Jekyll2017-09-20T20:58:16+00:00https://11011110.github.io/blog/11011110Geometry, graphs, algorithms, and moreDavid EppsteinGraphs with sparse crossings2017-09-19T23:07:00+00:002017-09-19T23:07:00+00:00https://11011110.github.io/blog/2017/09/19/graphs-with-sparse<p><a href="http://sigspatial2017.sigspatial.org/">ACM SIGSPATIAL 2017</a> (to be held in Southern California in November) doesn’t yet seem to have posted a list of accepted papers, but two of them are mine (one long, one short).
I already posted about the short one, “<a href="/blog/2017/06/29/stable-redistricting-in.html">Defining Equitable Geographic Districts in Road Networks via Stable Matching</a>”, so let me just say a little about the other one, now up on the arXiv: “Crossing Patterns in Nonplanar Road Networks” (with UCI grad student Sid Gupta, <a href="https://arxiv.org/abs/1709.06113">arXiv:1709.06113</a>).</p>
<p>To begin with, road networks are graphs with a vertex at each intersection of roads and an edge for each segment of road between the intersections. We tend to think of them as being planar graphs, but they’re not. Road segments cross each other, often without an intersection, when one segment is part of an overpass, underpass, or tunnel. Additionally, some pairs of road segments can look like they’re crossing in our data, even when they don’t cross in real life, because the data makes a segment of road look straight when actually it bends around the end of another road.</p>
<p style="text-align:center"><img src="/blog/assets/2017/HighFive.jpg" alt="High Five Interchange at the intersection of I-635 and U.S. Route 75 in Dallas, Texas, looking towards the southwest" title="cropped from https://commons.wikimedia.org/wiki/File:High_Five.jpg by fatguyinalittlecoat on flickr, under a CC-BY 2.0 license" /></p>
<p>But it would be nice if these graphs were planar (even if they aren’t), because then we could apply all the fast algorithms researchers have developed for planar graphs. Many of these algorithms don’t depend on the detailed properties of planarity, but only on the <a href="https://en.wikipedia.org/wiki/Planar_separator_theorem">planar separator theorem</a>. So it would be nice if these graphs obeyed a similar separator theorem.</p>
<p>And they do! Or at least, the graphs that fit the model of nonplanar road networks that we use in our new paper do have good separators. Our idea is that most of the time when roads cross they do so in isolation; a crossing road segment would often only cross one other road segment. If that were true everywhere, we’d get the <a href="https://en.wikipedia.org/wiki/1-planar_graph">1-planar graphs</a>, but there are a few segments that cross multiple others (think of the long tunnels under Boston, or complicated freeway interchanges such as the one in Dallas above). Even in those cases, though, the collection of road segments that cross each other has a restricted patterns. In a long tunnel, for instance, a single segment may cross many others, but each of those other crossed segments will typically be much shorter and have no other crossings.</p>
<p>We can understand these crossing patterns more clearly by drawing another graph, the <em>crossing graph</em>, which has a vertex for each road segment and an edge connecting it to each other road segment that it crosses. Most vertices are isolated (they don’t cross anything) and most edges are isolated (they connect two road segments that cross each other and nothing else). There are occasional larger connected components, but these still have a simple structure. We thought going into this research that they would all be trees; that’s not true, but they do all have low <a href="https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)">degeneracy</a>. That means, in any cluster of crossing road segments, at least one of the segments is crossed only a small number of times. Remove that segment, and the remaining segments still have one that’s crossed only few times, and so on.
We tested the degeneracy of the road networks of many cities, and we never saw a number higher than 6 (in Buenos Aires). For most cities the number was even smaller, 3 or 4. And the number of non-tree clusters of crossing road segments was always a small fraction of all crossings. We tried including only the crossings we new were real (by using the identification of some road segments as bridges or tunnels in the data we used) or looking at all the crossings, even the artificial ones caused by differences between the straight segments of the data and the curvature of the actual road segments; it didn’t make much difference to our overall results.</p>
<p>We also show (by using the <a href="https://en.wikipedia.org/wiki/Crossing_number_inequality">crossing number inequality</a>) that road networks with <script type="math/tex">d</script>-degenerate crossing graphs are sparse (they have <script type="math/tex">O(nd^{1/2})</script> edges and <script type="math/tex">O(nd^{3/2})</script> crossings). It follows that they have small separators, inherited from the separators of their <a href="/blog/2017/08/17/two-new-preprints.html">planarizations</a>. So I think the graphs with degenerate crossing graphs make a good model to look for algorithms on: not so restrictive (like planar or 1-planar graphs) that it eliminates the real-world graphs we’re trying to model, but restrictive enough to have properties useful in algorithm design.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/b3Htx5KvtY4">G+</a>)</p>David EppsteinACM SIGSPATIAL 2017 (to be held in Southern California in November) doesn’t yet seem to have posted a list of accepted papers, but two of them are mine (one long, one short). I already posted about the short one, “Defining Equitable Geographic Districts in Road Networks via Stable Matching”, so let me just say a little about the other one, now up on the arXiv: “Crossing Patterns in Nonplanar Road Networks” (with UCI grad student Sid Gupta, arXiv:1709.06113).Linkage for the end of summer2017-09-15T20:57:00+00:002017-09-15T20:57:00+00:00https://11011110.github.io/blog/2017/09/15/linkage-end-summer<p>Fewer links than usual this time because of all my Japan posts.</p>
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<p><a href="https://oeis.org/A290966">The number of convex layers in a square grid</a> (<a href="https://plus.google.com/100003628603413742554/posts/32p3w4KWLXn">G+</a>). A new entry in the Online Encyclopedia of Integer Sequences.</p>
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<p><a href="http://2017.cccg.ca/program.html">The program from the Canadian Conference on Computational Geometry</a> (<a href="https://plus.google.com/100003628603413742554/posts/7CZF94dmRVS">G+</a>), now with video links for most of the talks.</p>
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<p><a href="https://algo2017.ac.tuwien.ac.at/awards/">Awards from ALGO 2017</a> (<a href="https://plus.google.com/100003628603413742554/posts/LFbnwZzjCqx">G+</a>), including the Nerode Prize to Fomin, Grandoni, and Kratsch for “measure and conquer”.</p>
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<p><a href="https://mathoverflow.net/q/280493/440">Perimeter-halving centers</a> (<a href="https://plus.google.com/100003628603413742554/posts/fgza6Avgkaw">G+</a>), a possibly-new triangle center defined by Joe O’Rourke based on distance to the nearest chord that bisects the perimeter.</p>
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<p><a href="https://www.pri.org/stories/2017-07-27/team-women-are-unearthing-forgotten-legacy-harvard-s-women-computers">The forgotten history of the Harvard Observatory computers</a> (<a href="https://plus.google.com/100003628603413742554/posts/VehtcPbKS81">G+</a>). “In the late 1800s, they were famous, only to be virtually forgotten during the next century. A recent discovery of thousands of pages of their calculations by a modern group of women has spurred new interest in their legacy.”</p>
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<p><a href="https://elifesciences.org/articles/27725">The readability of scientific texts is decreasing over time</a> according to a new research report (<a href="https://plus.google.com/100003628603413742554/posts/crBrYK9kVLz">G+</a>). Many scientists could do much more to make their work readable. But in the G+ comments, Mark Wilson points out that <a href="http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004205">they might not have much incentive to do so</a>.</p>
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<p><a href="https://en.wikipedia.org/wiki/Kawasaki%27s_theorem">Kawasaki’s theorem</a> characterizing flat-foldable single-vertex origami patterns (<a href="https://plus.google.com/100003628603413742554/posts/G4DVJfaTGB2">G+</a>), now a Good Article on Wikipedia.</p>
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<p><a href="https://arxiv.org/abs/1709.04228">The complete Graph Drawing proceedings, now on arXiv</a> (<a href="https://plus.google.com/100003628603413742554/posts/arRZovJMvnv">G+</a>). See G+ for discussion of why we still might need conventional publishers.</p>
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<p><a href="https://cacm.acm.org/magazines/2017/9/220429-divination-by-program-committee/fulltext">Divination by program committee</a> (<a href="https://plus.google.com/100003628603413742554/posts/UebF5u6go7X">G+</a>). As Moshe Vardi points out, after picking off the easy accepts and easy rejects, the remaining selections of conference submission acceptances are essentially random. So why don’t we save a lot of effort and produce a more scalable system of fewer bigger conferences by just taking them all?</p>
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<p><a href="http://www.thisiscolossal.com/2017/09/a-towering-4-story-organic-structure-built-from-material-as-thin-as-a-coin-by/">Minima Maxima by Marc Fornes</a> (<a href="https://plus.google.com/100003628603413742554/posts/1TyZtnNyQ6F">G+</a>), a tall self-supporting structure resembling a minimal surface built out of thin layers of aluminum composite.</p>
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</ul>David EppsteinFewer links than usual this time because of all my Japan posts.Miscellaneous photos from Tokyo2017-09-10T16:20:00+00:002017-09-10T16:20:00+00:00https://11011110.github.io/blog/2017/09/10/miscellaneous-photos-tokyo<p><a href="http://www.ics.uci.edu/~eppstein/pix/tokyomisc/">Here are my remaining photos from Tokyo</a>, the ones that didn’t fit into large enough thematic groups to post separately.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/tokyomisc/NationalArtCenter3-m.jpg" alt="National Art Center, Tokyo" style="border-style:solid;border-color:black;" /></p>
<p>This is the National Art Center, the building that housed both the <a href="/blog/2017/09/05/sculpture-in-tokyo.html">Giacometti</a> and half of the <a href="/blog/2017/09/07/sunshower-contemporary-art.html">Sunshower</a> exhibits. Later from the Tokyo City View we could see that this curvy glass-lined shape is only the front facade: the back side of the building, where all the exhibits are held, is a more conventional rectangular box. From here we walked to the Suntory Museum, which held a small curated set of Japanese arts and crafts with family-friendly exhibit design, including an amusing <a href="https://www.google.com/culturalinstitute/beta/asset/EQEeEDqgJWR29A?hl=en">16th century scroll depicting the story of Gon-no-kami, a mouse who marries a human woman</a>. Near the Suntory we had a very nice set lunch at <a href="http://www.tokyoweekender.com/2017/07/a-pricey-ingredient-takes-center-stage-at-this-midtown-establishment/">Artisan de la Truffe</a>.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/tokyomisc/JubanInariTreasureBoat-m.jpg" alt="Seven lucky gods in a treasure boat, Juban-Inari shrine, Tokyo" style="border-style:solid;border-color:black;" /></p>
<p>It is only by accident that we encountered the <a href="https://en.wikipedia.org/wiki/Seven_Lucky_Gods">Seven Lucky Gods</a> and <a href="http://todayintokyo.tumblr.com/post/122776275841/j%C5%ABban-inari-jinja-%E5%8D%81%E7%95%AA%E7%A8%B2%E8%8D%B7-%E7%A5%9E%E7%A4%BE-has-two-famous-frog">firefighting frog</a> at the Jūban Inari Shrine. We had been to the Mori Museum in Roppongi Hills for the other half of Sunshower, and were trying to return home to our hotel in <a href="/blog/2017/09/03/kagurazaka.html">Kagurazaka</a>. I think the easiest (if perhaps not fastest) route would have been to take the Toei-Oedo line from Roppongi to Iidabashi, with a lot of stops but not much walking or even any transfers. But for some reason the navigation software we tried refused to show us that route, instead directing us to walk a kilometer or so to a farther station in <a href="https://en.wikipedia.org/wiki/Azabu-J%C5%ABban">Azabu-Jūban</a>, past this shrine.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/tokyomisc/Univac120-m.jpg" alt="Univac 120 at the Museum of Science, Tokyo University of Science" style="border-style:solid;border-color:black;" /></p>
<p>This array of vacuum tubes is part of a <a href="https://en.wikipedia.org/wiki/Remington_Rand_409">Univac 120</a>. It’s part of an impressive collection of old calculating devices at the Museum of Science on the Tokyo University of Science campus. Jin Akiyama’s Math Experience Plaza is downstairs, accessed by an exterior stair in front of the entrance to the Museum. Unfortunately I couldn’t take photos there but it has a big collection of mathematical models of concepts including <a href="https://en.wikipedia.org/wiki/Brachistochrone_curve">descent curves</a>, hyperboloid gears, polyhedral dissections, and <a href="https://en.wikipedia.org/wiki/Category:Space-filling_polyhedra">space-filling polyhedra</a>.</p>
<p>As usual, follow the link at the top of the post for the rest of the photos.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/cuFE6SKBYVV">G+</a>)</p>David EppsteinHere are my remaining photos from Tokyo, the ones that didn’t fit into large enough thematic groups to post separately.Report from JCDCG³2017-09-09T15:53:00+00:002017-09-09T15:53:00+00:00https://11011110.github.io/blog/2017/09/09/report-from-jcdcg3<p>I suppose I should write down my recollections of <a href="http://www.jcdcgg.u-tokai.ac.jp/">JCDCG<sup>3</sup></a> before they fade too badly. It was only my second conference in Japan, and the first one was nearly 20 years earlier (PARAOPT V, at the invitation I think of Naoki Katoh). This time, the invitation was by Hiro Ito, but unfortunately he fell ill and couldn’t attend. Instead, the position of host was capably handled by Jin Akiyama. Here are Jin and Naoki with me and my wife Diana, photographed by Toshinori Sakai (who despite being behind the lens for most of the conference, mysteriously appears in many past conference photos):</p>
<p style="text-align:center"><img src="/blog/assets/2017/DavidDianaJinNaoki-m.jpg" alt="David Eppstein, Diana Eppstein, Jin Akiyama, and Naoki Katoh, at the banquet of JCGCG3, Agnes Hotel, Tokyo; photo by Toshinori Sakai" style="border-style:solid;border-color:black;" /></p>
<p>This was the 20th anniversary of the conference, and was dedicated in celebration of five of its regular contributors: Jin Akiyama, Vašek Chvátal, Mikio Kano, János Pach, and Jorge Urrutia. Along with those five, there were four more invited plenary speakers (Erik Demaine, Naoki Katoh, Evangelos Kranakis, and myself), and four days of double-session contributed talks.
There’s a 15Mb pdf <a href="http://www.jcdcgg.u-tokai.ac.jp/JCDCG3_2017_abstracts.pdf">collection of abstracts</a> online, but the booklet they handed out at the conference was, instead, a 75-page illustrated history of the conference (which doesn’t seem to be online). As usual the full proceedings will be published later.</p>
<p>I attended most of the conference (missing one early morning session) but that still means I saw fewer than half of the contributed talks. Some of the highlights for me:</p>
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<p>I thought almost everything was known about how many guards are needed in most natural variants of the <a href="https://en.wikipedia.org/wiki/Art_gallery_problem">art gallery problem</a>. But in the opening invited talk, Jorge Urrutia mentioned an open problem that was new to me: suppose you want to guard a simple orthogonal polyhedron by as few edge guards as possible. That is, you need to choose as few edges of the polyhedron as possible so that every point of the polygon is visible from one of the edges through the polyhedron interior. One way to do this is to consider the cross-sections parallel to one of the three coordinate planes (orthogonal polygons). From any point within any cross-section, look north, then sweep your gaze clockwise; you will either run into the east end of a south-facing wall or the north end of a west-facing wall. That is, the set of east ends of south walls and north ends of west walls is unavoidable; every point is visible from one such corner. In the same way you can find three other unavoidable sets of corners, such that each corner of the cross-section belongs to two of the four unavoidable sets. Doing the same thing for the two other coordinate planes generates a total of 12 unavoidable sets of edges of the polyhedron, with each edge belonging to two sets, so one of them gives a guard set with at most <script type="math/tex">E/6</script> guards where <script type="math/tex">E</script> is the total number of edges. Very recently, Urrutia has managed to improve this slightly, to <script type="math/tex">11E/72.</script> But the only known lower bound is <script type="math/tex">E/12-O(1),</script> from a big cube with many small cubical bumps on one of its faces, so there’s still a big gap to close.</p>
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<p>In my own talk on <a href="/blog/2017/07/26/forbidden-configurations-in.html">forbidden configurations</a>, I asked whether Erdős and Szekeres knew about the lower bound for the <a href="https://en.wikipedia.org/wiki/Happy_ending_problem">happy ending problem</a> when they conjectured the formula for the problem in 1935, or not until they actually published the lower bound in 1960. I got conflicting responses: Pach told me he asked them this question, and that the conjecture was purely youthful optimism after seeing the numbers of points needed to force a triangle, quadrilateral, or pentagon. But Chvátal told me instead that Szekeres wrote (perhaps in the 60th-birthday festschrift for Erdős) that they knew the lower bound but held off on publishing it because they thought it was too easy. So I still don’t know.</p>
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<p>Min Yan provided some interesting ways of transforming any tiling of the plane or a sphere into a subdivided pentagonal tiling, on the way to classifying all tilings of the sphere by congruent pentagons. They’re probably easier to draw than to describe in text; the thick black edges below show the existing tiling and the thin colored edges show the new edges added in the subdivision.</p>
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<p style="text-align:center"><img src="/blog/assets/2017/pentagonal-tile-subdivns.svg" alt="Pentagonal subdivisions of a plane tiling" /></p>
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<p>Low-dimensional topologist Genevieve Walsh, slumming as a discrete geometer, asked which topological triangulations of the sphere can be realized by acute triangles. The answer in the plane was known by Maehara in 2003, who showed that a disk can be acutely triangulated by a given triangulation if and only if no triangle or quadrilateral encloses a point. Walsh’s new theorem is that on the sphere, a triangulation can be made acute if and only if no triangle or quadrilateral has points on both of its sides. The proof involves the <a href="https://en.wikipedia.org/wiki/Circle_packing_theorem">circle packing theorem</a> and an equivalence between acute sphere triangulations and hyperbolic convex polyhedra with right dihedrals.</p>
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<p>János Pach’s talk concerned the <a href="https://en.wikipedia.org/wiki/Crossing_number_inequality">crossing lemma</a>, that a drawing of a graph with <script type="math/tex">n</script> vertices and <script type="math/tex">m\ge 4n</script> edges has <script type="math/tex">\Omega(m^3/n^2)</script> crossings, and its extension to hypergraphs. In the hypergraph case one can’t just take the input as an abstract graph, because there are hypergraphs with many edges and no crossings (for instance repeat many edges of a planar graph) but instead one should consider “simple topological multigraphs”, multigraphs drawn with at most one crossing per edge pair, no crossings of incident edges, and no empty digons. The usual probabilistic proof of the crossing lemma (take a random sample of the edges small enough to reduce the number of edges to <script type="math/tex">4n</script>, and compare the expected number of crossings from the sample with the fact that you still have at least a linear number of crossings) doesn’t work, because random sampling can destroy the no-empty-digon property. Instead he proves the multigraph version of the lemma by a connection between crossing number and bisection width (somewhat related to <a href="/blog/2017/08/17/two-new-preprints.html">my new GD paper on width</a>?) that he had proved with Shahrokhi and Szegedy in 1996:</p>
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<script type="math/tex; mode=display">\mathrm{bw} = O\left(\sqrt{\mathrm{cr} + \sum_v \mathrm{deg}(v)^2} \right).</script>
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<p>It is known that you can collapse a (hollow) cube down onto one of its square faces, keeping that face and the opposite face flat and parallel, and folding the four faces between them. However to do so you have to use a “rolling fold”, one whose position on a folded face continuously shifts; it’s not possible to do this as <a href="https://en.wikipedia.org/wiki/Rigid_origami">rigid origami</a>. Chie Nara spoke on a higher dimensional generalization: you can fold a hypercube onto one of its square faces, keeping the opposite face parallel to it. Again, she used rolling folds, but Tomohiro Tachi asked whether the extra flexibility of the higher dimensions might allow that to be avoided. (Nara didn’t know, and I don’t either.) The next talk also involved Nara, flattening, and an interesting class of 3d polyhedra that I hadn’t previously seen: “semi-orthogonal polyhedra”, the polyhedra in which all faces are either parallel to or perpendicular to a common line.</p>
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<p>Elena Khramtcova started her talk by handing out paper regular-hexagons and tape, and asked us to fold and tape the hexagons to form a convex polyhedron. For instance, it’s not hard to make a regular octahedron this way, or an (irregular) triangular prism; see the nets and gluing patterns below. Three hexagons glue together to form a flat piece of paper, so the vertices of the polyhedra must be missing either one or two of those three hexagons, with six missing altogether. It was distracting enough that I don’t remember exactly what results she presented, but they were a partial classification of the combinatorial types of shapes you can form, with some cases still open. Soon after, Jason Ku solved one of the open cases: a rectangular pyramid with a sharp angle (two missing hexes) at its apex and blunter angles (one missing hex) at the other four vertices. I’ll leave its solution as a puzzle for readers.</p>
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<p style="text-align:center"><img src="/blog/assets/2017/hexnets.svg" alt="Nets and gluing patterns for hexagonally-tiled triangular prism and regular octahedron" /></p>
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<p>Naoki Katoh spoke about how to make square grids rigid by adding diagonal cross-braces. He started with a small brain teaser: if you cross-brace the four corner squares of a <script type="math/tex">3\times 3</script> grid, is the result rigid?
If you think of these braces as edges in a bipartite graph, whose vertices are rows and columns of squares in the grid, then a minimally rigid set of braces corresponds to a spanning tree in the graph. That is, the rigidity matroid for this case is a graphic matroid. But the problem becomes more complicated when the square grid can have holes in it rather than forming a simply-connected polyomino. Again, he provided an activity to involve the audience: a game involving a mechanical grid of squares, with a set number of cross-braces. Two players alternate in placing the cross-braces, with one trying to make the structure rigid and the other trying to keep it flexible. With a minimal number of cross-braces and a large grid, this is too easy for the flexible player to win (just place four braces making a 4-cycle in the underlying bipartite graph, ignoring what the other player does) but there should be some larger number of braces that makes it more of an even struggle.</p>
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<p>Aaron Williams offered a similar distraction for his talk: he handed out <a href="https://arduboy.com/">Arduboys</a> loaded with <a href="https://community.arduboy.com/t/mazezam-puzzle-game/3723">his implementation of MazezaM</a> while he talked about his proof that puzzle solutions can be exponentially long (a step towards <script type="math/tex">\mathsf{PSPACE}</script>-completeness). The puzzle is polynomial for a fixed number of rows (because the number of states is <script type="math/tex">\mathrm{cols}^{\mathrm{rows}}</script>) raising the question of its parameterized complexity. I didn’t get very far with the Arduboy because I wanted to pay some attention to the actual talk. Somewhere out there is a photo of several of us showing off the Arduboys.</p>
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<p>Rudolf Fleischer studied some problems involving what happens when you repeatedly do <a href="https://en.wikipedia.org/wiki/Faro_shuffle">perfect shuffles</a> of a card deck, but change whether you use in-shuffles or out-shuffles. For deck sizes of size a power of two, these shuffles can be described (in terms of their action on the indexes of the cards) as either a binary rotation, or a rotation followed by flipping the low-order bit. The Cayley graph of these operations is a directed graph resembling, but not quite the same as, the <a href="https://en.wikipedia.org/wiki/Cube-connected_cycles">cube-connected cycles</a>, the Cayley graph of the action on binary words by rotations and low-order flips (as separate operations). I wonder what is known about the structure of these graphs.</p>
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<p>Several of the invited talks were historical reminiscences rather than technical, but Mikio Kano found time to tell us about some problems related to the <a href="https://en.wikipedia.org/wiki/Ham_sandwich_theorem">ham sandwich theorem</a>, involving line partitions of a tricolored set of points with no majority color into subsets with the same <a href="http://ftw.usatoday.com/2015/11/j-k-rowling-revealed-the-american-word-for-muggle-and-people-are-furious-about-it">no-maj</a> property, a problem Sakai also spoke about in a contributed talk. Together they showed that when the number of points is a multiple of four, one can find a partition of this type in which each side is still a multiple of four, allowing a recursive partition of the plane into convex regions with only four points each, no three the same color.
However a counterexample from Jan Kynčl shows that it’s not always possible to evenly trisect large multiple-of-six no-maj point sets by a convex partition.</p>
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<p>Jin Akiyama gave another in-talk demonstration, asking us to cut paper envelopes by two trees, one on each side, spanning the four envelope corners. If you cut only one tree, you get a shape that tiles the plane. If you cut both trees, you get four shapes that can be reassembled in two different ways, to form either of the two tiles generated by one of the two trees. By combining the same trees in different pairs on different envelopes, one can get a sequence of dissections from one shape into another, which he showed off over the course of a story about his life: not being able to decide between mathematics and sailing, he goes fishing and catches a fish with shrimp as bait (shrimp turns into fish), but a cat steals the fish (fish turns into cat), the cat chokes and its body is made into a shamisen (cat turns into shamisen), and he sells the shamisen to a geisha (shamisen turns into geisha), who wisely advises him to use the money to go to the university and study hard (geisha turns into university).</p>
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<p>Hiroshi Nishiyama spoke on odd-depth trees: rooted spanning trees of a given <a href="https://en.wikipedia.org/wiki/Rooted_graph">rooted graph</a> in which all leaves have odd distance from the root. (The root does not count as a leaf, even if it has degree one.) In bipartite graphs it can be solved by <a href="https://en.wikipedia.org/wiki/Matroid_intersection">matroid intersection</a>: find the largest forest (one matroid) in which every vertex other than the root on the same side of the bipartition has at most two neighbors (the other matroid). If it provides exactly two neighbors to all the root-side vertices, you can extend it to an odd-depth tree, and otherwise no. But for biconnected non-bipartite graphs it’s <script type="math/tex">\mathsf{NP}</script>-complete.</p>
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<p>Masanori Fukui, Koki Suetsugu, and Akira Suzuki contributed a paper on <a href="https://en.wikipedia.org/wiki/Goishi_Hiroi">Goishi Hiroi</a>, a Hamiltonian-path-like puzzle from a 1727 Japanese puzzle book. Given a collection of stones on a grid you need to find a path of axis-aligned segments that visits all the stones. The path can jump over empty spaces or stones that are already on the path but not other stones, and can’t make any U-turns. They show that it’s <script type="math/tex">\mathsf{NP}</script>-complete, but the reduction involves unrealistic problem setups with disconnected islands of stones. It’s still open how hard it is when the initial set of stones forms a single contiguous group.</p>
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<p>Yasuaki Kobayashi, Koki Suetsugu, and Hideki Tsuiki contributed another paper on a family of puzzles with the intriguing property that they’re <a href="https://en.wikipedia.org/wiki/Graph_isomorphism_problem"><script type="math/tex">\mathsf{GI}</script>-complete</a> rather than having a higher or lower complexity. They’re called “lattice puzzles” (<a href="https://cults3d.com/en/game/lattice-puzzle">here is an example</a>) and they consist of a collection of metal or plastic strips, with slots of different heights that allow one strip to connect to another perpendicular strip. The goal is to connect all of the strips in this way to form a square lattice of strips. If you start out knowing which strips are horizontal and which vertical, then you can think of it as finding a permutation of the horizontal strips that causes their slots to match up with the slots in another permutation of the vertical strips. And if the slots have only two heights then this turns out to be exactly the problem of finding an isomorphism between a bipartite graph determined from the slots of one height on the horizontal strips and another bipartite graph determined from the complementary height on the vertical strips.</p>
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</ul>
<p>Overall, quite a fun conference. I hope it doesn’t take me quite so long to return next time!</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/D1djb5YkmFv">G+</a>)</p>David EppsteinI suppose I should write down my recollections of JCDCG3 before they fade too badly. It was only my second conference in Japan, and the first one was nearly 20 years earlier (PARAOPT V, at the invitation I think of Naoki Katoh). This time, the invitation was by Hiro Ito, but unfortunately he fell ill and couldn’t attend. Instead, the position of host was capably handled by Jin Akiyama. Here are Jin and Naoki with me and my wife Diana, photographed by Toshinori Sakai (who despite being behind the lens for most of the conference, mysteriously appears in many past conference photos):Sunshower: Contemporary Art from Southeast Asia2017-09-07T22:42:00+00:002017-09-07T22:42:00+00:00https://11011110.github.io/blog/2017/09/07/sunshower-contemporary-art<p>Here’s <a href="http://www.ics.uci.edu/~eppstein/pix/sunshower/">one more batch of photos of art</a> from Japan before my last catch-all batch of pictures. It’s from an exhibit shared between two museums: “<a href="http://www.nact.jp/english/exhibitions/2017/sunshower/">Sunshower: Contemporary Art from Southeast Asia</a>”, at the National Art Center and Mori Museum. The Mori was especially fun because it is on around the 50th floor of a tall building on a hill, and you can get combined tickets to the <a href="http://www.roppongihills.com/tcv/en/about/index.html">Tokyo City View</a> observation area, from which you can see much of the city. Unfortunately, while clear enough to see, the weather wasn’t nice enough to make photography of the view particularly interesting.</p>
<p>Anyway, the title of the exhibit is pretty self-explanatory: it’s a collection of contemporary art from all over southeast Asia. The exhibit continues through late October and is worth seeing if you’re a fan of contemporary art or have any interest in the politics of that part of the world. The piece below is by Thai artist Surasi Kusolwong, who filled a large room about a meter deep with balls of colored yarn and then (perhaps as a statement, perhaps to encourage visitors to stay and interact with the yarn) hid ten gold necklaces somewhere in the yarn. If you find a necklace, you get to keep it.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/sunshower/SurasiKusolwongGoldenGhost1-m.jpg" alt="Surasi Kusolwong, Golden Ghost (Reality Called, So I Woke Up), 2014, in the Sunshower exhibit at the Mori Museum, Tokyo" style="border-style:solid;border-color:black;" /></p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/NmXzbWtMyge">G+</a>)</p>David EppsteinHere’s one more batch of photos of art from Japan before my last catch-all batch of pictures. It’s from an exhibit shared between two museums: “Sunshower: Contemporary Art from Southeast Asia”, at the National Art Center and Mori Museum. The Mori was especially fun because it is on around the 50th floor of a tall building on a hill, and you can get combined tickets to the Tokyo City View observation area, from which you can see much of the city. Unfortunately, while clear enough to see, the weather wasn’t nice enough to make photography of the view particularly interesting.Sculpture in Tokyo2017-09-05T22:25:00+00:002017-09-05T22:25:00+00:00https://11011110.github.io/blog/2017/09/05/sculpture-in-tokyo<p>I had a few days in Tokyo before <a href="http://www.jcdcgg.u-tokai.ac.jp/">JCDCG<sup>3</sup></a> to run around being a tourist, visiting art museums and such. <a href="http://www.ics.uci.edu/~eppstein/pix/sculptok/">The next batch of my photos from Tokyo</a> is all of sculptures that I saw in this time. Despite being from vastly different times (<a href="https://en.wikipedia.org/wiki/J%C5%8Dmon_period">Jōmon</a> to modern) I felt that they fit together well in their use of stylization and form.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/sculptok/TokyoUniversityOfScience2-m.jpg" alt="Rabbit-eared creature at the Tokyo University of Science" style="border-style:solid;border-color:black;" /></p>
<p>This strange and much-repaired creature was one of the first things I saw after leaving the train from the airport. It’s part of a set on the grounds of the Tokyo University of Science, at the base of the steps where we took the conference group photo.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/sculptok/FemaleHaniwa-m.jpg" alt="Female haniwa at the Tokyo National Museum in Ueno, Tokyo" style="border-style:solid;border-color:black;" /></p>
<p>If you want a quick overview of Japanese culture through the ages, or just to see a lot of shiny swords, the place to go is the Honkan (Japanese Gallery) and Heiseikan (Japanese Archaeology Gallery) of the Tokyo National Museum in Ueno. This female <a href="https://en.wikipedia.org/wiki/Haniwa">haniwa</a> figure comes from the same time as the late Roman empire, but to my eyes with her crisp uniform she could be a modern policewoman or traffic inspector. The Gallery of Hōryū-ji Treasures in the same museum is also worth seeing, especially the big array of dozens of Buddha and Kannon figures, but it was too dark for me to take any photos.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/sculptok/GiacomettiMonumentalHead2-m.jpg" alt="Alberto Giacometti, Monumental Head, in the National Art Center, Roppongi, Tokyo" style="border-style:solid;border-color:black;" /></p>
<p>The main show at the National Art Center in Roppongi, while we were there, was a retrospective of the sculptures of <a href="https://en.wikipedia.org/wiki/Alberto_Giacometti">Alberto Giacometti</a>. That doesn’t sound very Japanese, but there is a Japanese connection: one of his close friends and favorite subjects was philosopher Isaku Yanaihara. The captions described Giacometti as working very hard to capture his subjects exactly as he saw them, but if so his vision must have been a bit off-kilter, with rough surfaces, exaggeratedly small or (in this case) large scale, skeletonized forms, and a deep perspective that encourages many of his works to be viewed head-on rather than from an angle.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/3e3dLy6zpK3">G+</a>)</p>David EppsteinI had a few days in Tokyo before JCDCG3 to run around being a tourist, visiting art museums and such. The next batch of my photos from Tokyo is all of sculptures that I saw in this time. Despite being from vastly different times (Jōmon to modern) I felt that they fit together well in their use of stylization and form.Kagurazaka2017-09-03T18:55:00+00:002017-09-03T18:55:00+00:00https://11011110.github.io/blog/2017/09/03/kagurazaka<p>I’m now back from my trip to Tokyo for <a href="http://www.jcdcgg.u-tokai.ac.jp/">JCDCG<sup>3</sup></a> (where the running joke was speculation on what the next G to be added to the conference name would stand for), and still somewhat jet-lagged. But I’ve now put online <a href="http://www.ics.uci.edu/~eppstein/pix/kagurazaka/index.html">the first of several batches of photos</a> from my trip, which I will continue posting as I finish processing them. This set is from <a href="https://en.wikipedia.org/wiki/Kagurazaka">Kagurazaka</a>, the neighborhood of both the conference and my hotel. Below are two of my favorites; see the link for more.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/kagurazaka/AkagiShrineCraftFair-m.jpg" alt="Craft fair at the Akagi Shrine, Kagurazaka, Tokyo, Japan" style="border-style:solid;border-color:black;" /></p>
<p>This craft fair was set in the grounds of the Akagi Shrine, one of many shrines and temples throughout the city. The shrine itself is modern in architecture (plate glass and wood) and well-used; there was a long line at the prayer bell. You can tell from the torii and the urban background in this shot that it’s set in Tokyo or somewhere similar, but it felt to me that the craft fair participants could have been anywhere in the world, although there was a distinct Japanese flavor to some of their crafts. We found it impossible to resist a tea-towel decorated with surfing anteaters, as the anteater is the UCI mascot.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/kagurazaka/AlleyWithLanterns-m.jpg" alt="Alley with lanterns in Kagurazaka, Tokyo, Japan" style="border-style:solid;border-color:black;" /></p>
<p>I’m not sure this alley has a name (many of them don’t seem to) but it’s just uphill from another alley that some maps label with 見番横丁, “kenban-yokocho”. This appears to mean something like “viewing alley” (someone who knows Japanese please correct me), and is so called because in older times you could view the geisha there. At a bend in the viewing alley there’s a basement restaurant that serves only garlic-based dishes (with decorations from the annual garlic festival in Gilroy, California), and the entrance to an even smaller side-alley with steps down to the bathhouse behind our hotel. There are still geisha working in some of the fancier local restaurants; one of them sang and played the shamisen for us at a dinner for some of the conference participants.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/2MYW4sztpo5">G+</a>)</p>David EppsteinI’m now back from my trip to Tokyo for JCDCG3 (where the running joke was speculation on what the next G to be added to the conference name would stand for), and still somewhat jet-lagged. But I’ve now put online the first of several batches of photos from my trip, which I will continue posting as I finish processing them. This set is from Kagurazaka, the neighborhood of both the conference and my hotel. Below are two of my favorites; see the link for more.Linkage from Japan2017-08-31T10:36:00+00:002017-08-31T10:36:00+00:00https://11011110.github.io/blog/2017/08/31/linkage-from-japan<ul>
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<p><a href="https://mateturismo.wordpress.com/">World guidebook for mathematical tourism</a> (in Spanish; <a href="https://plus.google.com/100003628603413742554/posts/4vNytSNuT68">G+</a>). Too bad there are no entries for Tokyo, where I am as I post this.</p>
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<p><a href="https://arxiv.org/abs/1708.01559">What is the least symmetric triangle in the plane?</a> (<a href="https://plus.google.com/100003628603413742554/posts/3eMxjXKNktf">G+</a>, <a href="https://mathstodon.xyz/@shonk/238838">via</a>). An answer can be obtained by constructing a certain arrangement of great circles on the unit sphere and finding a point of the sphere that is farthest from any great circle.</p>
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<p><a href="https://en.wikipedia.org/wiki/Tardos_function">Tardos function</a> (<a href="https://plus.google.com/100003628603413742554/posts/11Tr2nC138k">G+</a>). New Wikipedia article on the function used to poke holes in Norbert Blum’s claimed proof of <script type="math/tex">\mathsf{P}\ne\mathsf{NP}</script>.</p>
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<p><a href="https://christchurchartgallery.org.nz/collection/2005-048">Titokowaru and Te Whiti discuss the question, ‘What is Peace?’</a> (<a href="https://plus.google.com/100003628603413742554/posts/9qFUYknyA2M">G+</a>). New Zealand lithographer <a href="https://en.wikipedia.org/wiki/Marian_Maguire">Marian Maguire</a> mixes Māori and ancient Greek art styles in a dialogue on war, peace, and passive resistance between Socrates and Māori leader <a href="https://en.wikipedia.org/wiki/T%C4%ABtokowaru">Tītokowaru</a>.</p>
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<p><a href="https://www.insidehighered.com/quicktakes/2017/08/21/thailand-files-charges-against-conference-attendees">Participants in a Thai academic conference arrested for violating public-assembly laws</a> (<a href="https://plus.google.com/100003628603413742554/posts/YEoQiJygoRS">G+</a>). Relevant for ISAAC, this December in Phuket, Thailand?</p>
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<p><a href="https://medium.com/popular-choice/why-a-dutch-court-stopped-high-school-students-from-exchanging-schools-1315303a48b6">Why a Dutch court stopped high school students from exchanging schools</a> (<a href="https://plus.google.com/100003628603413742554/posts/4f9dbfUdz1C">G+</a>), and how the mechanism used to allocate scarce resources can have a big effect on how the participants behave.</p>
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<p><a href="http://www.lizpowley.com/explorations-bubble-painting/">Bubble painting</a> (<a href="https://plus.google.com/100003628603413742554/posts/LPYFno6brjn">G+</a>). What you get when you make a foam of bubble bath solution and paint, spread it on paper, and then let the bubbles dry and pop.</p>
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<p><a href="http://blog.geomblog.org/2017/08/on-free-speech-gerrymandering-and-self.html">For a democratic society to remain democratic, it must reject democratically chosen acts whose effects are inimical to democracy</a> (<a href="https://plus.google.com/100003628603413742554/posts/YYF1phLgmoH">G+</a>). Suresh Venkatasubramanian uses Popper’s Paradox of Tolerance (for a tolerant society to remain tolerant, it must be intolerant of intolerance) as an argument against partisan gerrymandering.</p>
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<p><a href="http://www.cutoutfoldup.com/981-torus-from-villarceau-circles.php">Paper model of the Villarceau circles on a torus</a> (<a href="https://plus.google.com/100003628603413742554/posts/PuNkdSFChBo">G+</a>, <a href="https://mathlesstraveled.com/2017/01/03/paper-torus-with-villarceau-circles/">via</a>).</p>
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<p><a href="http://www.nature.com/doifinder/10.1038/nature.2017.22474">Mysteries of turbulence unravelled</a> (<a href="https://plus.google.com/100003628603413742554/posts/EC4pY8EvPy2">G+</a>). Simulations follow how swirls in a fluid transfer and dissipate energy.</p>
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<p><a href="http://www.sciencemag.org/news/2017/08/biologists-forgotten-experiment-started-preprint-revolution-5-decades-ago">A biology preprint exchange from the early 1960s</a> (<a href="https://plus.google.com/100003628603413742554/posts/3wSywHddvyF">G+</a>), killed off by publishers banning shared papers from being published.</p>
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<p><a href="https://www.instagram.com/p/BN0ylyxlkNL/">Kyouhei Kasai</a>, Tokyo instagrammer (<a href="https://plus.google.com/100003628603413742554/posts/5XZyziGmvCw">G+</a>). While in Japan I saw a tiny gallery exhibit of some of his film work.</p>
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<p><a href="https://www.youtube.com/watch?v=G_uybVKBacI">Braids in higher dimensions</a> (<a href="https://plus.google.com/100003628603413742554/posts/a5kLuSJyvKH">G+</a>). One-dimensional curves cannot form nontrivial braids in four dimensions, but two-dimensional surfaces can. Zsuzsanna Dancso explains how to visualize these in one-dimension lower, by interpreting the braiding dimension as time.</p>
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<p><a href="https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/">Separating fact from speculation in math history</a> (<a href="https://plus.google.com/100003628603413742554/posts/jhd4W95jaKQ">G+</a>). Evelyn Lamb clears up some of the nonsense and hype surrounding Wildberger’s new paper on Babylonian tablet <a href="https://en.wikipedia.org/wiki/Plimpton_322">Plimpton 322</a>.</p>
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<p><a href="http://isohedral.ca/heesch-numbers-part-4-edge-to-edge-pentagons/">Edge-to-edge Heesch numbers</a> (<a href="https://plus.google.com/100003628603413742554/posts/epQofHrYv8i">G+</a>). Craig Kaplan finds pentagons that can be surrounded by edge-to-edge copies of themselves but cannot tile the whole plane.</p>
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</ul>David EppsteinWorld guidebook for mathematical tourism (in Spanish; G+). Too bad there are no entries for Tokyo, where I am as I post this.Two new Graph Drawing preprints2017-08-17T22:42:00+00:002017-08-17T22:42:00+00:00https://11011110.github.io/blog/2017/08/17/two-new-preprints<p>I have two new preprints on arXiv, from my two papers <a href="https://gd2017.ccis.northeastern.edu/accepted-papers/">accepted to Graph Drawing</a>.</p>
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<p>“Triangle-free penny graphs: degeneracy, choosability, and edge count” (<a href="https://arxiv.org/abs/1708.05152">arXiv:1708.05152</a>) is a conference-paper version of <a href="/blog/2017/02/19/triangle-free-penny.html">an earlier post here</a>, which showed that the triangle-free penny graphs are 2-degenerate. The main article is short but also expands on a comment I made on that post, a bound on the largest possible number of edges in these graphs that is close to the known lower bound. In an appendix, I also included material from two later posts here: <a href="/blog/2017/06/13/how-many-edges.html">some analogous results for squaregraphs</a>, and a counterexample showing that <a href="/blog/2017/06/18/the-malyshev-graphs.html">the same results do not extend to arbitrary 2-degenerate triangle-free planar graphs</a>.</p>
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<p>“The effect of planarization on width” (<a href="https://arxiv.org/abs/1708.05155">arXiv:1708.05155</a>) is an extended version of <a href="https://cstheory.stackexchange.com/q/35974/95">my answer to a cstheory question by Bart Jansen</a>. Bart asked whether it is possible to draw <script type="math/tex">K_{3,n}</script> and then replace every crossing by a vertex, in such a way that the resulting planar graph has low pathwidth. The answer is no: every planarization of a drawing of <script type="math/tex">K_{3,n}</script> has pathwidth <script type="math/tex">\Omega(n)</script>. The proof idea from that answer turns out to generalize to treewidth as well as pathwidth, and I also looked at the same question for many other graph width parameters. Some of these parameters behave like treewidth and pathwidth, blowing up when you planarize even as simple a graph as <script type="math/tex">K_{3,n}</script>, while others stay bounded when you planarize a bounded-width graph. Here’s one of the figures from the planarization paper, a carving decomposition of <script type="math/tex">K_{3,3}</script> and its planarization:</p>
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<p style="text-align:center"><img src="/blog/assets/2017/carving-decomposition.svg" alt="Carving decomposition of K_{3,3} and its planarization" /></p>
<p>Graph Drawing has a policy of maintaining a shadow proceedings on arXiv.org, with the same content as the official proceedings, so I think we should be seeing quite a few more of these from other authors as the early-September proceedings deadline approaches.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/BKV2aEL3dxm">G+</a>)</p>David EppsteinI have two new preprints on arXiv, from my two papers accepted to Graph Drawing.Linkage2017-08-15T21:56:00+00:002017-08-15T21:56:00+00:00https://11011110.github.io/blog/2017/08/15/linkage<ul>
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<p><a href="https://en.wikipedia.org/wiki/Simplicial_depth">Simplicial depth</a> (<a href="https://plus.google.com/100003628603413742554/posts/hF39ynQEHZj">G+</a>). Measuring how central a point is within a cloud of points by how many other triangles surround it. New Wikipedia article on an important topic in robust statistics.</p>
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<p><a href="http://boingboing.net/2017/08/02/nation-of-scottish-bankers.html">Catholic outrage at giant robot spider</a> (<a href="https://plus.google.com/100003628603413742554/posts/QzhxhmsFEXk">G+</a>). The semi-official CCCG excursion to see this show made for a fun and memorable evening.</p>
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<p><a href="http://blogs.ams.org/beyondreviews/2017/01/26/citations/">Why automatic citation-counting can be difficult</a> (<a href="https://plus.google.com/100003628603413742554/posts/URnssDcxF7j">G+</a>), why it’s important to read and check your references yourself rather than just copying them from earlier publications, and why it’s a good idea to get your bibtex data from a high-quality source like MathSciNet (whose blog the linked post is from).</p>
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<p><a href="https://gd2017.ccis.northeastern.edu/accepted-papers/">Graph Drawing accepted papers</a> and <a href="https://algo2017.ac.tuwien.ac.at/ipec/accepted-papers/">IPEC accepted papers</a> (<a href="https://plus.google.com/100003628603413742554/posts/Y5LwYgVksdR">G+</a>). I have two at GD (about which more later) and one at IPEC (<a href="/blog/2017/03/11/fast-k-best.html">previously</a>).</p>
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<p><a href="https://www.quora.com/How-do-you-find-the-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit?share=1">Deep math in the solution to a simple-looking equation</a> (<a href="https://plus.google.com/100003628603413742554/posts/WxYfTnL2nLN">G+</a>, <a href="https://mastodon.social/users/hntooter/updates/4026248">via</a>): find positive integers <script type="math/tex">x</script>, <script type="math/tex">y</script>, and <script type="math/tex">z</script> with</p>
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<script type="math/tex; mode=display">\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} = 4.</script>
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<p><a href="http://econpapers.repec.org/paper/wpawuwppe/9705001.htm">Arrow’s impossibility theorem for infinitely many voters</a> (<a href="https://plus.google.com/100003628603413742554/posts/9Z4AhRSKgwA">G+</a>). In this case, one can construct election decision rules that are unexpectedly nice: every election has an outcome, which matches the preferences of at least one voter, can’t be changed by any finite subset of voters, and behaves monotonically. But they can’t be implemented by any recursive algorithm, and require weak forms of the axiom of choice to even exist.</p>
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<p><a href="https://scholarlykitchen.sspnet.org/2017/07/25/cabells-new-predatory-journal-blacklist-review/">Rick Anderson reviews Cabell’s List</a> (<a href="https://plus.google.com/100003628603413742554/posts/V2QEt9bkdiH">G+</a>), planned as a replacement for the now-defunct Beall’s List of predatory open-access journals. But it doesn’t distinguish low standards from predatory behavior, and its subscription-only and non-transparent nature are also problematic.</p>
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<p><a href="https://www.quantamagazine.org/new-shapes-solve-infinite-pool-table-problem-20170808/">Shapes with the property that every billiard path either repeats or eventually covers the whole shape</a>, rather than chaotically filling some part of the shape and then completely bypassing some other part (<a href="https://plus.google.com/100003628603413742554/posts/PNvXarRRLk3">G+</a>). Not many are known, but now there are two new ones, both non-convex quadrilaterals with rational angles.</p>
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<p><a href="http://twistedsifter.com/videos/blowing-bubbles-physics-lesson/">Bubble-blowing clip from BBC’s <em>The Code</em></a> (<a href="https://plus.google.com/100003628603413742554/posts/CKyN5pgfdSc">G+</a>), with near-polyhedral shapes in the center of clusters of bubbles.</p>
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<p>Problem: US is way behind many other countries in connecting individual homes to high-speed internet. Solution: <a href="https://arstechnica.com/information-technology/2017/08/maybe-americans-dont-need-fast-home-internet-service-fcc-suggests/">Who needs fast internet when you have cell phones that are only 5x slower?</a> (<a href="https://plus.google.com/100003628603413742554/posts/4NbkQEzk32S">G+</a>)</p>
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<p><a href="https://www.citylab.com/transportation/2017/08/intricate-x-ray-maps-of-new-york-city-subway-stations/535965/?utm_source=SFFB">What NYC subway stations actually look like</a> (<a href="https://plus.google.com/100003628603413742554/posts/4zqViEpgaDn">G+</a>). Or would, if we had X-ray vision.</p>
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<p><a href="http://www.thisiscolossal.com/2017/08/the-phenomenon-of-crown-shyness-where-trees-avoid-touching/">The phenomenon of “crown shyness” where trees avoid touching</a> (<a href="https://plus.google.com/100003628603413742554/posts/Mi8G421tEmh">G+</a>). Still with no definitive explanation of how or why.</p>
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<p><a href="https://jeremykun.com/2017/08/14/notes-on-math-and-gerrymandering/">Notes on math and gerrymandering</a> (<a href="https://plus.google.com/100003628603413742554/posts/iddK7kjQgFK">G+</a>). Jeremy Kun’s notes from the Geometry of Redistricting workshop in Boston, with explanations of why a tangle of legal and political issues make it very difficult to impose simple mathematical solutions.</p>
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</ul>David EppsteinSimplicial depth (G+). Measuring how central a point is within a cloud of points by how many other triangles surround it. New Wikipedia article on an important topic in robust statistics.