Jekyll2019-09-16T01:18:08+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage2019-09-15T17:51:00+00:002019-09-15T17:51:00+00:00https://11011110.github.io/blog/2019/09/15/linkage<ul>
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<p><a href="https://www.washingtonpost.com/opinions/no-i-wont-start-spying-on-my-foreign-born-students/2019/08/29/01c80e84-c9b2-11e9-a1fe-ca46e8d573c0_story.html">Lee Bollinger, president of Columbia University: “No, I won’t start spying on my foreign-born students”</a> (<a href="https://mathstodon.xyz/@11011110/102718751083310854"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://windowsontheory.org/2019/08/30/update-on-the-safe-toc-initiative-guest-post-by-sandy-irani/">Update on the Safe ToC initiative</a> (<a href="https://mathstodon.xyz/@11011110/102725927728134229"><script type="math/tex">\mathbb{M}</script></a>). Sandy Irani describes progress in combatting harassment and discrimination at theoretical computer science conferences, and calls for volunteer advocates to serve as contact points at conferences.</p>
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<p><a href="http://isohedral.ca/escher-like-spiral-tilings/">Escher-like spiral tilings, by Craig Kaplan</a> (<a href="https://mathstodon.xyz/@11011110/102732614495435403"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=20854644">via</a>). Sadly with no angels, devils, fish, or geese, but maybe some talented artist will take up that challenge.</p>
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<p><a href="https://arxiv.org/abs/1909.00263">How to peel self-intersecting onions</a> (<a href="https://mathstodon.xyz/@jeffgerickson/102734672335961160"><script type="math/tex">\mathbb{M}</script></a>). Gabriel Nivasch extends the <a href="/blog/2017/10/11/peeling-vs-shortening.html">conjectured equivalence</a> between <a href="https://en.wikipedia.org/wiki/Convex_layers">convex layers</a> and the <a href="https://en.wikipedia.org/wiki/Curve-shortening_flow#Related_flows">affine curve-shortening flow</a> to non-convex and self-intersecting curves. The generalized onion-peeling process alternates between steps that jump over grid points and steps that shrink curves to the shortest curve that passes between the same grid points. See also Gabriel’s animations of this process for <a href="https://mathstodon.xyz/@gnivasch/102741204463574303">grid points</a> and for <a href="https://mathstodon.xyz/@gnivasch/102753301344471044">random points</a>.</p>
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<p><a href="https://arxiv.org/abs/1909.00917">New stick number bounds from random sampling of confined polygons</a> (<a href="https://mathstodon.xyz/@shonk/102742716819892997"><script type="math/tex">\mathbb{M}</script></a>). Tom Eddy and Clayton Shonkwiler do knot theory on large numbers of random 3d polygons, in the process finding polygonal representations of many knots with fewer segments than were known before.</p>
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<p><a href="https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974%c2%b3-80435758145817515%c2%b3-12602123297335631%c2%b3/">42 is the answer</a>. The question is: What is <script type="math/tex">(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3</script>?</p>
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<p><a href="https://doi.org/10.1177/2378023118823946">Who Counts as a Notable Sociologist on Wikipedia? Gender, Race, and the “Professor Test”</a> (<a href="https://mathstodon.xyz/@11011110/102755324164533099"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_Signpost/2019-08-30/Recent_research">via</a>). After authors Adams, Brückner, and Naslund factored out seniority and impact of sociologists, white men remained more likely than others to have Wikipedia articles. Surprisingly to me, the disparity happens at article creation, not deletion. So we should create more articles about women! Or be less quick to create them on borderline-notable men…</p>
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<p><a href="https://www.flickr.com/photos/132410114@N04/24230683269/">Permutations in the real world: 12784563</a> (<a href="https://mathstodon.xyz/@11011110/102763393307640430"><script type="math/tex">\mathbb{M}</script></a>). Actually I have sentimental reasons to prefer 15426378, but I couldn’t find a nice photo of that one.</p>
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<p><a href="https://simon.lc/the-history-of-tetris-randomizers">The history of Tetris randomizers</a> (<a href="https://mathstodon.xyz/@11011110/102766825773116819"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/182935/let-piece-I-J-L-TMathfloorMathrandom-4">via</a>). Truly uniformly random distributions tend to be more clustered than people expect (having runs of the same piece or of mostly the same piece). So later versions took measures to make the piece distribution less random and more non-clustered.</p>
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<p><a href="https://arxiv.org/abs/1908.07097">An <script type="math/tex">\Omega(n^2)</script> lower bound for random universal sets for planar graphs</a> (<a href="https://mathstodon.xyz/@11011110/102771230722757397"><script type="math/tex">\mathbb{M}</script></a>). Random subsets of a square act like grids in lots of ways. Here’s one, from the linked preprint: to draw all <script type="math/tex">n</script>-vertex planar graphs with chosen points as vertices, you need either a grid or a random point set of <script type="math/tex">\Theta(n^2)</script> points. The reason is that drawings of the nested triangles graph contain a sequence of <script type="math/tex">\Omega(n)</script> points (corners of bounding boxes of triangles) that’s monotone in both coordinate directions, and smaller random sets (or grids) don’t have such sequences.</p>
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<p><a href="https://codepen.io/collection/eErLu/">CSS polyhedra</a> (<a href="https://mathstodon.xyz/web/statuses/102775320989084287"><script type="math/tex">\mathbb{M}</script></a>). Visualizations of 3d rotating polyhedra, coded entirely in html/css and embeddable in other web pages.</p>
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<p><a href="https://www.maths.ox.ac.uk/node/30217">Random minimum spanning trees</a> (<a href="https://mathstodon.xyz/@11011110/102786143581334329"><script type="math/tex">\mathbb{M}</script></a>). Did you know that in random graphs with edge probability <script type="math/tex">1/n</script> (just below the appearance of the giant component) there are lots of components of size <script type="math/tex">n^{3/2}</script> that all look nearly the same as uniformly random spanning trees of a complete graph? And that the minimum spanning tree of a randomly-weighted complete graph, instead, looks like one of these components with a lot of others all glued onto it? Christina Goldschmidt describes her work in this area.</p>
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<p><a href="https://www.gwern.net/Turing-complete">A big list of unlikely or surprising Turing-complete systems</a> (<a href="https://mathstodon.xyz/@11011110/102789724968251958"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/183095/On-having-sufficient-complexity-to-allow-for-arbitrary-computation">via</a>). My favorite: <a href="https://github.com/tom-p-reichel/svg-is-turing-complete">SVG is Turing-complete</a> because it can be used to (slowly) simulate Rule 110 (and one hopes also simulate the weird boundary conditions needed to make Rule 110 Turing complete).</p>
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<p><a href="https://www.english.cam.ac.uk/cmt/?p=5751">Milton’s hand-annotated volume of Shakespeare’s plays discovered sitting on a library shelf in Philly</a> (<a href="https://mathstodon.xyz/@11011110/102791968928048975"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/183100/Miltons-Shakespeare">via</a>).</p>
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<p><a href="https://en.wikipedia.org/wiki/Vojt%C4%9Bch_Jarn%C3%ADk">Vojtěch Jarník, now a good article on Wikipedia</a> (<a href="https://mathstodon.xyz/@11011110/102798710399112671"><script type="math/tex">\mathbb{M}</script></a>). I teach his algorithm for minimum spanning trees in my classes, but lately in my research I’ve been citing him more for his work on the number of integer grid points on convex curves. He also did important work on Diophantine approximation and nowhere-differentiable functions.</p>
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</ul>David EppsteinLee Bollinger, president of Columbia University: “No, I won’t start spying on my foreign-born students” ().Linkage2019-08-31T14:43:00+00:002019-08-31T14:43:00+00:00https://11011110.github.io/blog/2019/08/31/linkage<ul>
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<p><a href="https://www.insidehighered.com/news/2019/08/14/phd-students-resent-expectation-they-bring-food-and-drinks-their-thesis-defenses">Should graduate students be expected to bring refreshments to their doctoral defenses?</a> (<a href="https://mathstodon.xyz/@11011110/102631002899869926"><script type="math/tex">\mathbb{M}</script></a>). It can be an imposition, it distracts from preparing from the defense, students on graduate fellowships may not easily afford it, and to some extent it looks like bribery of the examining committee. On the other hand, it’s traditional and it’s hard to break traditions, and a non-hungry committee is a non-hangry committee.</p>
<p>Some places have resorted to banning student-provided refreshments, and/or providing refreshments through other means. For <a href="/blog/2019/08/27/congratulations-dr-besa.html">our most recent defense</a>, what we ended up doing was less extreme: we made sure the candidate understood that we were not expecting treats and that he should only bring them if he wanted to (which it turns out he did) and we reimbursed the expenses.</p>
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<p><a href="https://carto.com/blog/zip-codes-spatial-analysis/">Stop using Zip Codes for geospatial analysis</a> (<a href="https://mathstodon.xyz/@11011110/102642311091282185"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/182600/humans-dont-behave-based-on-administrative-units">via</a>). The problem is that they are designed to be convenient for the post office (geographically compact) rather than convenient for demographers (they sometimes combine areas that have quite different groups of people in them). The linked article is by a group selling software for one alternative subdivision that works better, but recommends several others as well.</p>
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<p><a href="https://arxiv.org/abs/1908.05124">Minimal representations of order types by geometric graphs</a> (<a href="https://mathstodon.xyz/@11011110/102646793309346399"><script type="math/tex">\mathbb{M}</script></a>). How should one visualize the order-type of a set of points (which triples are clockwise, which counterclockwise, and which collinear)? Aichholzer et al consider points without collinear triples, and draw a graph over the points so that as points move the order-type changes only as points cross edges. Their graph’s embedding determines the order type and has at most a third as many edges as the complete graph.</p>
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<p><a href="https://mathoverflow.net/questions/338674/properties-of-the-affine-curve-shortening-flow">Properties of the affine curve-shortening flow</a> (<a href="https://mathstodon.xyz/@gnivasch/102651157230836596"><script type="math/tex">\mathbb{M}</script></a>). Gabriel Nivasch asks MathOverflow for references on the behavior of this process for self-intersecting curves.</p>
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<p><a href="https://www.popularmechanics.com/science/math/a28722621/mow-your-lawn-using-math/">Mow your lawn using the power of math</a> (<a href="https://mathstodon.xyz/@jhertzli/102655629487900352"><script type="math/tex">\mathbb{M}</script></a>). This piece on optimal lawn-mowing seems to be unaware of prior work on the same subject (Fekete et al, “The Lawnmower Problem”, CCCG 1993). But it has some interesting suggestions about the possibility of an art installation with grass covering a Möbius strip.</p>
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<p><a href="https://www.chronicle.com/article/Women-Only-STEM-Programs/246996">Trump administration threatens to shut down scholarships, societies, and even summer camps that promote women in STEM, claiming that they discriminate against men</a> (<a href="https://mathstodon.xyz/@11011110/102669219180128915"><script type="math/tex">\mathbb{M}</script></a>). Note: Not <em>The Onion</em>.</p>
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<p><a href="https://www.quantamagazine.org/to-make-two-black-holes-collide-try-three-20190815/">To make two black holes collide, try three</a> (<a href="https://mathstodon.xyz/@11011110/102675012007662181"><script type="math/tex">\mathbb{M}</script></a>). The recent discovery of gravity waves from black hole collisions has led to the realization that these collisions are surprisingly frequent: two-body problems are sufficiently stable, even under general relativity, that pairs of black holes should continue orbiting around each other for eons rather than crashing. As <em>Quanta</em> describes, one possible resolution is to add a third black hole to the mix, and get a (significantly less stable) three-body problem.</p>
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<p><a href="http://oeis.org/A236266">The greedy no-3-in-line sequence</a> (<a href="https://mathstodon.xyz/@11011110/102680040398389099"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://mathoverflow.net/questions/338528/on-the-first-sequence-without-collinear-triple">via</a>). This is what you get when you try to solve the <a href="https://en.wikipedia.org/wiki/No-three-in-line_problem">no-3-in-line problem</a> by choosing, for each non-negative integer <script type="math/tex">x</script>, the smallest non-negative <script type="math/tex">y</script> avoiding lines through previous points. <a href="http://oeis.org/A236266/graph">Its scatterplot</a> suggests that this greedy algorithm finds subsets of <script type="math/tex">[n]^2</script> of size <script type="math/tex">\Omega(n)</script>, even though this method is very different from the constructions for which this density has been proven.</p>
<p style="text-align:center"><img src="/blog/assets/2019/a236266.png" alt="Graph of sequence A236266, from http://oeis.org/A236266/graph" /></p>
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<p><a href="https://www.insidehighered.com/news/2019/08/15/dutch-academics-fear-language-rule-would-hinder-foreign-recruitment">Dutch academics fear language rule would hinder foreign recruitment</a> (<a href="https://mathstodon.xyz/@11011110/102687368627549981"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.bbc.com/news/world-europe-46030112">see also</a>, <a href="https://www.timeshighereducation.com/news/compulsory-dutch-looms-foreign-students-netherlands">see also</a>, <a href="https://www.dutchnews.nl/news/2019/08/foreign-students-may-face-dutch-language-lessons-times-higher-education/">see also</a>). Many Dutch universities have been teaching entire programs of study in English, and the Eindhoven University of Technology is English-only. A proposed law would outlaw that; not unreasonably, the Dutch universities are worried that this change would make it harder to recruit students from other countries.</p>
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<p><a href="https://www.nature.com/articles/d41586-019-02479-7">New database of self-citation</a> (<a href="https://mathstodon.xyz/@11011110/102697481360799420"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://retractionwatch.com/2019/08/24/weekend-reads-self-citation-farms-an-editor-refuses-to-retract-publishing-enters-politics/">via</a>). There are legitimate reasons to self-cite (as the article makes clear) but when senior researchers have a majority of citations from their own papers, and then get government awards for being highly cited, there’s likely a problem. Or more than one problem, both in the researcher’s citation habits and in the criteria for the awards.</p>
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<p><a href="https://mathstodon.xyz/@ccppurcell/102702702715651994">Why do academic talk slides conventionally cite the speaker using only an initial?</a> And how would one go about achieving that effect in BibTeX?</p>
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<p><a href="https://en.wikipedia.org/wiki/Pandrosion">Pandrosion</a> (<a href="https://mathstodon.xyz/@11011110/102713694251643571"><script type="math/tex">\mathbb{M}</script></a>), a woman in ancient Greek mathematics prior to Hypatia. New article on Wikipedia.</p>
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</ul>David EppsteinShould graduate students be expected to bring refreshments to their doctoral defenses? (). It can be an imposition, it distracts from preparing from the defense, students on graduate fellowships may not easily afford it, and to some extent it looks like bribery of the examining committee. On the other hand, it’s traditional and it’s hard to break traditions, and a non-hungry committee is a non-hangry committee.Grid majors2019-08-29T20:47:00+00:002019-08-29T20:47:00+00:00https://11011110.github.io/blog/2019/08/29/grid-majors<p>I’ve <a href="/blog/2012/12/16/grid-minors-in.html">written here about grid minors</a> before, <a href="/blog/2013/03/05/more-on-grid.html">more than once</a>. These are <a href="https://en.wikipedia.org/wiki/Lattice_graph">grid graphs</a> that you can get from a given graph <script type="math/tex">G</script> as a <a href="https://en.wikipedia.org/wiki/Graph_minor">minor</a> (that is, by removing edges or vertices from <script type="math/tex">G</script> and contracting others of its edges). So, in the other direction, a “grid major” of <script type="math/tex">G</script> must be a grid graph that has <script type="math/tex">G</script> as a minor. These are the subject of my new preprint, “Homotopy height, grid-major height and graph-drawing height” (<a href="https://arxiv.org/abs/1908.05706">arXiv:1908.05706</a>, with Therese Biedl, Erin Chambers, Arnaud De Mesmay, and Tim Ophelders, to appear at Graph Drawing).</p>
<p>You can define the terms in the paper’s title for any planar graph, but our results are cleanest for <a href="https://en.wikipedia.org/wiki/Maximal_planar_graph">maximal planar graphs</a>. If we define the height of a grid to be the smaller of its two dimensions, then the grid-major height of a graph <script type="math/tex">G</script> is just the smallest height of a grid that has <script type="math/tex">G</script> as a minor. The detailed definition of homotopy height is trickier to explain, but the rough idea is that (after choosing one face of your graph to be the outer one) you should sweep a path from one edge of the outer face, across the graph, to another edge of the outer face, trying as you do to keep the path as short as possible. Paths can move across triangular faces (changing the number of vertices by <script type="math/tex">\pm 1</script>) or path vertices can “slide” along edges (leaving the total number unchanged. The smallest length <script type="math/tex">\ell</script> such that you can do this sweep and always maintain length at most <script type="math/tex">\ell</script> turns out to equal the grid-major height, and both are lower bounds on the height of a conventional drawing of the graph (with its vertices at grid points and its edges as straight line segments).</p>
<p>For instance, the following drawing shows a height-<script type="math/tex">3</script> grid drawing of a maximal planar graph, a sequence of <script type="math/tex">5</script> paths of length at most <script type="math/tex">3</script> sweeping from the left to the right edge of the graph, and the subsets of vertices to contract in a <script type="math/tex">3\times 5</script> grid to produce the graph as a minor. Each column of the grid has the same vertices as the corresponding path in the walk. For this graph, the drawing height, homotopy height, and grid-major height are all equal, but it is possible for the drawing height to be much larger than the other two quantities.</p>
<p style="text-align:center"><img src="/blog/assets/2019/grid-majors.svg" alt="Grid drawing of a maximal planar graph, homotopy sequence for the graph, and grid major representation of the graph" /></p>
<p>Both homotopy height and grid-major height come in simple and non-simple variations. They are equal for both variations, and both variations give a valid lower bound on graph drawing height. In simple homotopy height the swept path is required to stay simple (as it is in the example here) while more generally we allow walks that can repeat vertices. (In particular they can have “spikes”, parts of paths that double back along a single edge.) Correspondingly, non-simple grid-major height does not restrict how <script type="math/tex">G</script> is formed as a minor of a grid while the simple variant requires the parts of the grid corresponding to a single vertex of <script type="math/tex">v</script> to be contiguous within each column of the grid (as they are in the example here).</p>
<p>Because grid-major height behaves nicely under graph minor operations, it is automatically fixed-parameter tractable, but in a non-uniform way. What this means is that for each <script type="math/tex">h</script> there are finitely many graphs that, when they are a minor of <script type="math/tex">G</script>, prevent <script type="math/tex">G</script> from having grid-major height <script type="math/tex">h</script>. We can find the grid-major height of <script type="math/tex">G</script> by looking for these “obstacles” and returning the smallest <script type="math/tex">h</script> for which no obstacles are found. And, whenever <script type="math/tex">h</script> is bounded, the time of the resulting algorithm is a polynomial whose exponent is independent of the size of <script type="math/tex">G</script>. So an algorithm exists for any fixed <script type="math/tex">h</script>, but we don’t know how to actually write it down as a program because we don’t know what the obstacles are or even whether the number of obstacles is computable.</p>
<p>Simple grid-major height is nicer from the algorithmic point of view, but still not entirely satisfactory. Our paper describes the existence of a simple grid major using the <a href="https://en.wikipedia.org/wiki/Logic_of_graphs">logic of graphs</a>. When <script type="math/tex">h</script> is bounded this gives us a linear-time algorithm via <a href="https://en.wikipedia.org/wiki/Courcelle%27s_theorem">Courcelle’s theorem</a>. The algorithm could in principle be written out explicitly as a program,
with <script type="math/tex">h</script> given as an input to the program rather than hardcoded into it. There is no problem with unknown lists of obstacles. But the logical description is huge and the dependence of Courcelle’s theorem on the size of the logical description is very high, so although the time is linear in the size of the graph it gets multiplied by a very quickly growing function <span style="white-space:nowrap">of <script type="math/tex">h</script>.</span></p>
<p>(<a href="https://mathstodon.xyz/@11011110/102706636079725648">Discuss on Mastodon</a>)</p>David EppsteinI’ve written here about grid minors before, more than once. These are grid graphs that you can get from a given graph as a minor (that is, by removing edges or vertices from and contracting others of its edges). So, in the other direction, a “grid major” of must be a grid graph that has as a minor. These are the subject of my new preprint, “Homotopy height, grid-major height and graph-drawing height” (arXiv:1908.05706, with Therese Biedl, Erin Chambers, Arnaud De Mesmay, and Tim Ophelders, to appear at Graph Drawing).Congratulations, Dr. Besa!2019-08-27T18:11:00+00:002019-08-27T18:11:00+00:00https://11011110.github.io/blog/2019/08/27/congratulations-dr-besa<p><a href="http://sites.uci.edu/juanbesa/">Juan Besa’s</a> successful dissertation defense was last Thursday, but I’ve been holding off posting about it here because one of his committee members took ill and couldn’t sign off on it until today.</p>
<p>Juan started his graduate work in embedded systems, but for the last several years he has been one of the theory students at UCI, advised by Mike Goodrich. <a href="https://www.ics.uci.edu/~eppstein/pubs/a-besa.html">My joint papers with him</a> have included <a href="/blog/2016/09/16/directing-traffic.html">algorithms for scheduling traffic signals to allow groups of vehicles to move through them quickly</a> and <a href="/blog/2018/05/09/sorting-when-sorted.html">maintaining an approximately sorted list</a> of items whose sorted order is changing dynamically in a context where comparisons are expensive.</p>
<p>He also has papers on which I was not involved, on chess knights’ tours with few crossings or bends (mentioned briefly with links <a href="/blog/2019/01/31/linkage.html">here</a>), <a href="https://arxiv.org/abs/1907.01630">embedding planar digraphs so each vertex has few alternations between incoming and outgoing edges</a> (to appear at ESA), and another one on low-width tree drawing that has yet to appear publicly as a paper but forms part of his dissertation.</p>
<p>Juan has dual Chilean and Spanish citizenship, and (having the usual double-barreled Spanish surname “Besa Vial”) had to deal with the difficulties of getting that name format understood in non-Spanish-speaking countries; he ended up choosing to truncate it to “Besa” in his recent publications. He is heading to England to work at a machine learning company there. With Brexit coming up quickly, I worry about how that will work out, but I hope it does work well for him.</p>
<p>Congratulations, Juan!</p>
<p>(<a href="https://mathstodon.xyz/@11011110/102692015309277308">Discuss on Mastodon</a>)</p>David EppsteinJuan Besa’s successful dissertation defense was last Thursday, but I’ve been holding off posting about it here because one of his committee members took ill and couldn’t sign off on it until today.Serpentine belts2019-08-22T21:19:00+00:002019-08-22T21:19:00+00:00https://11011110.github.io/blog/2019/08/22/serpentine-belts<p>Many car engines use a <a href="https://en.wikipedia.org/wiki/Serpentine_belt">serpentine belt</a>, passing across multiple pulleys and <a href="https://en.wikipedia.org/wiki/Tensioner">tensioner wheels</a>, to transmit mechanical power and timing information from the car’s crankshaft to its alternator, engine fan, water pump, air conditioning, steering pump, and other systems. Another (usually separate) belt, the <a href="https://en.wikipedia.org/wiki/Timing_belt_(camshaft)">timing belt</a>, similarly connects the crankshaft to the camshaft, which controls and drives the timing of the engine’s valves.
The <a href="https://commons.wikimedia.org/wiki/File:Belt_drive_systen_01.JPG">Volvo bus engine</a> below shows both of these belts:</p>
<p style="text-align:center"><img src="/blog/assets/2019/Volvo-bus-engine-belts.jpg" alt="Timing and serpentine belts of a Volvo bus engine; CC-BY-SA photo by Miya.m from https://commons.wikimedia.org/wiki/File:Belt_drive_systen_01.JPG" style="border-style:solid;border-color:black;" width="60%" /></p>
<p>But suppose you encounter such an engine with its belt removed. How can you tell where to run the belt to connect it all back together again? There may be many different orderings in which you can connect a given set of wheels by a belt. On a real engine (such as the one in a Ford Escort on which I loosely modeled the top image below), you might get a little more information from which of the wheels are ribbed (inside the belt) and which are smooth (outside the belt) but even that extra information won’t always give you a unique solution:</p>
<p style="text-align:center"><img src="/blog/assets/2019/escort.svg" alt="Two serpentine belts for the Ford Escort" /></p>
<p>Connecting wheels with belts like this is the topic of my latest preprint, “Existence and hardness of conveyor belts” (<a href="https://arxiv.org/abs/1908.07668">arXiv:1908.07668</a>, with a large group of co-authors merging groups from the University of Washington and the Bellairs workshops). The name isn’t really accurate, because most conveyor belts have simpler geometry, but that’s what the question was called in past work. At least we managed to avoid the misspelling “conveyer” of some of that work.</p>
<p>To be tight, a belt for a system of disjoint disks (representing the pulleys and wheels) must be a smooth simple closed curve, made by connecting arcs of the disks with <a href="https://en.wikipedia.org/wiki/Tangent_lines_to_circles#Tangent_lines_to_two_circles">bitangents</a> between pairs of disks. It was already known that some systems of disks have no belt, and conjectured that unit disks always have one. Our work shows that it’s <script type="math/tex">\mathsf{NP}</script>-complete to tell whether a belt exists, that they do exist for certain special systems of disks (e.g. when no vertical line crosses multiple disks), and that by adding a linear number of extra disks one can always make a belt exist.</p>
<p>The question of how many different belts can connect the same system of disks (posed to me by co-author Sara Billey) was to a large extent the inspiration for <a href="/blog/2019/03/12/counting-polygon-triangulations.html">my earlier paper on counting triangulations</a>. If counting simple polygons through a set of points is hard, then so is counting belts for tiny disks. And when I couldn’t prove that counting simple polygons is hard, I turned to other problems for which the proof was easier.</p>
<p>As part of this project I wrote <a href="/blog/assets/2019/drawbelts.py">a little Python program to draw systems of disks and their belts</a>, used for the image above and for some of the images in the paper. It doesn’t find which belts are possible itself; instead, you have to specify the belt by listing the disks that it touches in order (allowing repetitions although some of the versions of the problem that we studied do not) and specifying for each disk which direction it turns when the belt touches it (equivalently, whether it is inside or outside the belt). To use it, you’ll need my <a href="https://www.ics.uci.edu/~eppstein/PADS/">PADS Python library</a>, or at least <a href="https://www.ics.uci.edu/~eppstein/PADS/SVG.py">the package in it for generating SVG files</a>.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/102664433496197083">Discuss on Mastodon</a>)</p>David EppsteinMany car engines use a serpentine belt, passing across multiple pulleys and tensioner wheels, to transmit mechanical power and timing information from the car’s crankshaft to its alternator, engine fan, water pump, air conditioning, steering pump, and other systems. Another (usually separate) belt, the timing belt, similarly connects the crankshaft to the camshaft, which controls and drives the timing of the engine’s valves. The Volvo bus engine below shows both of these belts:Footprints in the snow2019-08-17T14:42:00+00:002019-08-17T14:42:00+00:00https://11011110.github.io/blog/2019/08/17/footprints-in-snow<p>Given an abstract optimization problem with multiple solutions, how much partial information about a solution do you have to know in order to uniquely identify that solution? That has been the topic of some of my earlier research, on <a href="/blog/2014/10/08/forced-creases-in.html">how many creases of an origami folding pattern you have to force to be mountain or valley folds in order to cause the remaining folds to go the way you want</a>. And it’s the topic of my new preprint “Tracking paths in planar graphs” (<a href="https://arxiv.org/abs/1908.05445">arXiv:1908.05445</a>, with Mike Goodrich, James Liu, and Pedro Matias).</p>
<p>There’s an old story about how to design the footpaths on a college campus: wait for it to snow, see where the heaviest sets of footprints cross the snow from building to building, and then once the snow melts place paths in those same places. But what if you live somewhere like Irvine where it never snows? Or what if you want to perform some other type of data analysis on a data set of the paths that people take? For instance, in order to design improvements to the road networks used by commuter traffic, it would be helpful to figure out where all the traffic actually goes each day. How can you collect that data?</p>
<p>Our paper takes the point of view that you can attach sensors to the network that record the times and identities of people passing by them, but that these sensors are expensive. The goal is (for a given network with designated start and destination vertices <script type="math/tex">s</script> and <script type="math/tex">t</script>) to place as few of these sensors as possible at graph vertices, in such a way that every simple <script type="math/tex">st</script>-path is uniquely identified by the sequence of sensors that it passes through.</p>
<p>The problem turns out to be <script type="math/tex">\mathsf{NP}</script>-complete, even on planar networks. But there’s a simple approximation ratio based on the idea that the optimal number of sensors is always going to be proportional to the number of faces in the network. Each face (in the sequence of biconnected components between <script type="math/tex">s</script> and <script type="math/tex">t</script>) has to have at least one sensor, to distinguish paths that go one way around the face from paths that go the other way around. It turns out that placing one sensor at a vertex shared by many faces doesn’t work — those faces still need a proportional number of additional sensors. And our approximation algorithm ensures that the number of sensors is at most proportional to the number of faces.</p>
<p>We also use <a href="https://en.wikipedia.org/wiki/Courcelle%27s_theorem">Courcelle’s theorem</a> to prove that the exact solution is fixed-parameter tractable in the clique-width of the graph. Like most or all uses of Courcelle’s theorem, the resulting algorithm is impractical, so it would be of interest to find a more direct algorithm, perhaps for a weaker parameter.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/102634545213040458">Discuss on Mastodon</a>)</p>David EppsteinGiven an abstract optimization problem with multiple solutions, how much partial information about a solution do you have to know in order to uniquely identify that solution? That has been the topic of some of my earlier research, on how many creases of an origami folding pattern you have to force to be mountain or valley folds in order to cause the remaining folds to go the way you want. And it’s the topic of my new preprint “Tracking paths in planar graphs” (arXiv:1908.05445, with Mike Goodrich, James Liu, and Pedro Matias).Linkage2019-08-15T16:07:00+00:002019-08-15T16:07:00+00:00https://11011110.github.io/blog/2019/08/15/linkage<ul>
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<p><a href="http://homepages.gac.edu/~jsiehler/games/pyramids-start.html">Tricolor pyramids</a> (<a href="https://mathstodon.xyz/@11011110/102546072670351246"><script type="math/tex">\mathbb{M}</script></a>). In this logic puzzle by @jsiehler, you have to 3-color hexagonal tiles avoiding 2-colored upright triangles. What interests me is not that, but the following: it’s the time-space diagram of a 3-state cellular automaton (with time flowing upward and each cell taking the color that makes the triangle below it work). But turned <script type="math/tex">120^\circ</script> it’s still the time-space diagram of the same automaton! I haven’t seen this sort of CA symmetry before.</p>
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<p><a href="https://boingboing.net/2019/08/02/publicsphere-v-elsevier.html">Elsevier sends copyright threat to site for linking to Sci-Hub</a> (<a href="https://mathstodon.xyz/@11011110/102551592262978036"><script type="math/tex">\mathbb{M}</script></a>). Apparently if you tell people they can find free copies of paywalled journal papers on pirate web site sci-hub, and link directly to sci-hub to make it even easier to find free copies of paywalled journal papers, you get a nasty letter from the corporate leeches who made it necessary to set up a pirate web site for free copies of paywalled journal papers . But if you <a href="https://en.wikipedia.org/wiki/Sci-Hub">go to Wikipedia</a> you can find the link in the infobox. And it turns out that <a href="https://eve.gd/2019/08/03/elsevier-threatens-others-for-linking-to-sci-hub-but-does-it-itself/">Elsevier journals themselves contain plenty of links to sci-hub</a> (<a href="https://boingboing.net/2019/08/03/zero">via</a>). Sci-hub sci-hub sci-hub.</p>
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<p><a href="https://mathstodon.xyz/@Breakfastisready/102554576764536048">@Breakfastisready recommends the new graphic novel “Prime Suspects”</a>, by Andrew and Jennifer Granville with illustrations by Robert J. Lewis: “It’s all about analytic number theory in metaphors.”</p>
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<p><a href="https://mati.naukas.com/">Mati y sus mateaventuras</a> (<a href="https://mathstodon.xyz/@11011110/102559401183096120"><script type="math/tex">\mathbb{M}</script></a>), blog of popularized mathematics stories by mathematician Clara Grima and illustrator Raquel Gu (in Spanish). The latest one (from a year ago; it hasn’t updated much recently) is <a href="https://en.wikipedia.org/wiki/Wythoff%27s_game">on Wythoff’s game</a>.</p>
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<p><a href="https://emanueleviola.wordpress.com/2019/08/04/because-of-pollution-conferences-should-be-virtual/">Manu suggests reducing our impact on the planet by making conferences virtual</a> (<a href="https://mathstodon.xyz/@11011110/102568139862446088"><script type="math/tex">\mathbb{M}</script></a>). Or we could, you know, publish our papers in journals instead of conferences, like everyone else, and not need to go to quite so many conferences.</p>
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<p><a href="https://blog.computationalcomplexity.org/2019/08/obstacles-to-improving-classical.html">Why modern integer factoring algorithms have the time bounds they do, and what would be needed to improve them</a> (<a href="https://mathstodon.xyz/@11011110/102577633871564197"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://www.insidehighered.com/quicktakes/2019/08/08/california-scientists-pull-support-elsevier-journals">The University of California’s fight with Elsevier spills over to editorships</a> (<a href="https://mathstodon.xyz/@11011110/102584940498877427"><script type="math/tex">\mathbb{M}</script></a>). 30 UC editors of Elsevier journals “will no longer provide editorial services” to Elsevier unless/until a satisfactory deal with Elsevier is reached.</p>
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<p><a href="https://senate.universityofcalifornia.edu/_files/reports/rm-jn-racialization-academic-espionage-concerns.pdf">A letter from the University of California Academic Council</a> (<a href="https://mathstodon.xyz/@11011110/102601263075559877"><script type="math/tex">\mathbb{M}</script></a>) expressing their alarm at “the increasingly racialized ways in which international scholars and students—especially those from China, Iran, and Russia—are being targeted in national conversations about academic espionage” and their support for the open exchange of research.</p>
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<p><a href="https://blogs.scientificamerican.com/roots-of-unity/the-longest-matrilineal-chain-in-math/">The longest matrilineal chain in math</a> (<a href="https://mathstodon.xyz/@11011110/102604836636431847"><script type="math/tex">\mathbb{M}</script></a>). Evelyn Lamb finds “five advisor-advisee chains of length four containing only women” in the Mathematics Genealogy Project, all starting with Olga Ladyzhenskaya and her student Nina Ivochkina. But her searches were haphazard so there may be longer ones still to find.</p>
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<p><a href="https://mathenchant.wordpress.com/2018/08/16/knots-and-narnias/">Knots and Narnias</a> (<a href="https://mathstodon.xyz/@11011110/102613921012752632"><script type="math/tex">\mathbb{M}</script></a>). Riffing on a video of a Bill Thurston lecture, Jim Propp explains that when a portal to another dimension has a knotted boundary, it can actually be a portal to several other dimensions.</p>
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<p><a href="https://gi.de/meldung/konrad-zuse-medaille-dorothea-wagner-erhaelt-hoechste-informatik-auszeichnung/">Dorothea Wagner wins the 2019 Konrad Zuse Medal</a> (<a href="https://mathstodon.xyz/@11011110/102616186862477955"><script type="math/tex">\mathbb{M}</script></a>). This is the highest award of the German Computer Science Society, and the first time since its establishment in 1987 that the winner is a woman. Dorothea’s research includes graph drawing, route planning, optimization, and social network analysis; see <a href="https://en.wikipedia.org/wiki/Dorothea_Wagner">her Wikipedia article</a> for more.</p>
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<p><a href="https://www.quantamagazine.org/new-proof-settles-how-to-approximate-numbers-like-pi-20190814/">Duffin–Schaeffer conjecture solved</a> (<a href="https://mathstodon.xyz/@11011110/102623251151301461"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://arxiv.org/abs/1907.04593">original paper</a>). This is about finding rational approximations to irrational numbers, like <script type="math/tex">\pi=355/113</script>. Given a criterion for how good an approximation you want, depending only on the denominator (for instance, allowing only prime denominators and seeking an approximation accurate to <script type="math/tex">\pm 1/p^{3/2}</script> for denominator <script type="math/tex">p</script>) the new theorem tells you when almost all irrationals have a good-enough approximation.</p>
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</ul>David EppsteinTricolor pyramids (). In this logic puzzle by @jsiehler, you have to 3-color hexagonal tiles avoiding 2-colored upright triangles. What interests me is not that, but the following: it’s the time-space diagram of a 3-state cellular automaton (with time flowing upward and each cell taking the color that makes the triangle below it work). But turned it’s still the time-space diagram of the same automaton! I haven’t seen this sort of CA symmetry before.Report from CCCG2019-08-10T15:34:00+00:002019-08-10T15:34:00+00:00https://11011110.github.io/blog/2019/08/10/report-from-cccg<p>After <a href="/blog/2019/08/07/report-from-wads.html">WADS</a>, I stayed in Edmonton for <a href="https://sites.ualberta.ca/~cccg2019/">CCCG</a>. The two conferences have not always been in the same places, but this year they were co-located, and the plan is to continue that pattern in odd years (when WADS is held). As far as I know there are no plans to move CCCG to Scandinavia for SWAT in the even years.</p>
<p>Like WADS, CCCG had three invited speakers. In past years, two were named the Paul Erdős Memorial Lecture and the Ferran Hurtado Memorial Lecture. This year, sadly, the third one has also been named, as the Godfried Toussaint Memorial Lecture.</p>
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<p>The Erdős Lecture was by Vida Dujmović, who spoke on her breakthrough work with several other Barbados workshop participants showing that <a href="https://arxiv.org/abs/1904.04791">every planar graph is a subgraph of a strong product of a path graph and a bounded-treewidth graph</a>, from which it follows that these graphs have bounded <a href="https://en.wikipedia.org/wiki/Queue_number">queue number</a>, that they can be embedded into 3d grids of linear volume, and many other nice properties. The timing of the lecture invitation to Vida was good, as the breakthrough happened after she had already agreed to speak!</p>
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<p>The Toussaint Lecture was by Joseph O’Rourke. Joe spoke on <a href="https://en.wikipedia.org/wiki/Net_(polyhedron)">polyhedral unfolding</a>, the problem of cutting the boundary of a polyhedron into a surface that can unfold into a simple polygon in the plane. One of the points of his talk was to rationalize some of the terminology in this area. The standard version of the problem asks for (in his new terminology) an <em>edge-unfolding</em>, a set of cuts along edges of the polyhedron, forming a spanning tree for its vertices, such that the resulting cut surface unfolds to a flat polygon. But one can also ask for an anycut-unfolding, using cuts that do not have to follow the edges. Or one can ask for an edge-unzipping or anycut-unzipping, in which the cuts must form a single (Hamiltonian) path through the vertices of the polyhedron. In this terminology <a href="http://www.openproblemgarden.org/op/d_urers_conjecture">Dürer’s conjecture</a> becomes the statement that every convex polyhedron has an edge-unfolding, and the example I recently posted of a <a href="/blog/2019/07/29/zipless-polycube.html">zipless polycube</a> shows that not every polycube has an edge-unzipping. Another well-known open question in this area asks whether every polycube whose boundary forms a topological sphere has an edge-unfolding. Joe conjectured that with high probability the convex hull of many random points on a sphere does not have an anycut-unzipping.</p>
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<p>Mark de Berg presented the Hurtado Lecture. His topic involved subexponential algorithms for disk intersection graphs and <a href="https://en.wikipedia.org/wiki/Unit_disk_graph">unit disk graphs</a>. At STOC 2018 he had a paper on <a href="https://arxiv.org/abs/1803.10633">finding the maximum independent set in disk graphs</a> in time <script type="math/tex">2^{O(\sqrt n)}</script>, matching the time for planar graphs. In planar graphs, you can use the <a href="https://en.wikipedia.org/wiki/Planar_separator_theorem">planar separator theorem</a>: for each of the <script type="math/tex">2^{O(\sqrt n)}</script> independent subsets of the separator, recurse on both sides. This turns out to work in disk graphs by replacing the usual size bound on the separator (it should have <script type="math/tex">O(\sqrt n)</script> vertices) with a decomposition into a union of cliques <script type="math/tex">C_i</script> with <script type="math/tex">\sum\log(\vert C_i\vert+1)=O(\sqrt{n})</script>. The separators can be found analogously to classical circle-packing methods for planar separators. Each clique can contribute one vertex to any independent set from which it follows that the separator again has <script type="math/tex">2^{O(\sqrt n)}</script> independent subsets. The same idea works for other problems like dominating sets in unit disk graphs (where the unit assumption is used to get a bounded contribution from each clique), and generalizes to fat objects in higher dimensions. The time bound is optimal assuming the <a href="https://en.wikipedia.org/wiki/Exponential_time_hypothesis">exponential time hypothesis</a>. And in FOCS 2018 de Berg obtained similar <a href="https://arxiv.org/abs/1807.06933">ETH-tight time bounds for the Euclidean traveling salesperson problem</a> by using separators of point sets with the property that few points are very close to the separator boundary.</p>
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<p>I can’t find links for all the contributed papers, but you can find them in the <a href="https://sites.ualberta.ca/~cccg2019/cccg2019_proceedings.pdf">complete proceedings</a>. Among them:</p>
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<p>In “Three-Coloring Three-Dimensional Uniform Hypergraphs”, Biniaz, Bose, Cardinal, and Payne prove that, for <script type="math/tex">n</script> points in the plane and a fixed triangle shape, one can <script type="math/tex">3</script>-color the points so that every scaled and translated copy of the triangle containing six or more points has more than one color. It was already known that if you change “six or more” to “two or more” you need four colors, and if you change it to “nine or more” you need only two colors.</p>
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<p>Audrey St. John’s talk on “Redundant Persistent Acyclic Formations for Vision-Based Control of Distributed Multi-Agent Formations” (with Burns, Klemperer, and Solyst) was beset by technical difficulties, but from it I learned that there is a theory of directed bar-and-joint frameworks, analogous to undirected rigidity theory, called “persistence theory”, and that the <a href="/blog/2013/12/07/kinematic-chains-and.html">pebble game</a> for testing rigidity of an undirected framework produces an orientation of the network that is persistent. She used the analogy of a flock of geese, walking in formation: each goose pays attention only to the other geese in front, but the whole formation can keep its shape as the leading goose moves arbitrarily. Her goal is to get robots to do the same thing.</p>
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<p>In “Chirotopes of Random Points in Space are Realizable on a Small Integer Grid”, Cardinal, Fabila-Monroy and Hidalgo-Toscano prove that, with high probability, random point sets in <script type="math/tex">\mathbb{R}^d</script> can be rounded to a grid of polynomial size without changing their order type.</p>
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<p>We had enough folding and unfolding papers to spill out over more than one section. Among them, I particularly liked “Efficient Segment Folding is Hard” by Klute, Parada, Horiyama, Korman, Uehara and Yamanaka. The question they asked is: given disjoint line segments on a piece of paper, when can you make a sequence of simple folds (that is, for a given fold line, folding all the layers of the paper that are crossed by the line), with each fold on a line through one of the segments that misses all the other segments? It turns out to be <script type="math/tex">\mathsf{NP}</script>-complete. If you do allow fold lines to pass through other segments, folding sequences can be infinite, and it’s unknown whether every set of segments has a finite sequence.</p>
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<p>Pilar Cano spoke on generating triangulations of point sets in an affine-invariant way (“Affine invariant triangulations” with Bose and Silveira). The main trick is to use covariance to choose a canonical affine transformation for the points, after which you can use Delaunay, minimum weight, or your favorite other triangulation algorithm. But there are necessarily some general position assumptions (as there already are for using Delaunay triangulation without the affine invariance): for points in a parallelogram, there is no affine-invariant way of choosing which diagonal to use.</p>
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<p>The excursion was to the <a href="https://royalalbertamuseum.ca/">Royal Alberta Museum</a>, where I skipped the special exhibit on Vikings (having gone to museum exhibits on them in Copenhagen a year earlier) and instead learned much about Great Plains geology and the historical mistreatment of the <a href="https://en.wikipedia.org/wiki/M%C3%A9tis">Métis</a>, local people descending both from First Nations and Europeans. (The First Nations themselves were of course also badly mistreated, but I had more of an idea of that already.)</p>
<p>From the business meeting, we heard that the acceptance ratio was a little higher than last year, but still approximately <script type="math/tex">75\%</script>. Two papers were withdrawn because the authors had visa issues, double the number from last year, and several others were presented by non-authors after their authors were unable to attend. One possible improvement would be to move the submission and acceptance dates earlier to provide authors more time to obtain visas. The main topic of discussion was the conference’s status as a conference: should papers at CCCG continue to count as publications (in which case why are they still limited to only six pages) or should they be considered as preliminary announcements of papers that can still be sent to other more prestigious symposia? One possible compromise involves giving authors a choice: either publish your paper in the proceedings or give up the proceedings slot but still present your work in some other way (possibly as a poster, as GD does).</p>
<p>(<a href="https://mathstodon.xyz/@11011110/102595225020207137">Discuss on Mastodon</a>)</p>David EppsteinAfter WADS, I stayed in Edmonton for CCCG. The two conferences have not always been in the same places, but this year they were co-located, and the plan is to continue that pattern in odd years (when WADS is held). As far as I know there are no plans to move CCCG to Scandinavia for SWAT in the even years.University of Alberta Botanic Gardens2019-08-09T11:46:00+00:002019-08-09T11:46:00+00:00https://11011110.github.io/blog/2019/08/09/university-alberta-botanic<p>The WADS excursion was to the University of Alberta Botanic Gardens.
Here are a few photos I took there:</p>
<div><table style="margin-left:auto;margin-right:auto">
<tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><a href="http://www.ics.uci.edu/~eppstein/pix/uabg/AgaKhanSourceFountain.html"><img src="http://www.ics.uci.edu/~eppstein/pix/uabg/AgaKhanSourceFountain-m.jpg" alt="University of Alberta Botanic Gardens, Aga Khan Garden, Source Fountain" width="300" style="border-style:solid;border-color:black;" /></a></td>
<td style="padding:10px"><a href="http://www.ics.uci.edu/~eppstein/pix/uabg/AgaKhanJilauKhana.html"><img src="http://www.ics.uci.edu/~eppstein/pix/uabg/AgaKhanJilauKhana-m.jpg" alt="University of Alberta Botanic Gardens, Aga Khan Garden, Jilau Khana" width="405" style="border-style:solid;border-color:black;" /></a></td>
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<td style="padding:10px"><a href="http://www.ics.uci.edu/~eppstein/pix/uabg/AgaKhanMahtabi.html"><img src="http://www.ics.uci.edu/~eppstein/pix/uabg/AgaKhanMahtabi-m.jpg" alt="University of Alberta Botanic Gardens, Aga Khan Garden, Mahtabi" width="390" style="border-style:solid;border-color:black;" /></a></td>
<td style="padding:10px"><a href="http://www.ics.uci.edu/~eppstein/pix/uabg/WetlandWalkMaysDock.html"><img src="http://www.ics.uci.edu/~eppstein/pix/uabg/WetlandWalkMaysDock-m.jpg" alt="University of Alberta Botanic Gardens, Wetland Walk, May's Dock" width="385" style="border-style:solid;border-color:black;" /></a></td>
</tr></table></div>
<p>(<a href="https://mathstodon.xyz/@11011110/102588301669141700">Discuss on Mastodon</a>)</p>David EppsteinThe WADS excursion was to the University of Alberta Botanic Gardens. Here are a few photos I took there:Report from WADS2019-08-07T17:27:00+00:002019-08-07T17:27:00+00:00https://11011110.github.io/blog/2019/08/07/report-from-wads<p>I’m in Edmonton, Canada for <a href="http://wads.org/">WADS</a>, which just finished, and <a href="http://cccg.ca/">CCCG</a>, which is just about to begin.</p>
<p>The three invited talks at WADS were by Rasmus Pagh, Bob Tarjan, and me. Pagh spoke on methods for representing sets of elements by concise sketches so that the size of intersections or unions of the sets could be rapidly and accurately estimated. A famous method for this is <a href="https://en.wikipedia.org/wiki/MinHash">MinHash</a>, in which one represents a set by the <script type="math/tex">k</script> elements with the smallest values of some hash function; the size of the overlap in representations is then an accurate estimator for the <a href="https://en.wikipedia.org/wiki/Jaccard_index">Jaccard similarity</a> of pairs of sets. New to me were <script type="math/tex">1</script>-bit variations of MinHash, in which you can get almost as accurate a representation in much less space by mapping each element of the MinHash set to <script type="math/tex">\{0,1\}</script> by another hash function. This works well when the Jaccard similarity is bounded away from both <script type="math/tex">0</script> and <script type="math/tex">1</script>, and Pagh spoke about some recent research he and others had done on finding even more accurate methods when it is near <script type="math/tex">0</script> or near <script type="math/tex">1</script>.</p>
<p>Tarjan spoke about <a href="https://arxiv.org/abs/1812.06177">parallel algorithms for connected components in graphs</a>, an old area but one in which apparently there have been frequent published mistakes. He presented a modular analysis of the algorithms in this area according to some basic operations they perform (hooking together roots of trees on components, propagating that root information downwards through the trees, flattening the trees to make the information propagate more quickly, and the like) and showed that simple combinations of these operations lead to new, simple, efficient and more importantly provably-correct algorithms.</p>
<p>My talk, “Graphs in Nature”, was about finding graph-theoretic characterizations of surface-embedded graphs arising in natural processes, and using those characterizations to find algorithms to reconstruct synthetic geometric structures of the same type from their graphs. I also gave roughly the same talk a month earlier, at the Symposium on Geometry Processing in Milan. <a href="https://www.ics.uci.edu/~eppstein/pubs/Epp-WADS-19.pdf">I’ve put my talk slides online</a> in case anyone else is interested.</p>
<p>The best paper award went to Hüseyin Acan, Sankardeep Chakraborty, Seungbum Jo and Srinivasa Rao Satti for their paper “<a href="https://arxiv.org/abs/1902.09228">Succinct Data Structures for Families of Interval Graphs</a>”. I can’t tell you much about the talk because, unfortunately, I missed it. I didn’t know it was the best paper until the business meeting that evening, so I went to the other parallel session instead.</p>
<p>I think the contributed talk from Tuesday that most stood out to me was Bryce Sandlund’s, on offline dynamic graph algorithms. This is a type of problem <a href="https://doi.org/10.1006/jagm.1994.1033">I worked on long ago for minimum spanning trees</a> in which you get as input a whole sequence of edge insertions and deletions in a graph, and must produce as output the sequence of changes to the solution to whatever you’re trying to solve. <a href="http://doi.org/10.1007/978-3-030-24766-9_40">Bryce’s new paper with Peng and Sleator</a> solves similar problems for higher-order graph connectivity. The main idea is to hierarchically decompose the update sequence into intervals, and then represent the non-dynamic part of the graph within each interval by a smaller equivalent replacement graph whose size is proportional to the interval length. At the end of his talk, Bryce hinted that he could also solve incremental problems (where the updates are given one at a time rather than all in advance, but are only insertions) using similar methods in a forthcoming paper.</p>
<p>I was inspired by Caleb Levy’s talk on <a href="https://en.wikipedia.org/wiki/Splay_tree">splay trees</a> (in which he showed that <a href="https://arxiv.org/abs/1907.06309">inserting elements in either the preorder or postorder of another binary search tree takes linear time</a>) to ask the following question: we know either by time-reversing the tree rotation operations or from the <a href="https://en.wikipedia.org/wiki/Geometry_of_binary_search_trees">geometric model of dynamic search trees</a> that any given access sequence should have the same optimal cost as its reverse. So from the <a href="https://en.wikipedia.org/wiki/Optimal_binary_search_tree">dynamic optimality conjecture</a> it should also be true that (up to constant factors) splay trees have the same performance on the reverse of any access sequence as they do on the unreversed sequence. Can this be proven?</p>
<p>From the business meeting, we learned that attendance and paper submissions were down by around 15% from the previous WADS. The acceptance rate is roughly the same, just under 50%. I suspect the smaller size is because the location is not as appealing, but it turns out to be a perfectly pleasant place to have a conference: the weather in Edmonton is pleasant this time of year (except for the thunderstorm), and there are abundant restaurants, good coffee shops, and lodging within walking distance of the conference center. WADS alternates with SWAT, which next year will be in the Faroe Islands. And then WADS 2021 (and CCCG 2021) will be in Halifax, Nova Scotia, which is both more touristy than Edmonton and easier to reach from the east coast and Europe. So I suspect the numbers will improve again.</p>
<p>WADS is moving towards a more democratically elected steering committee formed from some combination of past PC chairs and at-large elections. They have already started implementing the <a href="https://www.ics.uci.edu/~irani/safetoc.html">SafeTOC recommendations</a> for combatting harassment and discrimination in theory conferences. And in a show of hands at the business meeting, the attendees were strongly in favor of moving towards double blind peer review for submissions. The conference is not really open access, though (its proceedings are published by Springer LNCS with special issues in Springer’s <em>Algorithmica</em> and Elsevier’s <em>Computational Geometry</em>) and there seems to be little pressure for that to change any time soon.</p>
<p>On to CCCG!</p>
<p>(<a href="https://mathstodon.xyz/@11011110/102578298917323647">Discuss on Mastodon</a>)</p>David EppsteinI’m in Edmonton, Canada for WADS, which just finished, and CCCG, which is just about to begin.