/Schlegel_diagram">Schlegel diagram</a> of the polytope (a subdivision of a convex polyhedron into smaller convex polyhedra). Does this happen for actual 4-polytopes? Yes! Actually, it's pretty ubiquitous among them.</p>
<p>The linked 4-polytope with the fewest vertices is a <a href="https://en.wikipedia.org/wiki/Duopyramid">duopyramid</a> formed from the convex hull of two equilateral triangles centered at the origin, one in the <i>xy</i>-plane and the other in the <i>zw</i>-plane. These two triangles are not actually two-dimensional faces of the duopyramid; instead, in the Schlegel diagram, they appear as two linked triangles. This polytope has nine tetrahedral facets; in the Schlegel diagram, they appear as one outer tetrahedron, two more adjacent to the top edge of the top linked triangle, two adjacent to the bottom edge of the bottom linked triangle, and four wrapping around the middle vertical edge connecting the two links.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/linked-duopyramid.png"></p>
<p>Similarly, a seven-vertex <a href="https://en.wikipedia.org/wiki/Neighborly_polytope">neighborly polytope</a> forms a complete graph on seven vertices. <a href="https://en.wikipedia.org/wiki/Linkless_embedding">As with every embedding</a> of the seven-vertex complete graph into space, it contains a knot.</p>
<p>What about <a href="https://en.wikipedia.org/wiki/Simple_polytope">simple 4-polytopes</a>? This means that every vertex has exactly four neighbors. The duopyramid doesn't have this property: its vertices all have five neighbors. The simple 4-polytope with the fewest vertices and facets in which I've found a link is a hypercube, with eight cubical facets.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/linked-hypercube.png"></p>
<p>It is also possible to form a trefoil knot in the <a href="https://en.wikipedia.org/wiki/4-6_duoprism">(4,6)-duoprism</a>, the Cartesian product of a square and a hexagon, and to form a link in the (3,6)-duoprism. These are simple polytopes with ten and nine facets respectively.</p>
<p>There are at least two interesting classes of 4-polytopes that don't have nontrivial knots or links, however. One of these is the class of <a href="https://en.wikipedia.org/wiki/Polyhedral_pyramid">polyhedral pyramids</a>: 4-dimensional pyramids with a 3-dimensional polyhedron base. Their graphs are <a href="https://en.wikipedia.org/wiki/Apex_graph">apex graphs</a>, embedded nicely with no knots; they have Schlegel diagrams in which the base forms the outside face and the apex of the pyramid is the only vertex inside it, connected to all the other vertices. So any system of closed curves must stay on the planar surface of the base with the exception of one pair of edges through the apex; that's not enough to make a knot or link.</p>
<p>The other is the class of stacked polytopes, formed by gluing simplices face-to-face. Their Schlegel diagrams are formed by repeatedly subdividing a tetrahedron into four smaller tetrahedra meeting at an interior point of the larger tetrahedron, and their graphs are the <a href="https://en.wikipedia.org/wiki/K-tree">4-trees</a>. For any collection of vertex-edge cycles in such a polytope, it's possible to undo one of the subdivision steps and either simplify the collection without changing its topological type, by shortcutting the subdivision vertex, or remove a cycle that forms a face of the polytope. So by induction there can be no knots.</p><a name='cutid1-end'></a>http://11011110.livejournal.com/301197.htmlknot theorygeometrypublic3http://11011110.livejournal.com/300993.htmlFri, 05 Dec 2014 07:27:36 GMTA strike against ERGMs
http://11011110.livejournal.com/300993.html
The <a href="https://en.wikipedia.org/wiki/Exponential_random_graph_models">exponential random graph model</a> is a system for describing probability distributions on graphs, used to model social networks. One fixes a set of vertices, and determines a collection of "features" among the edges of this fixed set (such as whether or not a particular edge or combination of a small number of edges), each with an associated real-valued weights. Then to determine the probability of seeing a particular graph, one simply looks at which features it has; the probability is exp(sum of feature weights), divided by a normalizing constant (the "partition function").<br /><br />This is a good model for several reasons: it is powerful enough that (with complicated enough features) it can describe any distribution. With simple enough features (e.g. just individual edges) it degenerates to the standard <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model">Erdős–Rényi–Gilbert model</a> of random graphs. It's easy to express features that model sociological theories of network formation, such as <a href="https://en.wikipedia.org/wiki/Assortative_mixing">assortative mixing</a> (more similar people are more likely to be friends) or <a href="https://en.wikipedia.org/wiki/Triadic_closure">triadic closure</a> (friends-of-friends are more likely to be friends). And by fitting the weights to actual social networks, one can learn something about the strengths of these effects.<br /><br />But on the other hand, there are some theoretical and practical obstacles to its use. It seems to be difficult to set up features and weights such that, when one generates graphs using the distribution they describe, the results actually look like social networks. If we go even a little bit beyond the Erdős–Rényi–Gilbert model we don't have closed form solutions to anything and have to use MCMC simulations to compute the partition function, fit weights, or generate graphs, and we don't know much about how quickly or slowly these simulations converge.<br /><br />And now, with my latest preprint "<a href="http://arxiv.org/abs/1412.1787">ERGMs are Hard</a>" (arXiv:1412.1787, with Michael Bannister and Will Devanny), the picture gets darker. We prove complexity-theoretic hardness results showing that with completely realistic features (in fact the ones for assortative mixing and triadic closure, but with unrealistic weights) we can't compute the partition function, we can't get anywhere close to approximating the partition function, we can't generate graphs with the right probabilities, and we can't even get anywhere close to the right probability distribution. And trying to escape the hardness by tweaking the features to something a little more complicated doesn't help: the same hardness results continue to be true when the features include induced subgraphs of any fixed type with more than one edge.<br /><br />The short explanation for why is that lurking inside these models are computationally hard combinatorial problems such as (the one we mainly use) finding or counting the largest induced triangle-free subgraphs. It was known that the maximization version of this problem was hard, but the reduction wasn't parsimonious (less technically, this means that the reduction can't be used to prove that the counting version of the problem is hard). So for that part we had to find our own reduction from another hard counting problem, counting perfect matchings in cubic bipartite graphs. Here it is in a picture.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/match2maxtf.png"></div><br /><br />Each vertex of the original graph turns into four triangles after the reduction. In this example, counting matchings in a cube turns into counting maximum triangle-free subgraphs of a <a href="https://en.wikipedia.org/wiki/Snub_cube">snub cube</a>. These subgraphs are formed by deleting all the snub cube edges that correspond to unmatched cube edges, and then deleting one more edge inside each four-triangle gadget. When I posted a drawing of <a href="http://11011110.livejournal.com/275610.html">a stereographic projection of a snub cube</a> a bit over a year ago, this is what it was for. Since that time, we've also been using the image of this reduction in the logo of <a href="http://www.ics.uci.edu/~theory/">our local research center</a>.<a name='cutid1-end'></a>http://11011110.livejournal.com/300993.htmlcomplexity theorysocial networkspaperspublic2http://11011110.livejournal.com/300600.htmlMon, 01 Dec 2014 02:20:01 GMTLinkage for the end of November
http://11011110.livejournal.com/300600.html
<ul><li><a href="https://www.insidehighered.com/news/2014/11/11/gamergate-supporters-attack-digital-games-research-association">Gamergate's attackers move on from (female) indie game developers to (female) game researchers</a> (<a href="https://plus.google.com/100003628603413742554/posts/K7V5MBJ88UJ">G+</a>)</li><br /><li><a href="http://retractionwatch.com/2014/11/17/fake-citations-plague-some-google-scholar-profiles/">Scammy publisher uses your name as the author of fake papers</a> (<a href="https://plus.google.com/100003628603413742554/posts/N4q6NrHusHq">G+</a>)</li><br /><li><a href="https://plus.google.com/101584889282878921052/posts/HTVRuPCTJXm">Escher-like impossible figures</a> by Regalo Bizzi based on a triangular grid (<a href="https://plus.google.com/100003628603413742554/posts/TpqDDw8oN14">G+</a>)</li><br /><li><a href="https://medium.com/the-open-company/trip-report-vegas-lights-cba073735683">James Turrell installation in Las Vegas</a> (<a href="https://plus.google.com/100003628603413742554/posts/VEtFfrQo4vR">G+</a>)</li><br /><li><a href="http://beesgo.biz/godot.html">Waiting for Godot: The Game</a> (by Zoe Quinn; <a href="https://plus.google.com/100003628603413742554/posts/bmkrFAKsUAA">G+</a>)</li><br /><li><a href="http://www.ams.org/samplings/feature-column/fc-2013-11">Fedorov's Five Parallelohedra</a>, a complete classification of the shapes that can tile space by translation (<a href="https://plus.google.com/100003628603413742554/posts/Gs3MRQePNAd">G+</a>)</li><br /><li><a href="http://www.cjr.org/behind_the_news/journalism_has_a_plagiarism_pr.php?page=all">On the high variance in journalistic standards for plagiarism</a> (<a href="https://plus.google.com/100003628603413742554/posts/d1NvmNbVMDY">G+</a>)</li><br /><li><a href="http://mathoverflow.net/a/187908/440">How many median graphs are there?</a> (<a href="https://plus.google.com/100003628603413742554/posts/TuceX7PT3hL">G+</a>)</li><br /><li><a href="http://www.siam.org/meetings/da15/">SODA/ALENEX/ANALCO 2015</a> preregistration closes Monday, Dec. 1 (<a href="https://plus.google.com/100003628603413742554/posts/DWK9RC62Lz1">G+</a>)</li><br /><li><div align="center"><lj-embed id="54" /></div><br />(<a href="https://plus.google.com/100003628603413742554/posts/6cRKBR8PYuU">G+</a>)</li><br /><li><a href="http://erkdemon.blogspot.com/2012/02/hexagonal-diamond-other-form-of-diamond.html">Hexagonal diamond</a>, a crystalline carbon structure even harder than true diamond (<a href="https://plus.google.com/100003628603413742554/posts/briWWcfq6za">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Laves_graph">The Laves graph</a>, an infinite symmetric 3-regular graph that forms yet another possible carbon crystal (<a href="https://plus.google.com/u/0/100003628603413742554/posts/gBzaBivAAbg">G+</a>)</li></ul>http://11011110.livejournal.com/300600.htmlcoloracademiaconferencesgeometrygraph theorypublic0http://11011110.livejournal.com/300302.htmlThu, 27 Nov 2014 07:25:54 GMTTrees that represent bandwidth
http://11011110.livejournal.com/300302.html
In my algorithms class today, I covered minimum spanning trees, one property of which is that they (or rather maximum spanning trees) can be used to find the bottleneck in communications bandwidth between any two vertices in a network. Suppose the network edges are labeled by bandwidth, and we compute the maximum spanning tree using these labels. Then between any two vertices the path in this tree has the maximum bandwidth possible, among all paths in the network that connect the same two vertices. (There may also be other equally good paths that aren't part of the tree.) So if you want to send all of your data on a single route in the network, and you're not worried about other people using the same links at the same time, and bandwidth is your main quality issue (a lot of ifs) then that's the best path to take. The bandwidth of the path is controlled by its weakest link, the edge on the path with the smallest bandwidth. If you want to quickly look up the bandwidth between pairs of vertices, you can do it in constant time using the nearest common ancestor in a <a href="https://en.wikipedia.org/wiki/Cartesian_tree">Cartesian tree</a> derived from the maximum spanning tree.<br /><br />Ok, but if you're so concerned about bandwidth then maybe you should use a more clever routing scheme that spreads your messages across multiple paths to get even more bandwidth. This can be modeled as a network flow, and the bottleneck to getting the most bandwidth is no longer a single edge. Instead, the <a href="https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem">max-flow min-cut theorem</a> tells you that the bottleneck takes the form of a cut: a partition of the graphs into two disjoint subsets, whose bandwidth is the sum of the bandwidths of the edges crossing the cut.<br /><br />Despite all this added complexity, it turns out that the bandwidth in this sort of multi-path routing scheme can still be described by a single tree. That is, there's a tree whose vertices are the vertices of your graph (but whose edges and edge weights are no longer those of the graph) such that the bandwidth you can get from one vertex to another by routing data along multiple paths in the graph is the same as the bandwidth of the single path between the same two vertices in the tree. More, the edges of the tree can be labeled by cuts in the graph such that the weakest link in the tree path between any two vertices is labeled by the minimum cut that separates those vertices in the original graph. This tree is called the <a href="https://en.wikipedia.org/wiki/Gomory%E2%80%93Hu_tree">Gomory–Hu tree</a> of the given (undirected and edge-weighted) graph. Using the same Cartesian tree ancestor technique, you can look up the bandwidth between any pair of vertices, in constant time per query.<br /><br />My latest arXiv preprint, <a href="http://arxiv.org/abs/1411.7055">All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs</a> (arXiv:1411.7055, with Cora Borradaile and new co-authors Amir Nayyeri and Christian Wulff-Nilsen) is on exactly these trees. For arbitrary graphs they can be found in polynomial time, but slowly, because the computation involves multiple flow computations. For planar graphs it was known how to find the Gomory–Hu tree much more quickly, in O(<i>n</i> log<sup>3</sup> <i>n</i>) time; we shave a log off this time bound. Then, we extend this result to a larger class of graphs, the graphs of bounded genus, by showing how to slice the bounded-genus surface up (in a number of ways that's exponential in the genus) into planar pieces such that, for every pair of vertices, one of the pieces contains the minimum cut. That gives us exponentially many Gomory-Hu trees, one for each piece, but it turns out that these can all be combined into a single tree for the whole graph.<br /><br />One curious difference between the planar and higher-genus graphs is that, for planar graphs, the set of cuts given by the Gomory–Hu tree also solves a different problem: it's the minimum-weight <a href="https://en.wikipedia.org/wiki/Cycle_basis">cycle basis</a> of the dual graph. In higher-genus graphs, we don't quite have enough cuts to generate the dual cycle space (we're off by the genus) but more importantly some of the optimal cycle basis members might not be cuts. So although the new preprint also improves the time for finding cycle bases in planar graphs, making a similar improvement in the higher genus case remains open.<a name='cutid1-end'></a>http://11011110.livejournal.com/300302.htmlgraph algorithmspaperspublic0http://11011110.livejournal.com/300115.htmlTue, 25 Nov 2014 08:12:35 GMTLIPIcs formatting tricks
http://11011110.livejournal.com/300115.html
If, like me, you're working on a SoCG submission, and this is the first time you've tried using the LIPIcs format that SoCG is now using, you may run into some minor formatting issues (no worse than the issues with the LNCS or ACM formats, but new and different). Here are the ones I've encountered, with workarounds where I have them:<br /><ul><li>The LIPIcs format automatically includes several standard LaTeX packages including <tt>babel</tt>, <tt>amsmath</tt>, <tt>amsthm</tt>, <tt>amssymb</tt>, and <tt>hyperref</tt>. So there's no point in including them yourself and (if you specify incompatible options) it may cause an error to include them yourself. I haven't needed to change the <tt>hyperref</tt> options, but if you do, see <a href="http://tex.stackexchange.com/questions/75542/incompatibility-between-lipics-and-hyperref">here</a>.</li><br /><li>You may like to use the <tt>lineno</tt> package so that, when reviewers give you comments on your submission, they can tell you more accurately which line they're referring to. If you try this with the LIPIcs format, you will notice that you don't get line numbers in the last paragraph of a proof, nor in a paragraph that contains a displayed equation (even if you correctly delimit the equation with <tt>\[...\]</tt> instead of using the obsolete <tt>$$...$$</tt>, which can also cause problems with <tt>lineno</tt>). The solution to the proof problem is to change the proof termination symbol (a triangle instead of the usual Halmos box) to use an explicit <tt>$...$</tt> instead of <tt>\ensuremath</tt>):<br /><br /><tt>\renewcommand\qedsymbol{\textcolor{darkgray}{$\blacktriangleleft$}}</tt><br /><br />The solution to the displayed equation problem is more complicated, but is given in <a href="http://phaseportrait.blogspot.com/2007/08/lineno-and-amsmath-compatibility.html">this blog post from 2007</a> (the update near the start of that post). Why this incompatibility hasn't been fixed in the last seven years is a different question.</li><br /><li>If you use the <tt>numberwithinsect</tt> document class option (pronounced "number with insect"), then the LIPIcs format numbers theorems, lemmas, etc by what section they are in: Lemma 2.1 for the first lemma in Section 2, etc. But if you also run past the page limit and use appendices, you may notice that the lemmas are being given numbers that re-use previously used numbers, because although the appendices have letters rather than numbers the lemmas use numbers: the first lemma in Appendix B is also called Lemma 2.1. The problem is that the LIPIcs style expands the <tt>\thesection</tt> macro (the one that gives the number or letter of the current section) at the time it defines <tt>\thetheorem</tt> (the one that gives the name of the current theorem or lemma). So when you use <tt>\appendix</tt> (or <tt>\begin{appendix}</tt> if you like to pretend that non-environment commands are really environments), <tt>\thesection</tt> gets changed but it's too late to make a difference to <tt>\thetheorem</tt>. The fix is to add the following lines after <tt>\appendix</tt>:<br /><br /><tt>\makeatletter<br />\edef\thetheorem{\expandafter\noexpand\thesection\@thmcountersep\@thmcounter{theorem}}<br />\makeatother</tt></li><br /><li>I've <a href="https://plus.google.com/100003628603413742554/posts/Fy34Vv4Xk6y">already written</a> about the problems with using the <tt>\autoref</tt> command of the <tt>hyperref</tt> package: because LIPIcs wants to use a shared numeric sequence for theorems, lemmas, etc., <tt>\autoref</tt> thinks they are all theorems. <a href="http://tex.stackexchange.com/questions/213821/using-cleveref-with-lipics-documentclass-fails-for-theorem-environments-sharing">Someone else also recently asked about this problem</a>. This is a more general incompatibilty between <tt>amsthm</tt> and <tt>hyperref</tt>, but LIPIcs also includes some of its own code for theorem formatting, which seems to be causing the fixes one can find for <tt>amsthm</tt> not to work. The solution is to fall back to the non-<tt>hyperref</tt> way of doing things: <tt>Lemma~\ref{lem:my-lemma-name}</tt> etc.</li><br /><li>Speaking of theorems: if you use <tt>\newtheorem</tt>, you probably want to have a previous line <tt>\theoremstyle{definition}</tt> so that whatever you're defining looks like the other theorems and lemmas.</li><br /><li>On the first page of a LIPIcs paper, the last line of text may be uncomfortably close to the Creative Commons licencing icon. I haven't found a direct workaround for this (although probably it's possible) but you can obtain better spacing by having a footnote (for instance one listing your grant acknowledgements) or a bottom figure on that page.</li><br /><li>If you're used to trying to fit things into a page limit with LNCS format, you may have learned to use <tt>\paragraph</tt> as a trick to save space over subsections. That doesn't work very well in LIPIcs, for two reasons. First, because it will give you an ugly paragraph number like 6.0.0.1 (the first numbered paragraph in an un-numbered subsubsection of an un-numbered subsection of section 6). You can work around this by using <tt>\paragraph*</tt>. But second, because unlike in LNCS the paragraph heading won't be folded into the paragraph that follows it, so you save a lot less space. I don't want to try to work around this one. And fortunately I haven't yet seen my coauthors adding code like <tt>\noindent{\bfseries Paragraph heading.} Paragraph text...</tt> (or worse <tt>\bf ...</tt>). Solution: find different tricks for your space-saving efforts. Like maybe write more tersely.</li></ul><br />Anyone else have any timely tips?<a name='cutid1-end'></a>http://11011110.livejournal.com/300115.htmltoolspublic5http://11011110.livejournal.com/299980.htmlTue, 25 Nov 2014 06:00:27 GMTThin folding
http://11011110.livejournal.com/299980.html
I have another new preprint on arXiv this evening: <a href="http://arxiv.org/abs/1411.6371">Folding a Paper Strip to Minimize Thickness, arXiv:1411.6371</a>, with six other authors (Demaine, Hesterberg, Ito, Lubiw, Uehara, and Uno); it's been accepted at <a href="http://www.buet.ac.bd/cse/walcom2015/">WALCOM</a>.<br /><br />The basic goal of this is to try to understand how to measure the thickness of a piece of paper that has been folded into a shape that lies flat in the plane. For instance, in designing origami pieces, it's undesirable to have too much thickness, both because it wastes paper causing your piece to be smaller than it needs to and because it may be an obstacle to forming features of your piece that are supposed to be thin.<br /><br />The obvious thing to do is to just count the number of layers of paper that cover any point of the plane, but can be problematic. For instance, if you have two offset accordion folds (drawn sideways below)<br /><pre>/\/\/\/\/\
\/\/\/\/\/</pre>then it's not really accurate to say that the thickness is the same as if you just had one of the two sets of folds: one of the two folds is raised up by the thickness of the other one so the whole folded piece of paper is more like the sum of the thicknesses of its two halves.<br /><br />In the preprint, we model the thickness by assuming that the flat parts of the paper are completely horizontal, at integer heights, that two overlapping parts of paper have to be at different heights, and that a fold can connect parts of paper that are at any two different heights. But then, it turns out that finding an assignment of heights to the parts of paper that minimizes the maximum height is hard, even for one-dimensional problems where we are given a crease pattern of mountain and valley folds as input, without being told exactly how to arrange those folds. The reason is that there can be ambiguities about how the folded shape can fit into pockets formed by other parts of the fold, and choosing the right pockets is difficult.http://11011110.livejournal.com/299980.htmlorigamipaperspublic0http://11011110.livejournal.com/299547.htmlSun, 16 Nov 2014 06:03:27 GMTLinkage
http://11011110.livejournal.com/299547.html
<ul><li><a href="http://blog.peerj.com/post/100580518238/whos-afraid-of-open-peer-review">An experiment in allowing journal reviewers to reveal their names</a> (the <a href="https://plus.google.com/100003628603413742554/posts/2dffs1kDkz4">G+</a> post has several additional links on academics including some well known graph theorists taking money to deliberately distort university rankings)</li><br /><li><a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/30/pumpkin-geometry-stunning-shadow-sculptures-that-illuminate-an-ancient-mathematical-technique?CMP=share_btn_fb">Pumpkin geometry</a>: stereographic projection of shadows from carved balls (<a href="https://plus.google.com/100003628603413742554/posts/GiB8FLHscM2">G+</a>; no actual pumpkins involved)</li><br /><li><a href="http://www.clintfulkerson.com/">Clint Fulkerson</a>: an abstract artist whose work feels somehow both geometric and organic (<a href="https://plus.google.com/100003628603413742554/posts/P19K5D3qdC5">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2014/11/spectacular-paper-pop-up-sculptures-designed-by-peter-dahmen/">Paper popups by Peter Dahmen</a> (<a href="https://plus.google.com/100003628603413742554/posts/57MJFLk9TK4">G+</a>)</li><br /><li><a href="http://www.planetjune.com/blog/polyhedral-balls-crochet-pattern/">Crochet Platonic polyhedra by June Gilbank</a> (<a href="https://plus.google.com/100003628603413742554/posts/UVXPvGuBJ3M">G+</a>)</li><br /><li><a href="http://tex.stackexchange.com/questions/187388/amsthm-with-shared-counters-messes-up-autoref-references/187395">Advice for combining autoref with shared counters for theorems and lemmas</a> (and in the <a href="https://plus.google.com/100003628603413742554/posts/Fy34Vv4Xk6y">G+</a> post, a plea for something similar that will work with the LIPIcs LaTeX format)</li><br /><li><a href="http://lifehacker.com/5914894/put-a-duvet-cover-on-with-minimum-effort-by-rolling-it-like-a-burrito">A topological trick with duvet covers</a> (<a href="https://plus.google.com/100003628603413742554/posts/dmqcGTrUWty">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Harborth%27s_conjecture">Harborth's conjecture</a> on graph drawing with integer edge lengths (<a href="https://plus.google.com/100003628603413742554/posts/GaNexRFKu84">G+</a>)</li><br /><li><a href="http://thorehusfeldt.net/2014/11/09/at-first-eppstein-liked-minimum-spanning-trees-4/">A cryptic crossword by Thore Husfeldt</a> featuring parameterized complexity and my name (<a href="https://plus.google.com/100003628603413742554/posts/bAWeuPdp9iA">G+</a>)</li><br /><li><a href="http://www.wired.com/2014/11/verdict-overturned-italian-geoscientists-convicted-manslaughter/">Giving scientific advice that turns out to be incorrect is not a crime</a> (<a href="https://plus.google.com/100003628603413742554/posts/c2TStXaVR5f">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Chv%C3%A1tal-Sankoff_constants">Chvátal-Sankoff constants</a> (the expected length of a random longest common subsequence; <a href="https://plus.google.com/100003628603413742554/posts/Ry6jQ7Jrwhi">G+</a>)</li><br /><li><a href="https://github.com/lizadaly/nanogenmo2014">Automatic Voynich</a> (<a href="https://plus.google.com/100003628603413742554/posts/7Eiv9TNQiTu">G+</a>)</li><br /><li><a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/nov/04/macaus-magic-square-stamps-just-made-philately-even-more-nerdy">Magic squares on stamps</a> (<a href="https://plus.google.com/100003628603413742554/posts/4FRqJvZtCCX">G+</a>)</li><br /><li><a href="http://snappizz.com/holyhedron">Polyhedra in which all faces are holy</a>, and an update on big Life spaceships (<a href="https://plus.google.com/100003628603413742554/posts/PiMnx4ZK2AN">G+</a>)</li><br /><li><a href="http://kevintwomey.com/calculators.html">Photos of mechanical calculators</a> (via <a href="http://www.wired.com/2014/11/kevin-twomey-low-tech/">Wired</a> and <a href="http://boingboing.net/2014/11/13/beautiful-detailed-photos-of.html">BB</a>; <a href="https://plus.google.com/100003628603413742554/posts/cFc4LuJg511">G+</a>)</li></ul>http://11011110.livejournal.com/299547.htmltoolswikipediaacademiageometryartpublic0http://11011110.livejournal.com/299376.htmlSun, 09 Nov 2014 09:12:54 GMTThe length of a 2048 game
http://11011110.livejournal.com/299376.html
Presumably many readers have seen and played <a href="http://gabrielecirulli.github.io/2048/">the 2048 game</a>, in which one slides tiles in different directions across a grid, causing certain pairs of tiles to combine with each other when their sum is correct (in this case, when the sum is a power of two):<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/8192.png"></div><br /><br />Or if you get bored with it there are many other variations, some of which have different combining rules, for instance there's one where <a href="http://cozmic72.github.io/987/">tiles only combine when their sum is a Fibonacci number</a>. I have the impression this all started with another of these games, <a href="http://threesjs.com/">threes</a>, which you would think allows tiles to combine when their sum is a power of three or twice a power of three (the ternary digits), but actually uses a different set, the numbers 1 or 2, and the numbers that are three times a power of two.<br /><br />At some point I switched from 2048 to 987 (the Fibonacci one) and noticed that the games were lasting longer. Or alternatively, I could reach higher tile values in 987 more easily despite the fact that it takes twice as long (the 987 game gives you 1's for each new tile, instead of 2's and 4's for 2048). Why is that?<br /><br />It's certainly not my skill or strategy; I think I'm a worse player at 987 than at 2048. I think there are actually two reasons why the games last longer. One is that the Fibonacci combining rule makes it less critical that the tiles be ordered a certain way, because if they're a little bit out of order they can still combine with each other. But the more important reason has to do with the mathematical limits on the length of a game.<br /><br />In 2048, generalized to arbitrary board sizes, the maximum tile you could possibly reach on a board with <i>n</i> cells is approximately 2<sup><i>n</i></sup>. That's because, to reach a tile with value 2<sup><i>n</i> + 1</sup>, you'd first have to get to a board whose total tile value was 2<sup><i>n</i> + 1</sup> − 2 (or maybe if you get lucky 2<sup><i>n</i> + 1</sup> − 4) and those numbers can only be expressed as a sum of powers of two by using many tiles (the number of nonzeros in their binary representation). In 987, there are also total tile values that require many tiles to express as a sum of Fibonacci numbers, but they're more widely spaced: they are formed by taking every other Fibonacci number (2, 5, 13, 34, 89, etc) and subtracting one to get 1, 4, 12, 33, 88, etc. These numbers grow roughly as 2.618<sup><i>n</i></sup>, so that's how big a tile value you could hope to get on a generalized 987 board. Another way of thinking about this is that we are counting the number of nonzeros in the <a href="https://en.wikipedia.org/wiki/Zeckendorf's_theorem">Zeckendorf representation</a> of the total tile value. The Zeckendorf representation is a number system that is based on Fibonacci numbers, which grow more slowly than powers of two, but only half of the bits of a number can be nonzero, so it takes higher numbers to get a lot of nonzero bits.<br /><br />This made me wonder: what about variants of 2048/987/etc that allow pairs of tiles to combine when the sum belongs to some other sequence of numbers, rather than powers of two of Fibonacci numbers? The sequence that Threes should have used, the ternary digits, would allow you to reach even larger tile values: the numbers that are hardest to express as a sum of ternary digits are of the form (3<sup><i>n</i></sup> − 1)/2, so again they grow exponentially but with a bigger base. Or what about if you allow tiles to combine whenever they are equal or a factor of two apart, so that the allowed tile values are the <a href="https://en.wikipedia.org/wiki/Smooth_number">3-smooth numbers</a>? Then, the sequence of hard-to-express totals grows much more quickly, as 1, 5, 23, 431, 18431, 3448733, 1441896119 (<a href="http://oeis.org/A018899">OEIS A018899</a>). These numbers look to me to be growing exponentially in <i>n</i><sup>2</sup> but I don't know why that should be.<br /><br />The primes don't make a good sequence of allowed tile values for games like this, because of parity issues (two odd primes can't add up to another prime). But what about the <a href="https://en.wikipedia.org/wiki/Semiprime">semiprimes</a>? According to <a href="https://en.wikipedia.org/wiki/Chen%27s_theorem">Chen's theorem</a>, every sufficiently large number can be decomposed into a sum of two primes or semiprimes, so there aren't any totals that could only be expressed using many tiles. Does this mean the game can go on forever?<br /><br />No! The semiprimes still have density zero in the set of all integers: the fraction of integers up to some number N that are semiprime goes to zero in the limit as N goes to infinity. And it turns out that every set of allowed tile values that allows the game to reach arbitrarily large tiles on a board of finitely many cells must have positive density. This is true even if you allow a more relaxed set of moves that ignores the board geometry and lets you combine arbitrary pairs of tiles no matter where they are on the board. The proof is by induction on the number of board cells. If a set <i>S</i> of allowable tile values lets the game reach arbitrarily large tile values on <i>n</i> − 1 cells, then the induction hypothesis shows that <i>S</i> must have positive density. Otherwise, let <i>M</i> be the maximum tile value reachable on <i>n</i> − 1 cells. Suppose that an <i>n</i>-cell board eventually reaches a position that includes a tile of value greater than <i>M</i>, and mark all the nonzero tiles at that point in time. Then the remaining game can be partitioned into moves on unmarked tiles, forming a "small game" on a board of at most <i>n</i> − 1 cells, together with moves that either merge two of the marked tiles or add a value from the small game to one of the marked tiles. (This removes a tile from the small game but every position reachable in this way could also be reachable in a small game without tile removal.) Only a finite number of merges are possible, so to reach arbitrarily large values it must be possible to add small-game values to marked tiles arbitrarily many times. But this means that the set <i>S</i> must have density at least 1/<i>M</i>, for otherwise there would be infinitely many gaps of length larger than <i>M</i>, too wide to cross by adding a small game value and too many to cross by merges.<br /><br />Therefore, the sequence of largest reachable tile values for the semiprime game on an <i>n</i>-cell board is a bona fide infinite sequence: there is no board that lets you play forever. What is that sequence and how quickly does it grow? I have no idea, but presumably much more quickly than exponential in <i>n</i><sup>2</sup>.<a name='cutid1-end'></a>http://11011110.livejournal.com/299376.htmlgame theorynumber theorypublic3http://11011110.livejournal.com/298806.htmlSat, 01 Nov 2014 17:47:32 GMTLinkage for (the day after) Halloween
http://11011110.livejournal.com/298806.html
<ul><li><a href="http://www.thisiscolossal.com/2014/10/diy-kinetic-origami-sculpture-designed-by-jo-nakashima/">Kinetic origami sculpture by Jo Nakashima</a> (<a href="https://plus.google.com/100003628603413742554/posts/a13GNcU9YY2">G+</a>)</li><br /><li><a href="http://corner.mimuw.edu.pl/?p=689">How pineapples help finding Steiner trees</a> (<a href="https://plus.google.com/u/0/100003628603413742554/posts/VU8ytgxCrN3">G+</a>)</li><br /><li><a href="http://www-kb.is.s.u-tokyo.ac.jp/~koba/icalp2015/">ICALP 2015</a> conference web site and call for papers (deadline Feb.17; <a href="https://plus.google.com/100003628603413742554/posts/bzAsJAXYtUf">G+</a>)</li><br /><li><a href="http://scholarlyoa.com/2014/10/02/an-editorial-board-mass-resignation-from-an-open-access-journal/">Mass resignation from an open access journal</a> (<a href="https://plus.google.com/100003628603413742554/posts/7oYRxJH6qiP">G+</a>)</li><br /><li><a href="http://arxiv.org/abs/1410.3820">A hardness result for organizing your Google Scholar profile</a> (<a href="https://plus.google.com/100003628603413742554/posts/e2BoRBFEkjy">G+</a>)</li><br /><li><a href="http://chronicle.com/article/Wikipedia-a-Professors-Best/149337/">Wikipedia, a Professor's Best Friend</a>, and a tangential note about <a href="https://en.wikipedia.org/wiki/Binary_logarithm">binary logarithms</a> (<a href="https://plus.google.com/100003628603413742554/posts/6qLy6Cy9UaQ">G+</a>)</li><br /><li><a href="http://www.npr.org/blogs/money/2014/10/21/357629765/when-women-stopped-coding">When women stopped coding</a>, an analysis of why and how long we've been seeing a decline in the number of women in computer science (<a href="https://plus.google.com/100003628603413742554/posts/JdabqryeXQQ">G+</a>)</li><br /><li><a href="http://www.reuters.com/article/2014/10/22/us-usa-unc-fraud-idUSKCN0IB2D520141022">Fake classes for athletes at UNC</a>, or why I'm happy UCI doesn't have a football team (<a href="https://plus.google.com/100003628603413742554/posts/5DYqPuT323U">G+</a>)</li><br /><li><a href="http://www.wads.org/">WADS 2015</a> conference web site and call for papers (deadline Feb.20; <a href="https://plus.google.com/100003628603413742554/posts/GASqUpYo24p">G+</a>)</li><br /><li><a href="http://www.datapointed.net/2014/10/maps-of-street-grids-by-orientation/">City maps colored by grid orientation</a> (<a href="http://www.metafilter.com/143874/Main-Street-ran-east-to-west-land-astride-platted-into-tidy-rectangles">MF</a>; <a href="https://plus.google.com/100003628603413742554/posts/3bNiPQPKXbW">G+</a>)</li><br /><li><a href="http://retractionwatch.com/2014/10/26/scientist-sues-pubpeer-commenters-subpoenas-site-for-names/">Scientist sues open-peer-review site commenters</a> (<a href="https://plus.google.com/100003628603413742554/posts/DcABTFF2P9X">G+</a>)</li><br /><li><a href="http://mobile.nytimes.com/2014/10/27/business/media/wikipedia-is-emerging-as-trusted-internet-source-for-information-on-ebola-.html">Wikipedia emerges as trusted internet source for ebola information</a> (<a href="https://plus.google.com/100003628603413742554/posts/WZNDGWcvZ6F">G+</a>)</li><br /><li><a href="http://tex.stackexchange.com/questions/208516/was-the-knuth-plass-line-breaking-output-ever-subjected-to-a-blind-experiment">Was the Knuth-Plass line breaking output ever subjected to a blind experiment?</a> (<a href="https://plus.google.com/u/0/100003628603413742554/posts/RchBefQFRnr">G+</a>)</li><br /><li><a href="http://www.quantamagazine.org/20141015-at-the-far-ends-of-a-new-universal-law/">At the far ends of a new universal law</a>, a popular-press article about the <a href="https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution">Tracy–Widom distribution</a> (<a href="https://plus.google.com/100003628603413742554/posts/H58sfwoDwVW">G+</a>)</li><br /><li><a href="http://crookedtimber.org/2014/10/31/brands-of-nonsense/">Brands of nonsense</a>, on university's attempts to apply corporate branding dogma to themselves (<a href="https://plus.google.com/100003628603413742554/posts/Ptx7LgKb4uE">G+</a>)</li></ul>http://11011110.livejournal.com/298806.htmlinformation visualizationcorporatizationwikipediaacademiaconferencesorigamipublic0http://11011110.livejournal.com/298505.htmlSun, 26 Oct 2014 23:21:48 GMTAnother 5-permutohedron
http://11011110.livejournal.com/298505.html
I've been playing with the <a href="http://11011110.livejournal.com/tag/permutohedron">Cayley graphs of the symmetric group</a> again, after accidentally running across the Wikipedia article on the <a href="https://en.wikipedia.org/wiki/Icositruncated_dodecadodecahedron">icositruncated dodecadodecahedron</a> and thinking "oh yeah, that's the polyhedron that geometrically represents the Cayley graph generated by a transposition and a 4-cycle".<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/600px-Icositruncated_dodecadodecahedron.png"></div><br /><br />Only, it's not. That was <a href="http://11011110.livejournal.com/117780.html">a different star polyhedron</a>. So is the icositruncated dodecadodecahedron also a permutohedron? And if so, what are the generators of the corresponding Cayley graph?<br /><br />To begin with, there are two different ways to form 3-regular (undirected) Cayley graphs. You can start with two generating permutations, one of which is self-inverse (an involution), or you can start with three generators, all three of which are involutions. In either case, the vertices are all possible permutations and the edges tell you which permutation you get when you multiply another permutation by a generator.<br /><br />Both of these constructions give rise not just to a graph but a topological surface, made out of three sets of polygons. In the two-generator case, one of the three sets of polygons are the cycles that you get by repeatedly applying the non-involution generator, and the other two are cycles that alternate between two generators. In the three-involution case, all three sets of polygons are formed by alternating between two out of three of the generators. What I really want to know is not just whether I have the right graph, but also whether I have the right sets of polygons (the red, blue, and yellow ones in the picture).<br /><br />First, let's consider the case that we have three involutions. You can visualize an involution as a matching on a set of vertices, so three involutions can be visualized as a 3-edge-colored multigraph on these vertices, and two out of three of them will form some kind of partition of the vertices into alternating paths and cycles. In the case of the icositruncated dodecahedron, two of the three sets of polygons are decagons (2<i>n</i>-gons, where <i>n</i> = 5 is the number of things being permuted). A little thought should convince you that in this case the corresponding two involutions have to form a single 5-vertex alternating path; otherwise there is no way of getting a factor of five into the order of the group they generate. So we have three involutions, two pairs of which form alternating paths. There are five combinatorially distinct ways of doing this, two of which cause the other pair of involutions to generate a 6-cycle:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/alternating-involutions.png"></div><br /><br />(The one labeled "10,10,10B", a 3-edge-coloring of the complete bipartite graph K<sub>2,3</sub>, has symmetries taking any color class to any other color class, and so should result in an especially symmetric Cayley graph.) But none of these generate the icositruncated dodecadodecahedron, nor do they generate the symmetric group. The reason is that all three generators have an even number of transpositions, so they all belong to the alternating subgroup of the symmetric group. (Sometimes a proper subgroup; for instance 10,10,10A is the dihedral group.) That means that we get at most 60 vertices, not 120, when we generate the Cayley graph of these generators and its corresponding system of polygons, giving us some other (abstract) polyhedron than the one we wanted.<br /><br />Ok, how about one involution and one non-involution? There, things work a bit better. Suppose that <i>g</i> is a permutation on a set of elements, consisting of two disjoint cycles of lengths <i>x</i> and <i>y</i>, and that <i>h</i> is an involution that transposes one pair of elements from one cycle to the other. Then it turns out that the product of these two permutations is a cycle on <i>x</i> + <i>y</i> elements, so <i>g</i> and <i>h</i> generate a transitive group. By composing <i>h</i> with powers of the cycle, we can find many other transpositions, enough that whenever gcd(<i>x</i>,<i>y</i>) = 1 we get the whole symmetric group. The sizes of the polygons in the Cayley graph generated by <i>g</i> and <i>h</i> are 2(<i>x</i> + <i>y</i>) (for the polygons that alternate between the two generators) and <i>xy</i> (for the polygons generated by <i>g</i> alone).<br /><br />Choosing (<i>x</i>,<i>y</i>) = (1,<i>n</i> − 1) gives us the Cayley graphs from the previous post, generated by a transposition and an (<i>n</i> − 1)-cycle. But choosing (<i>x</i>,<i>y</i>) = (2,3) gives us the cycle lengths of 6 and 10 that we want, in a genuine Cayley graph of the symmetric group. Is it the same as the graph of the polyhedron? I'm not sure, but I strongly suspect that it is.<br /><br />The good folks at <a href="http://rjlipton.wordpress.com/">Gödel's last letter</a> have a habit of ending their posts with open problems, so here's one: this construction using a two-cycle permutation and a transposition gives a big class of 3-regular graphs, in which the cycles that we know about can be made quite large. (Of course there could be other, shorter cycles.) Regular graphs in which all cycles are sufficiently large (larger than this) are automatically good expander graphs. Do some or all of these (<i>x</i>,<i>y</i>)-Cayley graphs also have good expansion?<a name='cutid1-end'></a>http://11011110.livejournal.com/298505.htmlxyz graphspermutohedronpublic0http://11011110.livejournal.com/298378.htmlWed, 15 Oct 2014 20:30:55 GMTLinkage
http://11011110.livejournal.com/298378.html
<ul><li><a href="http://chronicle.com/article/Why-Academics-Writing-Stinks/148989/">Why academic writing stinks</a> or, keep it simple (<a href="https://plus.google.com/100003628603413742554/posts/59CKJwHwcGb">G+</a>)</li><br /><li><a href="http://bpalop.blogspot.com/2014/10/hasta-luego-ferran.html">Sad news of the death of Ferran Hurtado</a> (<a href="https://plus.google.com/100003628603413742554/posts/UXwaU71WSEV">G+</a>)</li><br /><li><a href="http://tabletopwhale.com/2014/07/21/a-visual-compendium-of-glowing-creatures.html">A visual compendium of glowing creatures</a>, scientific illustration by Eleanor Lutz (<a href="https://plus.google.com/100003628603413742554/posts/Jx3y1B3tV3v">G+</a>)</li><br /><li><a href="https://www.insidehighered.com/news/2014/08/18/study-raises-questions-about-why-women-are-less-likely-men-earn-tenure-research">Women in computer science get tenure at significantly lower rates than men</a> even after normalizing for research productivity (<a href="https://plus.google.com/100003628603413742554/posts/ETZTMV2C9Q5">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Tietze%27s_graph">Tietze's graph</a>, Wikipedia article expanded with a new illustration of why it has the name it has: it was an earlier counterexample in the theory of coloring graphs on non-orientable surfaces (<a href="https://plus.google.com/100003628603413742554/posts/URafzajyVZ1">G+</a>)</li><br /><li><a href="http://www.dw.de/abdel-kader-haidara-awarded-germanys-2014-africa-prize-for-rescuing-timbuktu-manuscripts/a-17729553">Abdel Kader Haidara awarded Germany's 2014 Africa Prize for rescuing Timbuktu manuscripts</a> (<a href="https://plus.google.com/100003628603413742554/posts/GDsLQuTkQbC">G+</a>)</li><br /><li><a href="http://www.metafilter.com/143388/Adobe-Digital-Editions-4-spying-on-users">Adobe Digital Editions 4 spying on users</a> by sending a listing of the contents of your entire digital library in cleartext (<a href="https://plus.google.com/100003628603413742554/posts/TPhpWEQiuse">G+</a>)</li><br /><li><a href="http://www.wired.com/2014/09/nasa-invents-folding-solar-panel-inspired-origami/">NASA Invents a Folding Solar Panel Inspired by Origami</a> (<a href="https://plus.google.com/100003628603413742554/posts/CX8opYqER1r">G+</a>)</li><br /><li><a href="http://cstheory.stackexchange.com/q/26031/95">Optimal randomized comparison sorting</a>: a question on the CSTheory exchange observing that randomization can break the decision tree lower bound and asking what's known about upper bounds (<a href="">G+</a>)</li><br /><li><a href="http://blogs.scientificamerican.com/roots-of-unity/2014/08/31/look-ma-no-zero/">More on the strange scale-free Babylonian concept of number</a>, in which 1 and 60 were apparently indistinguishable (<a href="https://plus.google.com/100003628603413742554/posts/AjoeettK7go">G+</a>)</li><br /><li><a href="http://www.dataisnature.com/?p=2053">Robert le Ricolais’s Tensegrity Models</a>, architectural models that could as well be abstract art (<a href="https://plus.google.com/100003628603413742554/posts/HwhvPPYgBMm">G+</a>)</li><br /><li><a href="http://retractionwatch.com/2014/10/12/european-science-foundation-demands-retraction-of-criticism-in-nature-threatens-legal-action/">European Science Foundation demands retraction of criticism in Nature, threatens legal action</a> (<a href="https://plus.google.com/100003628603413742554/posts/EbpnnvdVLXE">G+</a>)</li><br /><li><a href="http://www.win.tue.nl/SoCG2015/">SoCG 2015 conference web site and call for papers</a> (<a href="https://plus.google.com/100003628603413742554/posts/RZMDa98Hn1S">G+</a>)</li></ul>http://11011110.livejournal.com/298378.htmlbloggingwikipediaacademiaconferencesartpublic0http://11011110.livejournal.com/297993.htmlTue, 14 Oct 2014 23:17:15 GMT2-site Voronoi triangle centers
http://11011110.livejournal.com/297993.html
The idea of a 2-site Voronoi diagram was pioneered by Barequet, Dickerson, and Drysdale in a WADS 1999 paper later published in <a href="http://dx.doi.org/10.1016/S0166-218X(01)00320-1">Discrete Applied Math. 2002</a>. The basic idea is that you have a function d(p,q;x) that tells you the distance from an unordered pair of sites {p,q} to another point x in the plane; given a collection of sites, you want to divide the plane up into regions so that the region containing a point x tells you which pair of sites is closest to x (or in some cases farthest from x). Since their work there have been several followup papers including <a href="http://arxiv.org/abs/1105.4130">one in which I was involved</a>.<br /><br />If three sites are given, then these sites determine three pairs of sites, three regions in the Voronoi diagram, and (usually) one or more Voronoi vertices where all three sites meet. Often, these vertices define <a href="https://en.wikipedia.org/wiki/Triangle_center">triangle centers</a>: special points defined from the triangle of the three sites, such that applying a similarity transformation to the plane commutes with constructing the given point. Technically, to get a triangle center out of a 2-site Voronoi diagram, the distance function has to be invariant under congruences of the plane, and if a triangle pqx is scaled by a factor of s then the distance d(p,q;x) has to scale by a factor of s<sup>e</sup> for some constant scaling exponent e (possibly zero). But this scaling requirement is true of many natural functions that you might want to use as distances.<br /><br />By now, thousands of different triangle centers have been studied. Which of them can be generated from 2-site Voronoi diagrams in this way? Here are some examples. (The X(...) numbers are references to Clark Kimberling's <a href="http://faculty.evansville.edu/ck6/encyclopedia/ETC.html">Encyclopedia of Triangle Centers</a>.)<br /><br /><ul><li>If d(p,q;x) is defined as the Euclidean distance from point x to line pq, with scaling exponent 1, then the level sets of this distance function are infinite slabs having pq as their centerline. For a given triangle pqr, the 2-site Voronoi diagram for this distance function has seven junctions. At the three points where two lines cross, four cells of the diagram meet, two for each of the two lines. And at four other points (one inside the triangle and three outside), all three cells of the diagram meet. The one inside the triangle is the <a href="https://en.wikipedia.org/wiki/Incenter">incenter</a> X(1), a triangle center that is the center point of a circle inscribed inside the triangle. The other three of these Voronoi vertices are the excenters, center points of circles that touch the three lines but are exterior to the triangle, and are not individually triangle centers.<br /><br />In the figure below, the level sets of the distances from the three lines are indicated by the red, blue, and yellow shading. The Voronoi diagram boundaries are the green line segments, and the incenter is the green Voronoi vertex inside the triangle. The excenters are outside the margins of the drawing.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/2site-incenter.png"></div><br /><br />Alternatively, we could have defined d(p,q;x) to be the Euclidean distance from point x to line segment pq, giving more complicated <a href="https://en.wikipedia.org/wiki/Stadium_(geometry)">stadium</a>-shaped level sets and only one Voronoi vertex at the incenter.</li><br /><li>If d(p,q,x) is the area of triangle pqx, then the scaling exponent is 2 and the level sets of the distance function are again slabs, with longer line segments having narrower slabs for the same distance. There is a Voronoi vertex at the <a href="https://en.wikipedia.org/wiki/Centroid">centroid</a> of the triangle X(2), because at that point all three of the triangles pqx, prx, and qrx have the same area as each other.</li><br /><li>If d(p,q;x) is the sum of distances px+qx, then the scaling exponent is 1, and the level sets of the distance function are ellipses. There is a Voronoi vertex at the <a href="https://en.wikipedia.org/wiki/Circumcenter">circumcenter</a> X(3); it is inside the triangle if it is acute, outside if it is obtuse, and at the midpoint of the hypotenuse if it is right.</li><br /><li>If d(p,q;x) is the Euclidean distance from x to the midpoint of pq, then the level sets of the distance function are just circles centered on this midpoint, and the scaling exponent is 1. The Voronoi vertex is at the <a href="https://en.wikipedia.org/wiki/Nine-point_center">nine-point center</a> of the triangle, X(5).</li><br /><li>If d(p,q,x) is the ratio of two quantities, the distance of x from line pq and the length of segment pq, then the scaling exponent is 0. The level sets of the distance function are slabs, as for the incenter and centroid, but scaled differently, and there is a Voronoi vertex at the <a href="https://en.wikipedia.org/wiki/Symmedian">symmedian</a> X(6).</li><br /><li>If d(p,q;x) is the angle subtended by segment pq, as viewed from x, then the level sets of the distance function are <a href="https://en.wikipedia.org/wiki/Lens_(geometry)">lunes</a> centered on the sides of the triangle, the scaling exponent is 0, and there is a Voronoi vertex at the <a href="https://en.wikipedia.org/wiki/Fermat_point">Fermat point</a> of the triangle, which for not-too-obtuse triangles coincides with X(13).</li><br /><li>If d(p,q;x) is the perimeter of triangle pqx, then the scaling exponent is 1 and the level sets of the distance function are ellipses. In this case there is a Voronoi vertex at the <a href="https://en.wikipedia.org/wiki/Isoperimetric_point">isoperimetric point</a> X(175).</li><br /><li>If d(p,q;x) is the ratio of the sum of distances px+qx to the length of segment pq (that is, the dilation of the path pxq) then the scaling exponent is 0 and the level sets are ellipses again. The associated triangle center is the <a href="http://11011110.livejournal.com/144178.html">dilation center</a> X(3513).</li><br /><li>If d(p,q;x) is the inradius of triangle pqx, then the scaling exponent is 1. I think the level sets of the distance function are hyperbolae but I haven't proved this. There is a Voronoi vertex at the <a href="http://faculty.evansville.edu/ck6/integer/unsolved.html">mysterious</a> <a href="http://www.jstor.org/stable/2690307">congruent incircles point</a> of the triangle, X(5394).</li></ul><br />Points X(31), X(32), X(75), and X(76) are also Voronoi vertices of differently-scaled distances from line pq. One can generate even more examples (although not necessarily interesting ones) by multiplying together two or more of these distance functions, or adding together functions with the same scaling exponent.<br /><br />Despite all these examples, it's not clear to me how to tell whether a triangle center is or isn't a 2-site Voronoi vertex, or whether there exists a center that is not a 2-site Voronoi vertex. The most obvious center for which I haven't found a good 2-site distance function is the <a href="https://en.wikipedia.org/wiki/Orthocenter">orthocenter</a> X(4), the point where the perpendiculars to each side through the opposite vertex meet. Is it a 2-site Voronoi vertex for some distance function, and if so, what is the function?<a name='cutid1-end'></a>http://11011110.livejournal.com/297993.htmlgeometrypublic4http://11011110.livejournal.com/297829.htmlThu, 09 Oct 2014 00:57:29 GMTForced creases in Miura folding
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I've <a href="https://plus.google.com/100003628603413742554/posts/hfkbtJmZ6na">posted before</a> about the Miura fold, a subdivision of a sheet of paper into paralellograms that gives a nice smooth motion from a compact folded state (in which the parallelograms are all folded on top of each other) to an unfolded state in which the paper is nearly flat. But this pattern can have <a href="http://www.sciencemag.org/content/345/6197/647.short">defects that interfere with its folding pattern</a>, in which small subunits of the pattern fold the wrong way: they still have the same creases but some of them are backwards. My latest arXiv preprint, "Minimum Forcing Sets for Miura Folding Patterns" (<a href="http://arxiv.org/abs/1410.2231">arXiv:1410.2231</a>, with Ballinger, Damian, Flatland, Ginepro, and Hull, to appear at SODA) studies the number of creases that need to be forced to go the right way, in order to be sure of getting the correct Miura fold for the rest of the creases as well.<br /><br />One of the key ideas in the paper comes from some previous work by undergraduate co-author Jessica Ginepro with her advisor Tom Hull (<span class="ljuser i-ljuser i-ljuser-type-P " lj:user="tomster0" ><a href="http://tomster0.livejournal.com/profile" target="_self" class="i-ljuser-profile" ><img class="i-ljuser-userhead" src="http://l-stat.livejournal.net/img/userinfo.gif?v=17080?v=123.1" /></a><a href="http://tomster0.livejournal.com/" class="i-ljuser-username" target="_self" ><b>tomster0</b></a></span>). They showed that the different ways of folding the Miura pattern (the right way, and the many wrong ways) have a one-to-one correspondence with the different ways of properly 3-coloring a grid graph. The correspondence uses a non-obvious <a href="https://en.wikipedia.org/wiki/Boustrophedon">boustrophedon</a> pattern of sweeping the grid in alternating order, left to right, right to left, left to right, etc., using the differences between consecutive colors in the sweep to determine the direction to fold each crease.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/miura/biject.png"></div><br /><br />Under this correspondence, a forcing set of creases in a Miura pattern becomes a forcing set of edges in the grid, such that if you label each selected edge with the difference (mod 3) of the colors of its endpoints, no other valid coloring has the same pattern of differences.<br /><br />Based on this correspondence, and some other combinatorial tools including domino tiling and planar feedback arc sets, we can show that the standard Miura folding pattern is the worst Miura, in the sense that it requires a bigger number of forced creases (approximately half of the creases) than any other way of folding the same pattern. In terms of the corresponding grid coloring, the Miura pattern forms a checkerboard of two colors, and any one of its grid squares can be recolored in a different color unless it has a forced edge incident to it, so there have to be a lot of forced edges.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/miura/checkerboard.png"></div><br /><br />In constrast, some other folding patterns for the same crease set can be forced by much smaller subsets of creases, proportional only to the square root of the total number of creases. For instance, fold a sheet of paper into a strip, folding each crease the same way so that the paper spirals around itself:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/miura/FalseMiura1.png"></div><br /><br />Then fold the strip in a spiral around itself again, so that the second set of folds are all parallel to each other but tilted a little rather than being perpendicular to the first set of folds:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/miura/FalseMiura2.png"></div><br /><br />This corresponds to a nice regular pattern of three colors in the grid in diagonal stripes, which is much more highly determined: if you fix the colors on the boundary of the grid, everything inside has its color forced. So, it has much smaller forcing sets (in fact, the smallest possible among all wrong ways of folding the Miura).<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/miura/stripes.png"></div><br /><br />More generally, for any pattern of folding the Miura (right or wrong), we can construct a minimum subset of the pattern that forces the rest, in polynomial time.<a name='cutid1-end'></a>http://11011110.livejournal.com/297829.htmlorigamipaperspublic1http://11011110.livejournal.com/297618.htmlWed, 01 Oct 2014 03:09:51 GMTLinkage for the end of September
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<ul><li><a href="http://www.algomation.com/">Algomation animated algorithms</a> (<a href="https://plus.google.com/100003628603413742554/posts/ifGuC6WWnhW">G+</a>)</li><br /><li><a href="http://player.vimeo.com/video/106226560">Rush hour</a> video, or, what our robot-driven future will be like (<a href="https://plus.google.com/100003628603413742554/posts/SpGwTwXX6up">G+</a>)</li><br /><li><a href="http://www.washingtonpost.com/blogs/answer-sheet/wp/2014/09/17/more-students-are-illegally-downloading-college-textbooks-for-free">The Washington Post rants about those evil student pirates</a>, but neglects to mention the free alternatives (<a href="https://plus.google.com/100003628603413742554/posts/GoN1H7rF3y2">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=TUHgGK-tImY">A song video about knots</a>, from <a href="https://ldtopology.wordpress.com/2014/09/18/a-song-about-a-knot/">the low-dimensional topology blog</a> (<a href="https://plus.google.com/100003628603413742554/posts/VipsGJPJHjb">G+</a>)</li><br /><li><a href="http://gamasutra.com/blogs/HermanTulleken/20140912/225495/20_Fun_Grid_Facts_Hex_Grids.php">Fun hex grid facts</a>, via <a href="http://www.metafilter.com/142931/THERE-IS-ONLY-ONE-MAGIC-HEXAGON">MF</a> (<a href="https://plus.google.com/100003628603413742554/posts/VRqtn49sKnU">G+</a>)</li><br /><li><a href="http://www.siam.org/meetings/da15/da15_accepted.pdf">SODA 2015 accepted papers</a> (<a href="https://plus.google.com/100003628603413742554/posts/EtQ3QChk1bc">G+</a>)</li><br /><li><a href="http://aperiodical.com/2014/09/katex-the-fastest-math-typesetting-library-for-the-web/">KaTeX</a>, a lobotomized but fast web math renderer (<a href="https://plus.google.com/100003628603413742554/posts/RAC6o9qaoq6">G+</a>)</li><br /><li><a href="https://medium.com/@cshirky/why-i-just-asked-my-students-to-put-their-laptops-away-7f5f7c50f368">Against laptops in lectures</a>, via <a href="http://www.metafilter.com/143006/Why-I-Just-Asked-My-Students-To-Put-Their-Laptops-Away-">MF</a> (<a href="https://plus.google.com/100003628603413742554/posts/irVC2gYXcLW">G+</a>)</li><br /><li><a href="http://www.dataisnature.com/?p=2048">David Wade’s ‘Fantastic Geometry’ – The Works of Wenzel Jamnitzer & Lorenz Stoer</a> on Dataisnature (<a href="https://plus.google.com/100003628603413742554/posts/QvmRaFPqvT8">G+</a>)</li><br /><li><a href="https://gist.github.com/gavinandresen/e20c3b5a1d4b97f79ac2">Something about how some data structures I helped develop could improve bitcoin mining</a> (<a href="https://plus.google.com/100003628603413742554/posts/SEgCxWu1rT5">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=09JslnY7W_k">63 and –7/4 are special</a>, numberphile video about prime factors of recurrence sequences (<a href="https://plus.google.com/100003628603413742554/posts/SDSanXttfwA">G+</a>)</li><br /><li><a href="http://www.ams.org/notices/201409/rnoti-p1040.pdf">Critique of Hirsch’s Citation Index</a>, article in the <i>Notices</i> about how the h-index doesn't provide much more information than the total citation count (<a href="https://plus.google.com/100003628603413742554/posts/4EXT17fkXHY">G+</a>)</li></ul>http://11011110.livejournal.com/297618.htmlbloggingpublic0http://11011110.livejournal.com/297428.htmlMon, 29 Sep 2014 03:16:59 GMTUniversity of Würzburg graffiti
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I neglected to pack my camera and its new lens with me for the trip to GD (oops), and anyway most of the time the weather wasn't very conducive to photography. But I did take a couple of cellphone snapshots of graffiti/murals on the University of Würzburg campus. This one, if Google translate is to be believed, proclaims Würzburg as the city of young researchers; it's on the wall of the Mensa where we ate lunch every day.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/uni-wuerzburg/CityOfYoungResearchers-m.jpg" border="2" style="border-color:black;" /></div><br /><br />And here's some advice to the students starting the new term, from just outside the conference hall:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/uni-wuerzburg/Study-m.jpg" border="2" style="border-color:black;" /></div>http://11011110.livejournal.com/297428.htmlgraffitiphotographypublic0http://11011110.livejournal.com/297186.htmlSun, 28 Sep 2014 01:29:03 GMTReport from Graph Drawing
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I'm currently in the process of returning<sup>*</sup> from Würzburg, Germany, where I attended the 22nd International Symposium on Graph Drawing (GD 2014) and was one of the invited speakers at the associated EuroGIGA/CCC Ph.D. school on graph drawing.<br /><br />The format for the Ph.D. school was three one-hour lectures in the morning and three hours of working on exercises in the afternoon, for two days. My contribution was a high-level overview of graph drawing methods that involve curves (an updated version of <a href="http://drops.dagstuhl.de/opus/volltexte/2013/4168/">my Dagstuhl survey</a>). You'll have to ask one of the students how well the school really went, but my impression is that it was a success. Among other things, it brought in 40 students, and perhaps helped GD itself achieve an unusually high number of participants.<br /><br />There were two best presentation awards from GD. One went to Vincent Kusters, for a talk on "Column planarity and partial simultaneous geometric embedding" with Evans, Saumell and Speckmann. Their main result is that if one is given two trees on the same vertex set, then it is possible to find planar drawings of them both, with straight line segments as edges, in which a large fraction of the vertices are in the same location in both drawings. (It is not always possible to place all vertices the same in both drawings.) There was also a best poster award, given to Thomas Bläsius, Fabian Klute, Benjamin Niedermann, and Martin Nöllenburg, for their work on "PIGRA", a tool for pixelated graphs. Instead of the usual poster format (a few boxes of text+figures, not very different from the set of slides from a short talk) they made an effort at presenting their work in a graphic-novel format, with a stick-figure narrator that looked like it was inspired by xkcd. There were two good invited talks, by Oswin Aichholzer on some problems related to the crossing numbers of complete graphs and by Jean-Daniel Fekete on visualizations of graphs based on adjacency matrices rather than node-link diagrams, as well as, of course, many interesting contributed talks.<br /><br />The other best presentation award went to Fidel Barrera-Cruz for his work with Penny Haxell and Anna Lubiw on morphing one triangulation to another by piecewise linear motions while preserving planarity at the intermediate stages. It was one of the few talks to include a demo of things happening dynamically rather than just static images, which worked very well for this topic. I can't find the paper itself online but it seems to also be part of <a href="https://uwspace.uwaterloo.ca/handle/10012/8518">Barrera-Cruz's Ph.D. thesis</a>.<br /><br />A few of the other talks that were memorable to me:<br /><br />Wednesday morning, Jan Christoph Athenstädt spoke about <a href="http://arxiv.org/abs/1408.6019">making overlaid drawings of two partitions</a> of a set of elements; the same problem can be interpreted as one of drawing a 2-regular hypergraph. It's easy to construct such a drawing by using membership in one partition as an x-coordinate and in the other partition as a y-coordinate, and drawing the sets as ovals around the rows and columns of the resulting grid layout, but this is problematic in some respects; for instance, it has empty regions where pairs of sets look like they intersect but actually don't. On the other hand if the partitions are to be drawn as pseudocircles (at most two crossing points per pair, intersecting only as needed) the problem turns into one of graph planarity, but is not always realizable. An intermediate level of constraint on these drawings is unfortunately NP-complete to recognize.<br /><br />On Wednesday afternoon, one of Philipp Kindermann and Fabian Lipp (I no longer remember which) spoke about their work with Sasha Wolff on boundary labeling. If you've ever used the <a href="http://www.ctan.org/pkg/todonotes">todonotes</a> package in LaTeX, you know what this is: a bunch of rectangular labels (notes to you or your co-authors reminding you of things that need to be changed) are to be placed in the margin of your LaTeX output, connected by "leaders" (polygonal paths) to the point where the change needs to be made. todonotes is a little brainless about how it places its notes and leaders, and their paper discussed <a href="http://www.ctan.org/pkg/luatodonotes">an improved system for the same problem</a>. Unfortunately it is implemented in LuaLaTeX, which I am still leery of using. (I don't think arXiv allows it and given the security issues of running a real programming language from user-contributed source code I'm not sure I'd want to advocate that they start; the same is true of the publishers I've worked with.)<br /><br />Thursday morning had a whole session on <a href="https://en.wikipedia.org/wiki/1-planar_graph">1-planarity</a>, a topic of some interest to me. Di Giacomo, Liotta, and Montecchiani showed that outer-1-planar graphs with maximum degree <i>d</i> can be drawn with the given embedding using line segments of O(<i>d</i>) slopes, or planarly (they are always planar graphs) using O(<i>d</i><sup>2</sup>) slopes, better than the higher polynomial for planar 3-trees and exponential known bound for planar graphs in general. Two additional papers concerned "fan-planar graphs", graphs drawn so that, when an edge is crossed, its crossing edges form a fan (adjacent on the same side to a single vertex). Apparently there is some connection between these graphs and confluent drawings, although I'm a little fuzzy on the details.<br /><br />Thursday afternoon, in a session including one of my papers on (impractical) algorithms for crossing minimization, we also had a user study presented by Sergei Pupyrev (with Kobourov and Saket) showing that for larger graphs <a href="https://cs.arizona.edu/people/kobourov/crossings.pdf">the number of crossings does not seem to be strongly correlated with usability of the drawing.</a> This is in contrast with many past works showing that for smaller graphs crossings are quite important. I'm always glad to see research in this style at GD – I think it helps keep us grounded, when otherwise we have a strong tendency to do theory that has no practical applications – but I can see why it is a bit rare: most of the questions afterwards were actually poorly-disguised complaints "you're doing it wrong". There's always something you could have done differently that might have shown a different result, and always someone who has strong opinions on that particular detail. So it's difficult to get such research accepted and the reactions can be a bit discouraging when it is.<br /><br />Friday morning Therese Biedl gave another entertaining talk, on transforming one style of drawing into another while preserving its height. But the moral of the story is that height is probably the wrong thing to optimize, at least for straight-line drawings, because of one of her examples: a planar 3-tree which could be drawn in linear area with height five, but required exponential area for its optimal height-four drawing.<br /><br />And finally one of the Friday afternoon talks also particularly caught my attention. Fabrizio Frati spoke on his work with Dehkordi and Gudmundsson on <a href="http://arxiv.org/abs/1408.2592">"increasing-chord graphs"</a>, a strengthening of greedy drawings. A path from s to t is greedy if the distance to t monotonically decreases along the path. It is self-approaching if every subpath is greedy in the same direction. And it is increasing-chord if it is self-approaching in both directions. It is unknown whether every point set in the plane forms the vertex set of a planar increasing-chord graph, but Frati and his co-authors showed that every point set can be augmented by a linear number of points so that the augmented set supports such a graph. The proof has two parts: a proof that a triangulation with all acute angles is always increasing-chord, and <a href="http://www.ics.uci.edu/~eppstein/pubs/p-pgood.html">an old result of mine</a> that every point set has an all-acute Steiner triangulation.<br /><br />In organizational news: We are soon to have elections for new GD steering committee members, as four members' terms are expiring (Ulrik Brandes, Seok-Hee Hong, Michael Kaufmann, and me). Next year, GD (renamed as the "International Symposium on Graph Drawing and Network Visualization" but with the same numbering and acronym) will be in Northridge, California, with Csaba Tóth organizing. (Csaba looked into instead having it at a nearby beach town but they were too expensive.) After dropping short submissions as a category this year (because of problems in past years where they were judged head-to-head against longer submissions and, unsurprisingly, lost) we will reinstate them next year as "GD notes and demos" with separate reviewing and shorter talks. Along with this year's best poster and best presentation awards, we are reinstating the best paper and test of time awards previously given in 2012; the test of time one will be for papers published in GD between 1994 and 1998 (inclusive). GD 2016 will be in Athens, Greece.<br /><br />Much German food and wine was served (this is in Franconia, one of Germany's wine-making areas and the home of drier wines than the other parts of Germany), old friends seen again, etc. I enjoyed my trip and am looking forward to a more conveniently-located GD next year.<br /><br /><sup>*</sup>Missed a connection and have to spend the night in Newark.<a name='cutid1-end'></a>http://11011110.livejournal.com/297186.htmlgraph drawingconferencestalkspaperspublic2http://11011110.livejournal.com/296778.htmlSun, 21 Sep 2014 01:05:13 GMTWhich polycubes have planar graphs?
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Joe Malkevitch recently asked me: which <a href="https://en.wikipedia.org/wiki/Polycube">polycubes</a> have planar graphs?<br /><br />By a polycube, I mean a set of unit cubes in the three-dimensional integer lattice whose dual graph (with vertices for cubes and edges for cubes that share a square with each other) is connected; Malkevitch had a more restrictive definition in mind in which the boundary of the union of the cubes is a connected manifold, but that turns out not to make a difference for this problem. The graph of a polycube is formed by the vertices and edges of the cubes in this set.<br /><br />The answer turns out to be: the polycubes that have a tree-like structure that can be formed by adding cubes one at a time, at each step adding a cube that touches the previous cubes in exactly one of its squares. One direction is easy: if you add cubes in this way, you end up changing the graph by replacing a quadrilateral face by five quadrilaterals, which preserves planarity.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/polycube-gluing-step.png"></div><br /><br />In the other direction, suppose you have a polycube that cannot be constructed in this way, and build it up by adding cubes one at a time in a preorder traversal of a spanning tree of the dual graph. At some point, you will add a cube <i>c</i> that is adjacent not only to its parent <i>p</i> in the tree, but to some other part of the already-constructed polycube. If you contract the parts of <i>c</i> and <i>p</i> that are not on their shared square, and contract the path in the polycube connecting those parts, you can get a <i>K</i><sub>3,3</sub> graph minor out of the graph of the polycube, showing that it must be nonplanar.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/loopy-polycubes-are-nonplanar.png"></div><br /><br />The same question is also interesting when you ask about the graph of the boundary of the polycube, but it has a simpler answer: if the boundary is connected, then it is planar if and only if the polycube is a topological ball.<a name='cutid1-end'></a>http://11011110.livejournal.com/296778.htmlmedia theorygraph theorypublic1http://11011110.livejournal.com/296537.htmlTue, 16 Sep 2014 06:20:03 GMTLinkage
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<ul><li><a href="http://www.thisiscolossal.com/2014/08/thomas-herbrich-smoke/">Unexpected shapes in smoke plumes</a>, as photographed by Thomas Herbrich (<a href="https://plus.google.com/100003628603413742554/posts/6CADmTYFxq3">G+</a>)</li><br /><li><a href="http://tcs.postech.ac.kr/isaac2014/accepted_papers.html">ISAAC 2014</a> and <a href="http://theory.utdallas.edu/COCOA2014/accepted-papers.html">COCOA 2014</a> accepted paper lists (<a href="https://plus.google.com/100003628603413742554/posts/BrPabmYReu4">G+</a>)</li><br /><li><a href="http://windowsontheory.org/2014/09/03/focs-2014-program-is-online/">FOCS 2014 program</a> and best paper winners (<a href="https://plus.google.com/100003628603413742554/posts/LnC49jZzLgt">G+</a>)</li><br /><li><a href="http://www.wired.com/2014/08/watch-804-wooden-balls-shape-shift-into-a-perfect-spiral/">Kinetic sculpture</a> made of wooden balls on threads, with some extensive software simulation behind its design (<a href="https://plus.google.com/100003628603413742554/posts/TBadJvQDmeY">G+</a>)</li><br /><li><a href="http://www.wired.com/2014/09/curvature-and-strength-empzeal/">How a 19th century math genius taught us the best way to hold a pizza slice</a>, or, a practical application of the theorem that when a flat surface is embedded in 3d, it remains flat in at least one direction (<a href="https://plus.google.com/100003628603413742554/posts/8MN9K3iXV7Z">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Centered_octahedral_number">Centered octahedral numbers</a> on Wikipedia (<a href="https://plus.google.com/100003628603413742554/posts/VHQFZG3ztR3">G+</a>)</li><br /><li><a href="http://aperiodical.com/2014/09/interesting-esoterica-summation-volume-9/">Interesting esoterica summation</a> (<a href="https://plus.google.com/100003628603413742554/posts/8Rx8UcAjksz">G+</a>)</li><br /><li><a href="http://dx.doi.org/10.1007/s00454-014-9627-0">A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing</a> (journal version of two of my old conference papers; <a href="https://plus.google.com/100003628603413742554/posts/WniiFV3VREv">G+</a>)</li><br /><li><a href="http://news.sciencemag.org/people-events/2014/09/researcher-loses-job-nsf-after-government-questions-her-role-1980s-activist">Researcher loses job at NSF after government questions her role as 1980s activist</a> (<a href="https://plus.google.com/100003628603413742554/posts/EAxPtzQiusE">G+</a>)</li><br /><li><a href="http://infosthetics.com/archives/2014/09/pi_visualized_as_a_public_urban_art_mural.html">Pi visualized as a public urban art mural</a> (<a href="https://plus.google.com/100003628603413742554/posts/K3jpjjv9ypa">G+</a>)</li><br /><li><a href="http://igorpak.wordpress.com/2014/09/12/how-not-to-reference-papers/">How not to reference papers</a> (a sad story by Igor Pak of academic misattribution; <a href="https://plus.google.com/100003628603413742554/posts/XN5S8t9d8QV">G+</a>)</li><br /><li><a href="http://jocg.org/v5n1p10/">Steinitz Theorems for Simple Orthogonal Polyhedra</a> (journal version of another of my papers; <a href="https://plus.google.com/100003628603413742554/posts/2Nz66ruDFHu">G+</a>)</li><br /><li><a href="http://sbseminar.wordpress.com/2014/09/14/editorial-board-of-journal-of-k-theory-on-strike-demanding-tony-bak-hands-over-the-journal-to-the-k-theory-foundation/">Editorial board of <i>Journal of K-theory</i> goes on strike</a> over publisher profiteering (<a href="https://plus.google.com/100003628603413742554/posts/g1MfZRTPDf2">G+</a>)</li></ul>http://11011110.livejournal.com/296537.htmlbloggingacademiageometryartpaperspublic0http://11011110.livejournal.com/296235.htmlSat, 13 Sep 2014 21:57:06 GMTBren Hall, East Stairs
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Just some test shots with my new travel lens (Canon's 17-40/F4 L, replacing a mysteriously nonfunctional and optically not as good 17-85IS).<br /><br /><div align="center"><table border="0" cellpadding="10">
<tr align="center" valign="middle">
<td><a href="http://www.ics.uci.edu/~eppstein/pix/brenstairs/1.html"><img src="http://www.ics.uci.edu/~eppstein/pix/brenstairs/1-m.jpg" border="2" width="160" height="240" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/brenstairs/2.html"><img src="http://www.ics.uci.edu/~eppstein/pix/brenstairs/2-m.jpg" border="2" width="240" height="160" style="border-color:black;" /></a></td>
</tr><tr align="center" valign="middle">
<td><a href="http://www.ics.uci.edu/~eppstein/pix/brenstairs/3.html"><img src="http://www.ics.uci.edu/~eppstein/pix/brenstairs/3-m.jpg" border="2" width="240" height="160" style="border-color:black;" /></a></td>
<td><a href="http://www.ics.uci.edu/~eppstein/pix/brenstairs/4.html"><img src="http://www.ics.uci.edu/~eppstein/pix/brenstairs/4-m.jpg" border="2" width="160" height="240" style="border-color:black;" /></a></td>
</tr></table></div>http://11011110.livejournal.com/296235.htmlarchitectureuciphotographypublic0http://11011110.livejournal.com/295979.htmlWed, 10 Sep 2014 05:15:24 GMTAlgorithmic representative democracy
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Did you ever wonder why different states of the US have the numbers of representatives in congress that they do? It's supposed to be proportional to population but that's not actually true: for instance the ratio of representatives to population is about 40% higher in Montana than California. <a href="https://en.wikipedia.org/wiki/United_States_congressional_apportionment#Apportionment_methods">What formula or algorithm do they use to pick the numbers?</a><br /><br />This has varied over the years but, Wikipedia tells me, currently it's the <a href="https://en.wikipedia.org/wiki/Huntington%E2%80%93Hill_method">Huntington–Hill method</a>. One way of describing this is by a simple but inefficient algorithm, with the following steps:<br /><br />1. Give each state a representative, since they all have to have at least one.<br /><br />2. Repeatedly, until there are the desired number of total representatives, prioritize the states by <img src="http://www.ics.uci.edu/~eppstein/0xDE/HuntingtonHill.gif" valign="middle"> and give one more representative to the state with the biggest priority.<br /><br />The problem of assigning seats to parties after a parliamentary election is very similar, using votes instead of population, but in that case it's ok for some parties to get zero seats. This causes the formulas that are used for prioritizing the parties to use different divisors, typically linear functions of the number of seats already assigned rather than the square root thing used here. This general type of apportionment method is called a <a href="https://en.wikipedia.org/wiki/Highest_averages_method">highest averages method</a>.<br /><br />The question asked by my most recent preprint (<a href="http://arxiv.org/abs/1409.2603">"Linear-time Algorithms for Proportional Apportionment", arXiv:1409.2603</a>, with Jack Cheng, to appear at ISAAC 2014) is: how quickly can you assign seats using these methods? Using the procedure described above, with a priority queue to do the prioritization, would take an amount of time slightly superlinear in the number of seats. But it turns out we can do quite a bit better: linear in the number of parties or states getting the seats. Probably this doesn't matter for actual elections, the slow part of which is collecting all the votes. But it might be useful if you want to run a lot of simulated elections, or to use apportionment algorithms for problems where the number of things being apportioned is much larger than the number of congressional representatives.<br /><br />There's also a nice way of viewing these problems more abstractly: suppose we have <i>n</i> different infinite arithmetic progressions, and an input parameter <i>k</i> How quickly can we find the <i>k</i>th smallest value in the disjoint union of the progressions? Answer: O(<i>n</i>) arithmetic operations, independently of <i>k</i>. For the parliamentary apportionment problem, you get these sequences by turning the priorities upside down, with the linear function of the number of representatives as the numerator and the number of votes as the denominator. For the congressional problem, this gives something that is not exactly an arithmetic progression, but it's close enough to one that the same algorithms work with only minor modification.<br /><br />Incidentally, there's a footnote on p. 3 of the preprint about two seemingly very relevant references, in Japanese, whose titles claim that they give linear time algorithms for related problems. Unfortunately despite attempts to contact both the authors of these references and the reviewer who used them as a reason to downvote our paper, we have been unable to obtain them <s>nor even to verify that they actually exist,</s> let alone to determine which variable their time is linear in and which apportionment methods they apply to. So we don't actually know whether our algorithms or results are really new. If anyone reading this has better access to these sources, we'd appreciate any help you could give us. ETA: I now have a copy of the IEICE Trans. D one, but haven't yet examined it. Apparently part of the difficulty is that there are two different IEICE Trans. D.'s.<a name='cutid1-end'></a>http://11011110.livejournal.com/295979.htmlvotingpaperspublic1http://11011110.livejournal.com/295706.htmlSat, 06 Sep 2014 06:47:45 GMTEfficiency of Rado graph representations
http://11011110.livejournal.com/295706.html
The <a href="https://en.wikipedia.org/wiki/Rado_graph">Rado graph</a> has interesting symmetry properties and plays an important role in the <a href="http://11011110.livejournal.com/295010.html">logic of graphs</a>. But it's an infinite graph, so how can we say anything about the complexity of algorithms on it?<br /><br />There are algorithmic problems that involve this graph and are independent of any representation of it, such as checking whether a first-order logic sentence is true of it (PSPACE-complete). But I'm interested here in problems involving the Rado graph where different ways of constructing and representing the graph lead to different algorithmic behavior. For instance: the Rado graph contains all finite graphs as induced subgraphs. How hard is it to find a given finite graph? Answer: it depends on how the Rado graph is represented.<br /><br />Historically the first way of constructing this graph involved the binary representations of the natural numbers: each vertex corresponds to a number, and vertices x and y are adjacent when the smaller of the two numbers is the index of a nonzero bit in the binary representation of the larger of the two numbers. For this representation it's very easy to find a copy of any given graph G as an induced subgraph: just find copies of the vertices of G one at a time. Each of the vertices you've already found, and its adjacency or nonadjacency to the next vertex, tells you one bit of the binary representation of the next number, and you just have to pad those bits with zeros in the remaining places to find the next vertex. The trouble is, these numbers grow very quickly, roughly as a tower of powers of two whose number of levels is the number of vertices in the graph. For instance, if you want to find an 5-vertex complete subgraph of the Rado graph, you can do it with the numbers 0, 1, 3, 11, 2059, but (according to <a href="https://oeis.org/A034797">OEIS A034797</a>) the smallest number you can use to extend this to a 6-vertex clique already has 620 decimal digits. And the one after that has more like 2^{2059} bits in its binary representation, too many to write down even in the biggest computers. So the algorithm for finding a given graph is easy to describe but not very efficient.<br /><br />An alternative construction just chooses randomly, for each pair of vertices, whether they form the endpoints of an edge. With infinitely many vertices, the result of these random choices is almost certainly the Rado graph. That's not a representation that can be used in a computer, but we could imagine an algorithm that had access to it as some sort of oracle. With this representation, an n-vertex graph should occur much more quickly: if G is such a graph, then the expected number of copies of G among the first N vertices of the Rado graph starts getting large when N is roughly 2^{n/2}. And that's the best you could hope for in any representation, because with fewer vertices there aren't enough n-tuples of vertices to cover all the different induced subgraphs that could exist. But finding a copy of G among these vertices would be difficult. Even for finding a clique, we don't know anything much better than trying all n-tuples of vertices and seeing which ones work. (Finding a clique of size approximately 2 log_2 N in an N-vertex random graph in polynomial time is a well known open problem even though we know that a clique of that size should usually exist.) Yet another construction of the Rado graph, based on the idea of Paley graphs, probably behaves similarly to the random construction but is difficult to prove much about.<br /><br />Here's a construction of an infinite graph in which induced subgraphs of any type are easy to find: instead of using binary numbers, use binary strings, of all possible lengths including zero. For any two strings s and t, connect them by an edge if s is shorter than t and the position of t indexed by the length of s is nonzero (or vice versa). Then you can build an n-vertex graph one vertex at a time, by using one bitstring of each length less than n, with the bits in each string given by the adjacencies to the earlier vertices with no padding. The copy of an n-vertex graph G will be somewhere in the first 2^n vertices (not 2^{n/2}), and the names of these vertices can be calculated and written down by an algorithm in time O(n^2) (matching the description complexity of G in terms of its adjacency matrix). But this is not the Rado graph. For instance, for the two binary strings "0" and "1", there is only one vertex in the graph adjacent to one and not the other (the empty string) whereas the Rado graph has infinitely many such vertices. One could construct a copy of the Rado graph by interspersing this construction with a very small number of random vertices, small enough that they don't affect the complexity of this subgraph-finding algorithm, but that seems a bit of a cheat.<br /><br />One way that it's a cheat is that it doesn't use the full power of the Rado graph. The actual defining property of the Rado graph is that if you start building a given induced subgraph, vertex by vertex, you can never make a mistake: it's always possible to add one more vertex. Or, more abstractly, if you have any two sets A and B of vertices in the Rado graph, there's always another vertex v that's adjacent to everything in A and nothing in B. By choosing A to be the set of already-placed vertices that are adjacent to the next vertex, and B to be the set of already-placed vertices that are not adjacent to the next vertex, you can use this property to find each successive vertex in an arbitrary induced subgraph. The graph of binary strings described above does not have this property, because when A={"0"} and B={"1"} there's no vertex v that matches.<br /><br />Is it possible to construct the Rado graph in such a way that the extension property becomes as easy as the subgraph property was for the graph of binary strings? The short answer is that I don't know. One attempt at an answer would be to build it in levels, much like the binary string graph can be divided into levels by the length of its strings. In the kth level, we include a collection of vertices that extends all subsets of k vertices from the previous level. But what is this collection? If there are N vertices in the previous level, then the vertices of the kth level can be described by N-bit bitstrings specifying their adjacencies. We want to choose as small as possible a set of bitstrings with the property that all k-tuples of previous vertices can be extended; a more geometric way to describe this is that we want to find a small set D of points in the N-dimensional hypercube that hits every (N-k)-dimensional subcube. Exactly this problem was one of the ones I studied in my recent paper "<a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p20">grid minors in damaged grids</a>". But the proof in that paper that D can be relatively small (Theorem 14) uses the probabilistic method, meaning essentially that it chooses a random set of the right number of hypercube points. So as a way of constructing Rado graphs in which the extension property is efficient, it is not an improvement over the method of choosing edges randomly. But maybe this nondeterministic proof that a good set exists can lead the way to a deterministic and efficient construction?<a name='cutid1-end'></a>http://11011110.livejournal.com/295706.htmlgraph algorithmspublic2http://11011110.livejournal.com/295544.htmlMon, 01 Sep 2014 05:59:14 GMTLinkage
http://11011110.livejournal.com/295544.html
More Google+ links from the last couple of weeks:<br /><ul><li>An interview with Haida artist <a href="https://en.wikipedia.org/wiki/Jim_Hart_%28artist%29">Jim Hart</a> (<a href="https://plus.google.com/100003628603413742554/posts/dC2GkH8wQVK">G+):<br /><lj-embed id="52" /></li><br /><li><a href="http://chance.amstat.org/2012/11/interview-with-persi-diaconis/">Persi Diaconis discusses mathematics and magic</a> (<a href="https://plus.google.com/100003628603413742554/posts/dXapc9JXucR">G+</a>)</li><br /><li><a href="http://cstheory.stackexchange.com/questions/25509/edit-distance-in-sublinear-space">A still-unsolved question about whether it's possible to compute edit distance in sublinear space and polynomial time</a> (<a href="https://plus.google.com/100003628603413742554/posts/6qtvh6gAJht">G+</a>)</li><br /><li><a href="http://www.nytimes.com/interactive/2014/08/13/us/starbucks-workers-scheduling-hours.html">A New York Times story about how scheduling software makes part-time workers' lives harder</a>. Or does it? The <a href="http://www.metafilter.com/141920/Working-anything-but-9-to-5">MF discussion of the article</a> makes it clear that managers have been doing the same things with lower tech for a long time. (<a href="https://plus.google.com/100003628603413742554/posts/8PmLWMEpyAM">G+</a>)</li><br /><li><a href="http://makingsocg.wordpress.com/2014/08/22/colocation-with-stoc-2016/">Kerfuffle over SoCG colocation with STOC</a>, later <a href="http://makingsocg.wordpress.com/2014/08/27/mea-culpa-and-good-news/">resolved</a> (<a href="https://plus.google.com/100003628603413742554/posts/VWq1DHWHwW6">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Barrier_resilience">Barrier resilience</a> on Wikipedia (<a href="https://plus.google.com/100003628603413742554/posts/8B1xmfEEzxb">G+</a>)</li><br /><li><a href="http://www.ted.com/talks/robert_lang_folds_way_new_origami">Robert Lang talks</a> about the way mathematics done purely for its aesthetic value (in this case mathematical origami) can turn around and have practical applications. (<a href="https://plus.google.com/100003628603413742554/posts/4iWzqDnwNDm">G+</a>)</li><br /><li><a href="http://www.newyorker.com/magazine/2014/09/01/troll-slayer">The Troll Slayer</a>. New Yorker profile of classics professor Mary Beard, who knows better than most exactly how long men have been silencing women. (<a href="https://plus.google.com/100003628603413742554/posts/aqEde6zwK7Z">G+</a>)</li><br /><li><a href="http://www.kokomotribune.com/news/nation_world_news/article_7028faa8-2d42-11e4-b5a1-0019bb2963f4.html">A study on how social media causes us to self-censor our opinions</a> (<a href="https://plus.google.com/100003628603413742554/posts/X1ZH2ZemiiM">G+</a>)</li><br /><li><a href="http://www.ocregister.com/articles/internet-633032-jordan-policy.html">My UCI colleague Scott Jordan takes a position advising the FCC about net neutrality</a> (<a href="https://plus.google.com/100003628603413742554/posts/WDA8kJfvrq4">G+</a>)</li><br /><li><a href="http://zacharyabel.com/sculpture/">Sculpture by Zachary Abel</a>, one of my new co-authors on the <a href="http://11011110.livejournal.com/295261.html">flat-folding paper</a> (<a href="https://plus.google.com/100003628603413742554/posts/LEs7sm7BVcy">G+</a>)</li></ul></a>http://11011110.livejournal.com/295544.htmlunsolvedbloggingcorporatizationmagicacademiaartorigamipublic2