Jekyll2018-02-16T04:45:34+00:00https://11011110.github.io/blog/11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage2018-02-15T20:44:00+00:002018-02-15T20:44:00+00:00https://11011110.github.io/blog/2018/02/15/linkage<ul>
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<p><a href="http://www.atlasobscura.com/places/anonymouse-shops-for-mice">Tiny mouse shops in Malmö</a> (<a href="https://plus.google.com/100003628603413742554/posts/HCPvveUSEN3">G+</a>, <a href="https://plus.google.com/105473622219622697310/posts/g9kmDi62rSG">via</a>). One of the local attractions you can look forward to visiting if you <a href="http://csconferences.mah.se/swat2018/index.html">submit your algorithms papers to SWAT</a> (submission deadline February 18).</p>
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<p>A tip for breaking out of the filter bubble Google+ is trying to impose on you, and see all the posts from all your contacts (<a href="https://plus.google.com/100003628603413742554/posts/RQgGCveuXbD">G+</a>).</p>
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<p><a href="https://tinyurl.com/y8hdw6lm">Spherical cross-section of a hyperbolic buckyball honeycomb flythrough</a> (<a href="https://plus.google.com/100003628603413742554/posts/cutvK9Vk3r8">G+</a>, <a href="https://plus.google.com/+RoiceNelson/posts/YSk9g3UuXPX">via</a>).</p>
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<p><a href="https://arxiv.org/abs/1705.07621">The grasshopper problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/MPjYjsgBEuR">G+</a>, <a href="https://mathstodon.xyz/@esoterica">via</a>). Strange gear-like shapes with the property that if you choose a uniformly random point within it and then jump in a random direction a unit step away, your probability of landing within the shape is maximized.</p>
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<p><a href="http://laurajdt.tumblr.com/post/168646338790/winchysteria-ossacordis-crockpotcauldron">The orangutan story</a> (<a href="https://plus.google.com/100003628603413742554/posts/ZYodTt2pEXr">G+</a>, <a href="https://www.metafilter.com/172247/BUT-WHAT-ABOUT-THE-ORANGUTAN">via</a>). Disagreements at theory conferences seem so much more staid.</p>
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<p><a href="https://www.nytimes.com/2018/01/31/business/dealbook/xerox-fujifilm.html">Xerox ends its life as an independent company</a> (<a href="https://plus.google.com/100003628603413742554/posts/E1CvekvxSd8">G+</a>, <a href="https://plus.google.com/112589180065090941598/posts/Efjkfas9RdG">via</a>). The NYT article features a photo of my high-school friend Kathy Van Stone, as a young teen in the PARC Smalltalk experiment.</p>
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<p><a href="https://www.youtube.com/watch?v=YdFHYW1aYAQ">Beyond the OEIS: Fingerprint databases for theorems</a> (<a href="https://plus.google.com/100003628603413742554/posts/hb9pARsV4db">G+</a>, <a href="https://plus.google.com/+LuisGuzmanJr/posts/FwT9G2AcG1i">via</a>). A talk by Sara Billey.</p>
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<p><a href="https://medium.com/@kma500/whos-important-a-tale-from-wikipedia-a370dc6ef078">Who’s Important? A tale from Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/Fu5pZ3jBkhz">G+</a>). Kirsten Menger-Anderson writes on women in mathematics, Erdős numbers, and Wikipedia notability.</p>
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<p><a href="https://blog.plover.com/math/petersen-graph.html">The many faces of the Petersen graph</a> (<a href="https://plus.google.com/100003628603413742554/posts/Cp7ui1kE7Tu">G+</a>). Mark-Jason Dominus finds a drawing I didn’t know, with the symmetries of a square. To deal with vertices landing on top of each other, he uses dumbell-shaped vertices that can cross each other without interacting. Also with a mini-rant about GraphViz not helping.</p>
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<p><a href="http://www.sagradafamilia.org/en/geometry/">The geometry of the Sagrada Familia</a> (<a href="https://plus.google.com/100003628603413742554/posts/BUkM5ctAdrN">G+</a>). I was inspired to find this by a recent post on a different <a href="https://weburbanist.com/2018/01/21/fractal-chapel-tree-inspired-columns-branch-out-to-open-up-interior-space/">chapel with a fractal-forest interior</a>, but Gaudí did it earlier and better.</p>
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<p><a href="https://blogs.scientificamerican.com/roots-of-unity/how-to-look-at-art-a-mathematician-s-perspective/">Why you should take chopsticks with you to art galleries and museums</a> (<a href="https://plus.google.com/100003628603413742554/posts/ZtmRJooy4iA">G+</a>). Evelyn Lamb on Annalisa Crannell’s studies of geometric perspective in art.</p>
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</ul>David EppsteinTiny mouse shops in Malmö (G+, via). One of the local attractions you can look forward to visiting if you submit your algorithms papers to SWAT (submission deadline February 18).#MeToo in theoretical computer science2018-02-14T10:06:00+00:002018-02-14T10:06:00+00:00https://11011110.github.io/blog/2018/02/14/metoo-in-theory<p>This guest post was sent to me by a colleague who wishes to remain anonymous. It is also crossposted on <a href="http://blog.geomblog.org/2018/02/a-metoo-testimonial-that-hits-close-to.html">Geomblog</a> and <a href="http://3dpancakes.typepad.com/ernie/2018/02/metoo-guest-post.html">Ernie’s 3d pancakes</a>,
and to several posts on Google+ (<a href="https://plus.google.com/+SureshVenkatasubramanian/posts/XndbETsRpJT">1</a> <a href="https://plus.google.com/118250784898510714238/posts/CVRSAq7uqzi">2</a> <a href="https://plus.google.com/+JeffErickson/posts/EUV8HK48H8x">3</a> <a href="https://plus.google.com/100003628603413742554/posts/apqLeqjob45">4</a>).</p>
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<p>Every #MeToo story over the last several months has made me pause. My heart races and my concentration fails. The fact that the stories have largely focused on the workplace adds to my difficulty.</p>
<p>Do I speak out too?</p>
<p>I have shared a few stories with colleagues about things that have happened to me in school and at work. But these stories have been somewhat lighthearted events that have been easy to share without outing the perpetrators.</p>
<p>For example, I have told a story about a university employee telling me, in so many words, that I should be barefoot and pregnant and not in the office. What I didn’t share is that the same employee, later that year – despite the fact that our common boss knew about this story because I did indeed report it – was awarded a best employee award. How do you think that made me feel? Like my experience didn’t matter and that such comments are condoned by our department. Why didn’t I share that information widely? Because I was worried that folks would then be able to figure out who the culprit was. And isn’t that even worse? Shouldn’t it be the sexist who is worried and not the woman who, yet again, is made to feel like she doesn’t belong?</p>
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<p>Let me tangent a bit. For years I have not flown. Ostensibly I stopped flying because of the contribution to the climate crisis. When I travel, I go by train. It takes longer, but has been surprisingly pleasant. And when travel takes 3-4 times as long, you don’t do it as often, further reducing your carbon footprint. Of course, that means that I don’t go to conferences unless they are nearby.</p>
<p>But when I really think about it, is this really the reason I stopped going to conferences? A conference I would normally go to was held nearby a few years ago and I didn’t go. Sure, I suffered a grievous injury two weeks before, but I hadn’t even registered. I had planned to not go long before that injury.</p>
<p>So, really, why do I no longer attend conferences? Partly I don’t feel that I need to anymore, now that I have tenure. When I stopped attending conferences, I was able to “coast into” tenure. Letter writers would remember me. I essentially stopped going to conferences and workshops as soon as I possibly could. </p>
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<p>Back to the beginning, or close to.</p>
<p>I was nervous at the first conference I attended as a graduate student. One of the reasons I was nervous was that I was athletic at the time and planned on daily runs while I was attending – I was worried that it might be viewed as a waste of time. My advisor, who also went to the conference, found out about my athleticism and suggested we run together. This was a relief to me. That is, until we were running and he started talking about his lackluster sex life with his wife. I responded by picking up the pace and feigning an illness on the remaining days. On the last day of the conference we were out for dinner with a large group of people and dinner went late into the night. I excused myself, as I had a 4AM bus to catch. My advisor walked me out of the restaurant and awkwardly said something about wanting me to stay and that we should talk. I stuck to leaving, knowing that I needed some sleep before the long trip home the next day. He said we should talk when we were back in the office. Honestly, at the time I thought he was going to complain about my talk or my professional performance in some way. I worried about it all through the weekend until we met next. I brought it up at the end of our meeting, asking what he wanted to talk about, naively expecting professional criticism. When he said I must surely know, in a certain voice, I knew he wasn’t talking about work. I feigned ignorance, and he eventually brushed it off and said not to worry. In the coming months, he would cancel meetings and otherwise make himself unavailable. After a half year I realized I wouldn’t be able to be successful without having a supportive advisor and, despite first planning to quit grad school, found a new advisor and moved on. That former advisor barely made eye contact with me for the remainder of my time in graduate school.</p>
<p>Fast forward many years. I was at a small workshop as a postdoc. A senior and highly respected researcher invited me to dinner. I was excited at the opportunity to make a stronger connection that would hopefully lead to a collaboration. However, at dinner he made it very clear that this was not professional by reaching across the table and stroking my hands repeatedly. I don’t even recall how I handled it. Perhaps I should have expected it – a grad school friend of mine had a similar, and probably worse, interaction with this same researcher. Shortly after I got to my room at the hotel, my hotel room phone rang. It was him. He wanted to continue our conversation. I did not.</p>
<p>Perhaps a year later, still as a postdoc, I was at a party and a colleague from another university was there too. At the end of the party, we were alone. We flirted, mutually. Flirting led to kissing, kissing led to him picking me up in a way that asserted how much stronger he is than me, which led to my utter discomfort, which led to me saying no, stop, repeatedly. Which he didn’t listen to. Which led to a calculation in my head. I could either resist and risk physical injury or I could submit. I chose to submit, without consent.</p>
<p>For the record, that is called rape.</p>
<p>For a long while, I suppressed it. I pretended in my own head that it didn’t happen that way, that it was consensual. I even tried to continue working with him – always in public places, mind you. The wall in my mind gradually broke down over the years until several years later, we were at the same workshop where the doors of the rooms didn’t have locks. You could lock them from the inside, but not the outside. I remember worrying that he would be lurking in my room and took to making sure I knew where he was before I ventured back to sleep.</p>
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<p>So why would I continue to go to workshops and conferences when that is the environment I know I will face? Even if I felt safe, if 95% of the attendees are men, how many look at me as a colleague and how many look at me as a potential score? When I was going up for tenure, I thought long and hard about listing the senior-and-highly-respected researcher on a do-not-ask-for-a-letter list. But where would it stop? Do I include all the people who hit on me? All the people who stared at my breasts or commented on my body? All the people who I had been given clear signals that they didn’t see me as a colleague and equal member of the research community, but as a woman meant to be looked at, hit on, touched inappropriately.</p>
<p>Should I have quit grad school when I had the chance? We all know it isn’t any better in industry. Should I have pursued another discipline? No discipline, it seems, is immune to sexualization of women. But I think the situation is uniquely horrible in fields where there are so few women. At conferences in theoretical computer science, 5-10% of the attendees are women, as a generous estimate. The numbers aren’t in women’s favor. The chances that you will get hit on, harassed, assaulted are much higher. There is a greater probability that you will be on your own in a group of men. You can’t escape working with men. It is next to impossible to build a career when you start striking men off your list of collaborators in such a field. That is not to say there aren’t wonderful men to work with. There are many men in our field that I have worked with and turned to for advice and spent long hours with and never once had detected so much as a creepy vibe. But you can’t escape having to deal with the many others who aren’t good. When you meet someone at a conference, and they invite you for a drink or dinner to continue the conversation, how do you know that they actually want to talk about work, or at least treat you as they would any colleague? How do you make that decision?</p>
<p>I hung on until I no longer needed to go to conferences and workshops to advance my career to the stability of tenure. But surely my career going forward will suffer. My decision is also hard on my students, who go to conferences on their own without someone to introduce them around. It is hard on my students who can’t, for visa difficulties, go to the international conferences that I am also unwilling to go to, so we roll the dice on the few domestic conferences they can go to.</p>
<p>And now I am switching fields. Completely. I went to two conferences last summer. The first, I brought the protective shield of my child and partner. The second, I basically showed up for my talk and nothing else. I wasn’t interested in schmoozing. It’ll be difficult, for sure, to establish myself in a new field without fully participating in the expected ways.</p>
<p>Is all this why I am switching fields? Not entirely, I’m sure, but it must have played a big role. If I enjoyed conferences as much as everyone else seems to, and didn’t feel shy about starting new collaborations, I might be too engrossed to consider reasons to leave. And certainly, the directions I am pursuing are lending themselves to a much greater chance of working with women.</p>
<p>Why am I speaking out now? The #MeToo moment is forcing me to think about it, of course. But I have been thinking about this for years. I hope it will be a relief to get it off my chest. I have been “getting on with it” for long enough. 1 in 5 women will deal with rape in their lifetime. 1 in 5! You would think that I would hear about this from friends. But I hadn’t told anyone about my rape. And almost no one has told me about theirs. I think it would help, in the very least therapeutically, to talk about it.</p>
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<p>I thought about publishing this somewhere, anonymously, as a “woman in STEM”. I considered publishing it non-anonymously, but was shy to deal with the trolls. I didn’t want to deal with what many women who speak out about their experiences face: have their life be scrutinized, hear excuses being made on behalf of the predators, generally have their experiences denied. But I think by posting it here, many people in theoretical computer science will read it, rather than a few from the choir. I am hoping that you will talk to each other about it. That you will start thinking of ways to make our community better for others. In all my years of going to conferences and workshops, of all the inappropriate comments and behaviors that others have stood around and witnessed, never once did any of the good ones call that behavior out or intervene. Maybe they did so in private, but I think it needs to be made public. Even the good ones can do better.</p>
<p>What can you do?</p>
<p>While you couldn’t have protected me from being raped, you can think about the situations we are expected to be in for our careers – at workshops in remote locations, where we’re expected to drink and be merry after hours. I hope not many of us have been raped by a colleague, but even if you haven’t, it doesn’t take many instances of being hit on or touched inappropriately to begin to feel unsafe.</p>
<p>I remember being at a conference and, standing in a small group, an attendee interrupted a conversation I was having to tell me that my haircut wasn’t good, that I shouldn’t have cut my hair short. I tried to ignore it, and continue my conversation, but he kept going on about it. Saying how I would never attract a man with that haircut. No one said anything. Speak up. Just say – shut up! – that’s not appropriate. Don’t leave it up to the people who have to deal with this day in day out to deal with it on their own. Create a culture where we treat each other with respect and don’t silently tolerate objectification and worse.</p>
<p>I regret never reporting my first graduate advisor’s behavior, but is it my fault? I had no idea who to report it to. I had no idea either in undergrad who I would report such behavior to. Where I am now is the first place I’ve been that has had clear channels for reporting sexual harassment and other damaging situations. The channels are not without problems, but I think the university is continuing to improve them. Perhaps we should have a way of reporting incidents in our field. I have a hard time believing, given that myself and a grad school friend had similar experiences with the same senior-and-highly-respected researcher, that others in the field don’t know that he is a creep. It is up to you to protect the vulnerable of our community from creeps and predators. Keep an eye on them. Talk to them. Don’t enable them. As a last resort, shame and isolate them.</p>David EppsteinThis guest post was sent to me by a colleague who wishes to remain anonymous. It is also crossposted on Geomblog and Ernie’s 3d pancakes, and to several posts on Google+ (1 2 3 4).Small hyperbolic tiles2018-02-03T19:26:00+00:002018-02-03T19:26:00+00:00https://11011110.github.io/blog/2018/02/03/small-hyperbolic-tiles<p>This is the <a href="https://en.wikipedia.org/wiki/Binary_tiling">binary tiling</a>, a tiling of the hyperbolic plane by congruent convex pentagons. (It’s usually shown more like a quadtree, like the top image of the Wikipedia article, but the tiles in that version aren’t convex and aren’t polygons.) We now have a <a href="https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/">complete classification of convex pentagons that tile the Euclidean plane</a>, but as this example shows the hyperbolic case is more complicated.</p>
<p style="text-align:center"><img src="/blog/assets/2018/binary-tiling.svg" alt="Binary tiling" /></p>
<p>The article on Wikipedia is quite new, added after another editor saw Ian Agol’s nice answer to <a href="https://mathoverflow.net/q/291452/440">a MathOverflow question asking for the smallest monohedral tile on the hyperbolic plane</a>. Agol used the binary tiling to show the area can be made arbitrarily small (essentially, you can make the horizontal dimension of the tiles in this image arbitrarily small without much changing the vertical dimension nor the overall pattern of the tiling), and Douglas Zare added that, by making tiles with <script type="math/tex">k</script> arcs on the top and <script type="math/tex">k+1</script> on the bottom, you can also make the diameter arbitrarily small. The ones in the image above are drawn fairly wide, in the Poincaré halfplane model of the hyperbolic plane, in order to make their angles 60–120–60–120–120; the thinner ones have messier angles.</p>
<p>It turns out that these tilings have been known for quite a while, at least since a <a href="http://real-j.mtak.hu/9373/">1974 paper of Böröczky</a> (but beware, the download is a huge pdf and the paper is in Hungarian). The tiling is less symmetric than it looks, as can be seen by encoding each tile’s situation in the tiling by a binary sequence that describes whether the tiles directly above it extend left or right. There are countably many tiles in a tiling, but uncountably many sequences, so (like the Penrose tiles) there are actually infinitely many different tilings with the same tiles. Two tiles map to each other by a symmetry of their tiling when they have the same sequence, and belong to the same tiling when their sequences differ by a finite shift and a change of finitely many bits. The repeating sequences (the ones corresponding to binary representations of rational numbers) come from tilings with an infinite cyclic group of symmetries (but with unbounded fundamental domains) while the irrational numbers give completely aperiodic tilings.</p>
<p>Here are two related questions I don’t know the answer to:</p>
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<p>How can you draw the tiles of this tiling using only a compass and straightedge within the hyperbolic plane? Using the Euclidean model as in this image rather than staying within hyperbolic geometry is cheating. (What I actually want to know is how to draw it in <a href="https://www.cinderella.de/tiki-index.php">Cinderella</a>.)</p>
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<p>Do there exist arbitrarily-small-diameter monohedral tiles on the sphere? (There do exist arbitrarily-small-area tiles, as the MathOverflow question already noted.)</p>
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<p>(<a href="https://plus.google.com/100003628603413742554/posts/QKj17nSEsZV">G+</a>)</p>David EppsteinThis is the binary tiling, a tiling of the hyperbolic plane by congruent convex pentagons. (It’s usually shown more like a quadtree, like the top image of the Wikipedia article, but the tiles in that version aren’t convex and aren’t polygons.) We now have a complete classification of convex pentagons that tile the Euclidean plane, but as this example shows the hyperbolic case is more complicated.Parallel matching in one-crossing-minor-free graphs2018-02-01T20:25:00+00:002018-02-01T20:25:00+00:00https://11011110.github.io/blog/2018/02/01/parallel-matching-in<p>One of the great things about having new colleagues is the potential new energy, new collaborations, new research directions, and new synergies they bring.
<a href="http://www.ics.uci.edu/~vazirani/">Vijay Vazirani</a>, now a Distinguished Professor here at UCI, has brought all four.
Vijay has been working on problems related to <a href="https://en.wikipedia.org/wiki/Matching_(graph_theory)">graph matching</a> for almost 40 years, he tells me, starting with his discovery of the <a href="https://en.wikipedia.org/wiki/Hopcroft–Karp_algorithm#Non-bipartite_graphs">Micali–Vazirani matching algorithm</a> in his first year as a graduate student. Our new preprint, “NC Algorithms for Perfect Matching and Maximum Flow in One-Crossing-Minor-Free Graphs” (<a href="https://arxiv.org/abs/1802.00084">arXiv:1802.00084</a>) continues both that theme and a theme in my own research of graph structure theory.</p>
<p>Parallel algorithms for graph matching were a hot topic in 1988 when Vijay published a paper that became the predecessor to this one, showing that one could <a href="https://ecommons.cornell.edu/handle/1813/6700">count the perfect matchings in <script type="math/tex">K_{3,3}</script>-minor-free graphs</a> in <a href="https://en.wikipedia.org/wiki/NC_(complexity)">NC</a>, a complexity class describing parallel algorithms with polylog runtime and polynomially many processors. That may sound somewhat specialized, but this is actually a natural graph class to look at for this problem, because these graphs have <a href="https://en.wikipedia.org/wiki/Pfaffian_orientation">Pfaffian orientations</a> allowing their matchings to be counted using matrix determinants, something that’s not true of any graph family that includes <script type="math/tex">K_{3,3}</script>. Earlier, it had been shown by Vijay and others that matchings could be found by randomized parallel algorithms, in all graphs, but finding them deterministically in NC was and is a big open problem. Counting and finding are the same for randomized algorithms, by Vijay’s <a href="https://en.wikipedia.org/wiki/Isolation_lemma">isolation lemma</a>, but nobody knows how to make that work deterministically. In his paper on <script type="math/tex">K_{3,3}</script>-free graphs, Vijay posed a special case of this question: can we find matchings in these graphs deterministically?</p>
<p>Since then, parallel algorithms and parallel matching went through a long lull, but now they’re hot again. Derandomized versions of the isolation lemma have led to matching algorithms in QNC (polylog time but with a bigger quasipolynomial number of processors) beginning with the <a href="https://arxiv.org/abs/1601.06319">bipartite algorithm of Fenner, Gurjar, and Thierauf</a> at STOC 2016, its <a href="https://arxiv.org/abs/1704.01929">generalization to all graphs by Svensson and Tarnawski</a>, and <a href="https://arxiv.org/abs/1708.02222">further generalizations by Gurjar, Thierauf, and Vishnoi</a>. Vijay got back into the picture himself, with his new paper with Nima Anari showing that <a href="https://arxiv.org/abs/1709.07822">matchings could be found in planar graphs and bounded-genus graphs in NC</a>. Most of this work is so new that I only have arXiv links for it, but I expect it to start appearing in good conferences soon.</p>
<p>Our new paper builds on the planar algorithm of Anari and Vijay, extending it to all one-crossing-minor-free graphs families. “One-crossing-minor-free” means that these are minor-closed graph families in which one of the forbidden minors can be drawn in the plane with only one crossing, as <script type="math/tex">K_{3,3}</script> can.
So our paper solves Vijay’s open question from 1988, but extends well beyond <script type="math/tex">K_{3,3}</script>-free graphs to many other minor-closed graph families.
If you have a planar forbidden minor, you get bounded-treewidth graphs, and the one-crossing-minor-free graph families all have an associated structure theorem in which their graphs can be built by gluing together bounded-treewidth and planar pieces.
I’d previously studied <a href="/blog/2010/07/11/one-crossing-minor-free-flows.html">flow problems in the same graphs</a>, using the concept of <a href="/blog/2012/07/27/mimicking-networks.html">mimicking networks</a>, small networks that you can use to replace the pieces in this decomposition without affecting the overall result. The main idea of the new paper is to define and use a similar concept of mimicking networks for flow instead of matching.</p>
<p>What do we mean by a mimicking network for matching? The idea is that you have a graph <script type="math/tex">G</script> in which a small set <script type="math/tex">T</script> of its vertices are designated as “terminals” and the rest are nonterminals. We want to find another graph <script type="math/tex">G'</script>
such that, when any larger graph contains <script type="math/tex">G</script> and connects to it only through the terminals, we can replace <script type="math/tex">G</script> by <script type="math/tex">G'</script> and not affect whether it has a perfect matching. The right definition of this turns out to be that every matching in <script type="math/tex">G</script> that covers all nonterminals and some terminals corresponds to a matching in <script type="math/tex">G'</script> that again covers all nonterminals and the same set of terminals, and vice versa. Mimicking networks for flow are easy to construct (find an appropriate family of minimum cuts and merge all subsets of vertices that these cuts do not separate). For matching, we don’t have a similarly explicit construction, only an existence proof that the mimicking networks always have bounded size. But fortunately, we can go through a case analysis to find them all for sets of at most three terminals, which (with a lot of care in the rest of the algorithm) turns out to be all we need. Here they are:</p>
<p style="text-align:center"><img src="/blog/assets/2018/matching-mimic.svg" alt="Matching-mimicking networks on up to three terminals" /></p>
<p>These networks are all outerplanar, which means that we can safely glue them into a triangular face of a planar graph without destroying its planarity, a property our algorithm needs. They also have a subtler property: for each subset of terminals, the matching that covers that subset and all nonterminals is unique. This turns out to be a key step in proving that they can mimic, not just the existence of perfect matchings, but the weights of minimum-weight perfect matchings. We use this property to extend our NC algorithm to the problem of finding minimum-weight perfect matchings in one-crossing-minor-free graphs with polynomially-bounded weights.</p>
<p>Despite solving a 30-year-old open problem, this work raises plenty of new open problems. Chief among them are: how big do matching-mimicking networks need to be, as a function of the number of terminals? Can we construct them explicitly? And is it always possible to mimic the weights of the minimum-weight perfect matching, for larger numbers of terminals than three? So even in this problem area there’s plenty of scope for continued research and collaboration.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/64LmQMyfvtD">G+</a>)</p>David EppsteinOne of the great things about having new colleagues is the potential new energy, new collaborations, new research directions, and new synergies they bring. Vijay Vazirani, now a Distinguished Professor here at UCI, has brought all four. Vijay has been working on problems related to graph matching for almost 40 years, he tells me, starting with his discovery of the Micali–Vazirani matching algorithm in his first year as a graduate student. Our new preprint, “NC Algorithms for Perfect Matching and Maximum Flow in One-Crossing-Minor-Free Graphs” (arXiv:1802.00084) continues both that theme and a theme in my own research of graph structure theory.Linkage2018-01-31T22:19:00+00:002018-01-31T22:19:00+00:00https://11011110.github.io/blog/2018/01/31/linkage<ul>
<li>
<p><a href="http://csconferences.mah.se/swat2018/index.html">The Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018</a> (<a href="https://plus.google.com/100003628603413742554/posts/Z82dHbrvwHm">G+</a>). This year it will be in Malmö, Sweden, on June 18–20. The submission deadline is February 18, coming soon. I’m PC chair. Send papers, please.</p>
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<p><a href="https://www.tbray.org/ongoing/When/201x/2018/01/15/Google-is-losing-its-memory">Google memory loss</a> (<a href="https://plus.google.com/100003628603413742554/posts/BR853VjAg6j">G+</a>, <a href="https://news.ycombinator.com/item?id=16153840">via</a>). Tim Bray observes that Google searches are becoming worse as a comprehensive index to the web; there’s lots of old stuff that they used to find but have forgotten.</p>
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<p><a href="https://www.linkedin.com/pulse/how-duolingo-achieved-5050-gender-ratio-new-software-engineer-sohn/">How Duolingo achieved a 50:50 gender ratio for new software engineer hires</a> (<a href="https://plus.google.com/100003628603413742554/posts/N59QzMYmUQ8">G+</a>).</p>
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<p><a href="http://retractionwatch.com/2018/01/19/judge-orders-journal-identify-peer-reviewers-crossfit-lawyer/">Judge orders journal to identify peer reviewers</a> (<a href="https://plus.google.com/100003628603413742554/posts/9mwZT631yUw">G+</a>). It’s a messy legal case in which a journal is accused of corrupt reviewing and is suing for slander, but it raises issues of how safe we are as reviewers in the supposed confidentiality of our identities.</p>
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<p><a href="http://www.ams.org/news?news_id=4011&utm_content=buffer43a4e&utm_medium=social&utm_source=plus.google.com&utm_campaign=buffer">The 2018 Mathematical Art Exhibition Awards at the Joint Mathematics Meetings</a> (<a href="https://plus.google.com/100003628603413742554/posts/1fqmPkgpj9z">G+</a>, <a href="https://plus.google.com/+AmsOrg/posts/geyFWLenLvK">via</a>).</p>
</li>
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<p><a href="https://twitter.com/DrJessicaLanger/status/952231957517295616">Pier Review</a> (<a href="https://plus.google.com/100003628603413742554/posts/bJedybg2jZk">G+</a>, <a href="http://retractionwatch.com/2018/01/20/weekend-reads-scientists-respond-badly-criticism-hidden-retractions-journal-cancels-issue/">via</a>). Jessica Langer found the best newspaper pun.</p>
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<p><a href="https://www.youtube.com/watch?v=5SfXqTENV_Q">Cannons and sparrows</a> (<a href="https://plus.google.com/100003628603413742554/posts/QE2yHhbNYM7">G+</a>, <a href="https://plus.google.com/113862074718836293294/posts/hF5UtChr4uE">via</a>). Günter Ziegler makes an appearance on numberphile, with a nice discussion of <a href="https://arxiv.org/pdf/1202.5504">equal-area equal-perimeter convex partitions</a>. There are no cannons or sparrows; the title refers to using heavy machinery to solve simple-looking problems.</p>
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<p><a href="https://arxiv.org/abs/1710.11103">Writing women in mathematics into Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/8K944L4YPSP">G+</a>). Marie A. Vitulli describes her experiences creating biographies of women mathematicians on Wikipedia.</p>
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<p><a href="https://en.wikipedia.org/wiki/Hypatia">Hypatia</a> (<a href="https://plus.google.com/100003628603413742554/posts/Ubu6Jd6Gsfa">G+</a>). Now a Good Article on Wikipedia, thanks to the efforts of Katolophyromai. But see <a href="https://www.jstor.org/stable/25678817">Clement et al., “Re-righting the History of Women in Science”, <em>Math Horizons</em> 2009</a> for a critique of the male-gaziness of some of our sources.</p>
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<p><a href="https://www.insidehighered.com/news/2018/01/25/study-finds-given-disciplines-perceived-gender-bias-not-math-biggest-predictor">Perceived gender bias in a discipline strongly affects women’s choices of what to study</a> (<a href="https://plus.google.com/100003628603413742554/posts/YhgabmRhMZk">G+</a>). In other news linked from the comments, <a href="http://www.latimes.com/local/lanow/la-me-ln-third-sea-lion-attack-san-francisco-20171215-story.html">sea lion attack leads to ban</a>.</p>
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<p><a href="https://lemire.me/blog/2018/01/09/how-fast-can-you-bit-interleave-32-bit-integers-simd-edition/">How fast can you bit-interleave 32-bit integers?</a> (<a href="https://plus.google.com/+DanielLemirePhD/posts/VEexd9UENAT">G+</a>). Daniel Lemire shows that special Intel-only instructions can be beaten by careful SIMD coding.</p>
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<p><a href="https://www.theregister.co.uk/2015/12/08/wikidata_special_report/">Trouble with Wikidata</a> (<a href="https://plus.google.com/100003628603413742554/posts/GKSj1U84n78">G+</a>). Its inter-language links are useful, but its proponents refuse to institute reliable sourcing requirements that are compatible with Wikipedia’s and have been pushing to force their data to be shown automatically as part of Wikipedia articles despite their poor sourcing, while the spammers close in.</p>
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<p><a href="https://plus.google.com/+johncbaez999/posts/A8sQSPmETHk">More secrets of the associahedra</a> (<a href="https://plus.google.com/100003628603413742554/posts/AY1GnZgbFn6">G+</a>). A neat connection between the face counts of different types in an associahedron and the coefficients of inverse power series, with an illustration that appears to be redrawn from <a href="/blog/2006/10/13/another-gratuitously-nonplanar-drawing.html">an old one of mine</a>.</p>
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<p><a href="http://www.thisiscolossal.com/2018/01/porous-boulder-like-sculptures-chiseled-from-italian-marble-by-sibylle-pasche/">Porous marble boulders sculpted by Sibylle Pasche</a> (<a href="https://plus.google.com/100003628603413742554/posts/PTP6iS4zEJh">G+</a>, <a href="https://plus.google.com/+Colossal/posts/XvRCSRm8JvK">via</a>). Some of her other sculptures look inspired by <a href="https://en.wikipedia.org/wiki/Hexagonal_tiling_honeycomb">hexagonally-tiled hyperbolic horospheres</a>.</p>
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<p><a href="https://plus.google.com/+DavidRoberts/posts/MacU2MwWqXt">Scottish solids again</a> (<a href="https://plus.google.com/100003628603413742554/posts/ZJ7PZFTiCG2">G+</a>). The latest piece on <a href="http://bit.ly/2Eeci1w">polyhedral unfolding in the <em>Notices</em></a> and its <a href="https://plus.google.com/+AmsOrg/posts/FFaFD8LQ7Ld">corresponding G+ post</a> repeat a myth, <a href="https://doi.org/10.1080/17498430.2012.670845">debunked by Lloyd in 2012</a>, that the neolithic Scots carved Platonic solids out of stone. David Roberts sets the record straight.</p>
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<p><a href="http://blog.excites.co.uk/post/164785963192/cuben">Geometric/minimalist art</a> by <a href="http://excites.co.uk/">Simon C. Page</a> (<a href="https://plus.google.com/100003628603413742554/posts/PyyrRjwMS1W">G+</a>).</p>
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</ul>David EppsteinThe Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 (G+). This year it will be in Malmö, Sweden, on June 18–20. The submission deadline is February 18, coming soon. I’m PC chair. Send papers, please.Linkage2018-01-15T17:39:00+00:002018-01-15T17:39:00+00:00https://11011110.github.io/blog/2018/01/15/linkage<ul>
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<p><a href="https://motherboard.vice.com/en_us/article/xw4gwd/public-domain-drought">2018 is the last year of America’s public domain drought</a> (<a href="https://plus.google.com/100003628603413742554/posts/c6ufSXKMKV7">G+</a>). Or would be the last, before Disney buys enough legislators to keep Mickey and Pooh locked up for another term. See also <a href="https://boingboing.net/2018/01/08/sonny-bono-is-dead.html">BoingBoing on why another copyright extension might not happen</a>.</p>
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<p><a href="https://arxiv.org/abs/1712.09678">Robert McCallum’s claimed proof that Reinhardt cardinals are inconsistent with ZF</a> and <a href="http://jdh.hamkins.org/discussion-of-mccallums-paper-on-reinhardt-cardinals-in-zf/">Joel Hamkins’ discussion post on the proof</a> (<a href="https://plus.google.com/100003628603413742554/posts/h5mnD8k3kUn">G+</a>, <a href="https://plus.google.com/+JoelDavidHamkins1/posts/MsEuL71hhGQ">via</a>). If it holds up this result would be big; this is also an interesting example of online and public peer-review of preprints. But Hamkins worries that this process may be too stressful to authors.</p>
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<p><a href="http://www.thisiscolossal.com/2018/01/the-dripping-and-undulating-ceramic-sculptures-of-toru-kurokawa/">The dripping and undulating ceramic sculptures of Toru Kurokawa</a> (<a href="https://plus.google.com/100003628603413742554/posts/bz2KPoCPcJi">G+</a>, <a href="https://plus.google.com/+Colossal/posts/JzXCEWiW85E">via</a>). In the comments, Roice Nelson suggests a resemblance to <a href="http://www.math.uni-tuebingen.de/user/nick/lawson/lawson.html">Lawson’s minimal surfaces</a>.</p>
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<p>A reminder that <a href="http://thecostofknowledge.com/">I am not refereeing for Elsevier</a>, an update on the <a href="https://www.nature.com/articles/d41586-018-00093-7">German university negotiations with Elsevier</a>, and a story about <a href="http://retractionwatch.com/2018/01/04/elsevier-knew-author-faked-reviews-kept-accepting-papers-year/">Elsevier being slow to act on reports of fake reviewers</a> (<a href="https://plus.google.com/100003628603413742554/posts/SKBatjMP6pT">G+</a>).</p>
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<p><a href="https://www.chronicle.com/article/PhDs-Are-Still-Writing/242039">Teaching doctoral students how to write for other specialists in their field doesn’t prepare them to write for a broader audience</a> (<a href="https://plus.google.com/100003628603413742554/posts/i9HajAK1ZVx">G+</a>). This is aimed at humanities students but applies equally well to more technical disciplines.</p>
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<p><a href="http://blog.geomblog.org/2018/01/report-on-double-blind-reviewing-in.html">Double blind reviewing in ALENEX 2018</a> (<a href="https://plus.google.com/100003628603413742554/posts/Xx5GHafWh4b">G+</a>, <a href="https://plus.google.com/+SureshVenkatasubramanian/posts/59ZgMZPRJR4">via</a>). See also <a href="http://mybiasedcoin.blogspot.com/2018/01/double-blind-alenex.html">Michael Mitzenmacher’s take on the subject</a>.</p>
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<p><a href="https://arxiv.org/help/stats/2017_by_area/index">More charts and figures than you probably care to see about the growth of arXiv</a> (<a href="https://plus.google.com/100003628603413742554/posts/CuMhMJBdKyk">G+</a>). The data structures and algorithms section is growing, but less rapidly than CS more generally.</p>
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<p><a href="http://www.ams.org/news?news_id=3878">Vi Hart and Matt Parker win the 2018 JPBM Communications Awards</a>; <a href="https://blogs.ams.org/blogonmathblogs/2018/01/08/elevating-the-art-of-maths-communication/">Anna Haensch writes about how they have elevated the art of mathematical communication</a> (<a href="https://plus.google.com/100003628603413742554/posts/Wvay6XZrudd">G+</a>, <a href="https://plus.google.com/+AmsOrg/posts/GvpVQiRCaxD">via</a>).</p>
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<p><a href="http://algrant.ca/projects/hinged-tessellations/">Al Grant’s hinged tesselations</a> (<a href="https://plus.google.com/100003628603413742554/posts/RhFVDAKtQBq">G+</a>).</p>
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<p><a href="https://www.youtube.com/watch?v=G1m7goLCJDY">The square-sum problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/6hfLCSdqTSU">G+</a>, <a href="https://plus.google.com/113862074718836293294/posts/L7zhCKrfYuQ">via</a>). A cute number-theoretic Hamiltonian path puzzle with an associated open problem: does this work for all sufficiently large numbers? See also <a href="https://mathoverflow.net/q/290505/440">MathOverflow on the connectivity of higher-order power-sum graphs</a>.</p>
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<p>By <a href="https://en.wikipedia.org/wiki/Monsky%27s_theorem">Monsky’s theorem</a> one cannot divide a square into an odd number of equal-area triangles, but it’s possible to use the <a href="https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue–Morse sequence</a> to get <a href="https://arxiv.org/abs/1708.02891">superpolynomially close to equal areas</a> (<a href="https://plus.google.com/100003628603413742554/posts/NrNtGBjiCzi">G+</a>). See also <a href="https://plus.google.com/u/0/+johncbaez999/posts/W9FaptoAnw4">John Baez’s more detailed post on the same topic</a>.</p>
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<p><a href="http://mathtourist.blogspot.co.uk/2017/11/crinkled-torus.html">Crinkled torus</a> folded from a single rectangular sheet of paper by William T. Webber (<a href="https://plus.google.com/100003628603413742554/posts/XyPZ5276nbF">G+</a>). This one looks a little warped, but it should be possible to create tori in this way whose geometry is exact (congruent flat triangles everywhere, with angles adding to exactly 2π at each vertex). On the other hand, I’m not convinced that it’s possible to get a torus that closes up exactly using equilateral triangles that meet six at a vertex; see <a href="/blog/2009/02/03/flat-equilateral-tori.html">my older post</a> for more.</p>
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<p><a href="http://www.win.tue.nl/~hermanh/doku.php?id=sound_of_space-filling_curves">Herman Haverkort turns Hilbert curves into music</a> (<a href="https://plus.google.com/100003628603413742554/posts/75Lyo4DJy9E">G+</a>, <a href="https://www.metafilter.com/171790/Math-rock-but-with-actual-math">via</a>).</p>
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<p><a href="https://www.newyorker.com/tech/elements/new-york-citys-bold-flawed-attempt-to-make-algorithms-accountable">New York City establishes a task force to look into issues of algorithmic fairness</a> (<a href="https://plus.google.com/100003628603413742554/posts/RPh2nBaj4gP">G+</a>, <a href="https://www.metafilter.com/171334/Opening-up-the-black-box">via</a>) but pushback from industry and staff is preventing them from seeing source code or even finding out which agencies use code to make resource allocation decisions. And if they saw the code, would they be able to figure out whether it makes its decisions fairly?</p>
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</ul>David Eppstein2018 is the last year of America’s public domain drought (G+). Or would be the last, before Disney buys enough legislators to keep Mickey and Pooh locked up for another term. See also BoingBoing on why another copyright extension might not happen.King tide at Virgin Creek2018-01-01T20:54:00+00:002018-01-01T20:54:00+00:00https://11011110.github.io/blog/2018/01/01/king-tide-virgin<p>I’m in Mendocino visiting my parents, as I often do over the holidays. We timed a visit to Virgin Creek Beach, north of Fort Bragg, for a particularly low tide on the afternoon of New Year’s Eve.
So <a href="https://www.ics.uci.edu/~eppstein/pix/kingtide/">here’s my first batch of photos for the New Year</a>, from that trip.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/kingtide/VirginCreek-m.jpg" alt="Virgin Creek" style="border-style:solid;border-color:black;" /></p>
<p>The low tide left a wide expanse of sand exposed, but also created plenty of tidepools filled with mussels, anemone, limpets, tiny whelks, hermit crabs, and even a sea star or two, recovering after a disastrous crash in their population from a 2013 outbreak of <a href="https://en.wikipedia.org/wiki/Sea_star_wasting_disease">sea star wasting disease</a>.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/kingtide/OrangeSeaStar-m.jpg" alt="Orange Sea Star" style="border-style:solid;border-color:black;" /></p>
<p>Although the last shots of the set look like sunset, they were actually from around 3:30 in the afternoon, around the peak low tide and 1 1/2 hours before sunset. (Don’t believe the times reported from my camera; I seldom bother to set its clock accurately.) A late afternoon storm blocked the sun and gave us some spectacular lighting.</p>
<p style="text-align:center"><img src="http://www.ics.uci.edu/~eppstein/pix/kingtide/Skyglow3-m.jpg" alt="Stormy light over Virgin Creek Beach" style="border-style:solid;border-color:black;" /></p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/iDZJxnjSutY">G+</a>)</p>David EppsteinI’m in Mendocino visiting my parents, as I often do over the holidays. We timed a visit to Virgin Creek Beach, north of Fort Bragg, for a particularly low tide on the afternoon of New Year’s Eve. So here’s my first batch of photos for the New Year, from that trip.Linkage for the end of the year2017-12-31T13:38:00+00:002017-12-31T13:38:00+00:00https://11011110.github.io/blog/2017/12/31/linkage-for-end<ul>
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<p><a href="https://theoutline.com/post/2369/everipedia-is-the-wikipedia-for-being-wrong">Everipedia is the Wikipedia for being wrong</a> (<a href="https://plus.google.com/100003628603413742554/posts/cN1L3wLfFRC">G+</a>). Linked not to diss a rival encyclopedia but to explain why Wikipedia’s strict standards for including only information that can be verified in reliable published sources (especially when it’s about a living person) are important.</p>
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<p><a href="https://plus.google.com/+lievenlebruyn/posts/3EbhxDh86N5">Recent gossip on Mochizuki’s purported proof of the abc conjecture</a>, and <a href="http://www.math.columbia.edu/~woit/wordpress/?p=9871">more gossip</a> (<a href="https://plus.google.com/100003628603413742554/posts/LmP9mWyfyVP">G+</a>). The story that his work has already been accepted to the journal that he’s the editor-in-chief of turns out to be exaggerated, but he did submit it there. And commenters are starting to point to specific points in the proof that seem problematic.</p>
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<p>The <a href="https://en.wikipedia.org/wiki/Matroid_parity_problem">matroid parity problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/i94AoWdUR1s">G+</a>), a powerful algorithmic problem on matroids with many applications in combinatorial optimization and graph drawing, among them finding large planar subgraphs and embedding graphs onto surfaces of maximum genus. New Wikipedia article.</p>
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<p><a href="https://www.quantamagazine.org/150-year-old-math-design-problem-solved-20150609/">Keevash’s solution to the combinatorial design problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/iAEgMwqLEDY">G+</a>). It’s a couple years old, but still a breakthrough: almost all parameters for combinatorial designs have solutions.</p>
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<p><a href="http://www.thisiscolossal.com/2017/12/japanese-tip-an-exhibition-of-8000-paper-sculptures-made-from-chopstick-sleeves/">Chopstick sleeve origami art exhibit</a> (<a href="https://plus.google.com/100003628603413742554/posts/fNgx83GycaW">G+</a>). Who could resist playing with the convenient piece of paper left over from wrapping your chopsticks? Not me.</p>
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<p><a href="https://dustingmixon.wordpress.com/2017/12/15/partisan-gerrymandering-with-geographically-compact-districts/">Partisan gerrymandering with geographically compact districts</a> (<a href="https://plus.google.com/100003628603413742554/posts/Mj8wkSDsCjc">G+</a>). Why looking at the shapes of congressional districts is inadequate as a test for whether they’re unfairly drawn — you have to look more carefully at the distribution of people and their preferences within those shapes. From <a href="https://arxiv.org/abs/1712.05390">a preprint by Boris Alexeev and Dustin Mason</a>.</p>
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<p><a href="https://www.youtube.com/watch?v=DIyruYQ-N4Q">Hexaflexaflakes</a> (<a href="https://plus.google.com/100003628603413742554/posts/UZvfwUtVH2H">G+</a>). Vi Hart cuts through some baloney regarding the number of folds needed to achieve a given degree of symmetry in paper snowflakes.</p>
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<p><a href="http://myjetpack.tumblr.com/post/165047890685/cartoon-for-new-scientist-also-my-new-book-of">Platonic solids on their summer holiday</a> (<a href="https://plus.google.com/100003628603413742554/posts/e6JQxRDo18N">G+</a>, from “You’re All Just Jealous Of My Jetpack” webcomic). With bonus hosohedron. It’s now the summer holidays in the southern hemisphere, right?</p>
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<p><a href="https://www.chronicle.com/article/Use-of-Free-Textbooks-Is/242086">Use of free textbooks is rising, but barriers remain</a> (<a href="https://plus.google.com/100003628603413742554/posts/4SMPBkJ6BhN">G+</a>). <em>The Chronicle of Higher Education</em> discusses factors behind the slow uptake of free textbooks (now up to 9% from a previous 5%) and efforts by publishers to recapture that market by publishing value-added non-free versions of the same texts.</p>
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<p><a href="https://www.justinobeirne.com/google-maps-moat/">Google Maps’s moat</a> (<a href="https://plus.google.com/100003628603413742554/posts/2iP6fS5wG5K">G+</a>). How integrating lower-level data such as the shapes of buildings and the precise locations of businesses lets Google maps build higher-level constructs such as “areas of interest” (business corridors within cities), and what this says about their dominance in mapping.</p>
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<p><a href="https://mathoverflow.net/q/288423/440">Vanishing line on Conway’s game of life</a> (<a href="https://plus.google.com/100003628603413742554/posts/aZvC9czVCDw">G+</a>). If you start with one-cell-thick infinite line of live cells in Conway’s game of life, it behaves like a one-dimensional cellular automaton whose time-space diagram looks like a Sierpinski triangle, but if it’s finite then it shrinks from its ends, and the Sierpinski triangle (at a larger scale) gets frozen into the debris left behind as it shrinks.</p>
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<p><a href="https://plus.google.com/+JoelDavidHamkins1/posts/UsuaKC7pdHp">Chaos Train</a> (<a href="https://plus.google.com/100003628603413742554/posts/WUrVJEgswmq">G+</a>). An amusing exercise on quantifier ordering by Joel David Hamkins. See also his three-quantifier <a href="https://plus.google.com/+JoelDavidHamkins1/posts/ec3trmgNzjc">Monkey Madness</a>.</p>
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<p><a href="https://www.youtube.com/watch?v=VvCytJvd4H0">Science YouTubers attempting a graph theory puzzle</a> (<a href="https://plus.google.com/100003628603413742554/posts/QeAYz2wCAyP">G+</a>). It’s the classic three utilities puzzle, on a mug. (The mug itself is <a href="https://mathsgear.co.uk/products/utilities-puzzle-mug">available for purchase</a>.)</p>
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<p><a href="http://math.mit.edu/~rstan/ec/ec1/">Enumerative Combinatorics, volume 1, second edition</a> (<a href="https://plus.google.com/100003628603413742554/posts/JwaaGnojcix">G+</a>, via David Roberts and Luis Guzman). Free online almost-final copy of Stanley’s classic book.</p>
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</ul>David EppsteinEveripedia is the Wikipedia for being wrong (G+). Linked not to diss a rival encyclopedia but to explain why Wikipedia’s strict standards for including only information that can be verified in reliable published sources (especially when it’s about a living person) are important.Folding polyominoes into (poly)cubes2017-12-30T21:06:00+00:002017-12-30T21:06:00+00:00https://11011110.github.io/blog/2017/12/30/folding-polyominoes-into<p>Here’s a trio of paper puzzles for you to print out, cut out, and fold. The goal is to fold each of these three shapes so that it covers the entire surface of a cube, whose sides are the same size as the squares in each shape. The uncolored areas in two of the shape and the thick black line in the third are holes that you should cut out from the middle of each shape before folding. They come from <a href="https://nbpuzzles.wordpress.com/2014/06/08/cube-folding/">a blog post by Nikolai Beluhov</a>, which also has a trickier variation of the top one in which illustrations printed onto certain squares of the shape should become visible once you fold it into a cube.</p>
<p style="text-align:center"><img src="/blog/assets/2017/Beluhov-folding-puzzles.svg" alt="Three polyominoes to be folded into cubes" /></p>
<p>These puzzles were the starting point for my latest preprint, “Folding polyominoes into (poly)cubes” (<a href="https://arxiv.org/abs/1712.09317">arXiv:1712.09317</a>, with Aichholzer, Biro, Demaine, Demaine, Fekete, Hesterberg, Kostitsyna, and Schmidt).
We published it in CCCG 2015, but I don’t seem to have posted about it at that time. The preprint is of the extended journal version. One complication with studying the problem mathematically (rather than just posting it as a puzzle) is that we have to specify more carefully what we mean by folding a polyomino to form the surface of a cube or polycube, rather than just letting the puzzler figure out which kinds of folds are needed to make the puzzle work. We ended up defining nine different models of what kinds of folds are allowed and (with a lot of case analysis) showing them all to be different from each other in terms of which shapes they allow to fold to cubes.</p>
<p>The corresponding algorithmic problems turn out to be a bit messy. We developed a dynamic programming algorithm for folding tree-like polyominoes to cover the surface of constant-sized polycubes in the simplest of our folding models, but weren’t able to handle more complicated folding models, primarily because it wasn’t clear how to extend an abstract mapping from the polyomino to the cube surface into a valid 3d embedding. The difficulties here are somewhat related to the <a href="https://en.wikipedia.org/wiki/Map_folding">map folding problem</a>, where it is unknown even how to determine whether a square grid with a fixed assignment of mountain or valley to its folds can be folded flat.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/WhxN1Nb4kkg">G+</a>)</p>David EppsteinHere’s a trio of paper puzzles for you to print out, cut out, and fold. The goal is to fold each of these three shapes so that it covers the entire surface of a cube, whose sides are the same size as the squares in each shape. The uncolored areas in two of the shape and the thick black line in the third are holes that you should cut out from the middle of each shape before folding. They come from a blog post by Nikolai Beluhov, which also has a trickier variation of the top one in which illustrations printed onto certain squares of the shape should become visible once you fold it into a cube.Factorial change-making2017-12-23T12:40:00+00:002017-12-23T12:40:00+00:00https://11011110.github.io/blog/2017/12/23/factorial-change-making<p>I think most people make change for amounts of money using a greedy algorithm: repeatedly start with an empty pile of money, and repeatedly add the largest-valued coin or bill that keeps the value of the pile at or under the desired total, until the total is reached. This can use more coins than necessary, though, for exotic coinage systems, or even some less-exotic ones like <a href="/blog/2009/07/27/greed-can-fail.html">US coins without the nickels</a>.</p>
<p style="text-align:center"><img src="/blog/assets/2017/MatsysMoneylender.jpg" alt="The Moneylender and his Wife, Quentin Matsys, 1514" /></p>
<p style="text-align:center;font-size:75%"><em><a href="https://commons.wikimedia.org/wiki/File:Quinten_Massijs_(I)_-_The_Moneylender_and_his_Wife_-_WGA14281.jpg">The Moneylender and his Wife</a></em>, <a href="https://en.wikipedia.org/wiki/Quentin_Matsys">Quentin Matsys</a>, 1514</p>
<p>There’s a simple test for whether the greedy algorithm is always optimal for a system of coins. It’s called the one-point test, and was first published by <a href="http://www.jstor.org/stable/169525">Magazine, Nemhauser, and Trotter in 1975</a>. It actually tests a stronger property, whether every prefix of the sequence is optimal for the greedy algorithm. To perform this test, look at each consecutive pair of coin values <script type="math/tex">x</script> and <script type="math/tex">y</script>. Round <script type="math/tex">y</script> up to a multiple <script type="math/tex">kx</script> of <script type="math/tex">x</script>. Then we could make change for <script type="math/tex">kx</script> non-greedily, using <script type="math/tex">k</script> copies of the <script type="math/tex">x</script> coin. Does the greedy algorithm do at least as well, using <script type="math/tex">y</script> and some smaller coins? If not, then we have found an example where greedy is non-optimal. But if a coin system passes this test for all of its coins, then the whole sequence and all of its prefixes are greedy-optimal.</p>
<p>This immediately shows that any sequence of coin values where each is an integer multiple of the next is greedy-optimal, because the greedy algorithm will always represent each tested value <script type="math/tex">kx</script> with the single coin <script type="math/tex">y</script>. And you can check that it’s true for US money as well (with the nickel, and with or without the $2 bill). But the no-nickel sequence <script type="math/tex">1,10,25</script> already fails the test, because when <script type="math/tex">x=25</script> and <script type="math/tex">y=10</script>, <script type="math/tex">ky=30</script> is not represented optimally by the greedy algorithm.</p>
<p style="text-align:center"><img src="/blog/assets/2017/SpadeCoins.jpg" alt="Han Dynasty spade coins" /></p>
<p style="text-align:center;font-size:75%"><a href="https://commons.wikimedia.org/wiki/File:Dinast%C3%ADaHan20100102052141SAM_2887.jpg">Han Dynasty spade coins</a> were not very round</p>
<p>The one-point test also shows that some systems of money that do not involve round multiples are still greedy-optimal. For instance, in the Fibonacci Republic they use coins of <script type="math/tex">1, 2, 3, 5, 8, 13, 21, 34,</script> and <script type="math/tex">55</script> cents, a Fibonacci dollar coin of <script type="math/tex">89</script> cents, and bills for the larger denominations of <script type="math/tex">144</script> cents etc. Each of these numbers is at most double the next, and we can represent each doubled Fibonacci number <script type="math/tex">ky=2F_i</script> in the one-point test greedily using the identity <script type="math/tex">2F_i=F_{i+1}+F_{i-2}.</script>
The nearby country of Mersenneland instead uses the numbers <script type="math/tex">M_i=2^i-1=1,3,7,15,31,\dots</script> for their money. Each is more than double but at most triple the next, and we can represent each number <script type="math/tex">ky=3M_i</script> in the one-point test greedily using the identity <script type="math/tex">3M_i=M_{i+1}+2M_{i-1}</script>.</p>
<p>Regardless of whether a coinage system is greedy-optimal or not, one can use another simple algorithm to find the amounts of money that would cause the greedy algorithm the most trouble.
Suppose that the sequence of coins is <script type="math/tex">c_i</script>, and we want to calculate the smallest amount of money <script type="math/tex">r_i,</script> that causes the greedy algorithm to use <script type="math/tex">i</script> coins. Then <script type="math/tex">r_{i+1}=r_i+c_j,</script> where <script type="math/tex">c_j</script> is the smallest coin value such that <script type="math/tex">r_i+c_j\lt c_{j+1}.</script> That is, we look for the first gap larger than <script type="math/tex">r_i</script> in the sequence of coin values, and we add <script type="math/tex">r_i</script> to the smaller of the two coins forming this gap.</p>
<p>In Factoria, the coin values are <script type="math/tex">1,2,6,24,</script> and <script type="math/tex">120</script> cents (the Factorial dollar), and they have bills for <script type="math/tex">6, 42, 336\dots</script> dollars.
This is a greedy-optimal coin system, because each coin or bill is an integer multiple of its predecessor.
Plugging in the factorial numbers to the formula above, the numbers that are hardest to express as sums of factorials (whether greedily or in any other way) are</p>
<p style="text-align:center"><script type="math/tex">1, 3, 5, 11, 17, 23, 47, 71, 95, 119, 239, 359, \dots</script> (<a href="https://oeis.org/A200748">OEIS A200748</a>).</p>
<p>In the breakaway republic of South Factoria, they’ve long been familiar with how hard it is to make change using factorials. So, wanting a currency that makes what was once difficult more easy, they use the numbers in A200748 as the values of their own money. This choice also has the advantage that it’s difficult to exchange South Factorial money for Factorial money. But is this a greedy-optimal coin system?</p>
<p>To see this, we need a clearer idea of how the numbers in A200748 are constructed. Each one is a factorial plus its predecessor.
A straightforward induction using the formulas for <script type="math/tex">c_i</script> and <script type="math/tex">r_i</script> shows that, for each integer <script type="math/tex">i</script>, there are exactly <script type="math/tex">i</script> differences equal to <script type="math/tex">i!</script>, running from <script type="math/tex">i!-1</script> to <script type="math/tex">(i+1)!-1</script>.
With this pattern in hand, we can show that sequence A200748 again passes the one-point test. When the sequence jumps from <script type="math/tex">i!-1</script> to <script type="math/tex">2i!-1</script>, the one-point test runs the greedy algorithm on <script type="math/tex">3(i!-1)=(2i!-1)+(i!-1)+1</script>, which it passes. And for any other jump in the sequence, from <script type="math/tex">xi!-1</script> to <script type="math/tex">(x+1)i!-1</script> for some integer <script type="math/tex">x\gt 1,</script> the one-point test runs the greedy algorithm on <script type="math/tex">2(xi!-1)=((x+1)i!-1)+((x-1)i!-1),</script> again passing. So South Factorial money is indeed greedy-optimal.</p>
<p>Which numbers are hard to change in South Factoria?
As before we can construct the sequence recursively, by adding the starting points of big gaps to previous values in the sequence. This gives us the numbers</p>
<p style="text-align:center"><script type="math/tex">1, 2, 7, 30, 149, 868, 5907, 46226, 409105,\dots</script> (<a href="https://oeis.org/A136574">OEIS A136574</a>).</p>
<p>This construction also gives us a new insight into number sequence A136574, which was originally defined in a more complicated way as the row sums of a triangle of numbers involving factorials.
Remember that that each term in this sequence is the previous term plus the smallest coin that comes before a big gap. And in the hard-to-change-factorially numbers A200748, the smallest coin that comes before a big gap is always one less than a factorial. So this gives us the nice recurrence <script type="math/tex">a(i)=a(i-1)+i!-1</script> for the values of A136574.</p>
<p>So far, the systems of coins we’ve been considering are all pretty sparse.
A dense system of coins (one with many coin values) will have a sparse system of hard-to-represent values, and vice versa. In the limit, for a system of coins with bounded gap size, the sequence of hard-to-represent values will be finite, and conversely for a system of finitely many coins the hard-to-represent values will be eventually periodic. So, to conclude with just a couple more examples, of greater density: if the coin values are all the square numbers (not greedy-optimal: <script type="math/tex">12 = 3\times 4 = 9 + 1 + 1 + 1</script> fails the one-point test) then the hard-to-represent-greedily values are</p>
<p style="text-align:center"><script type="math/tex">1, 2, 3, 7, 23, 167, 7223, 13053767,\dots</script> (<a href="https://oeis.org/A006892">OEIS A006892</a>).</p>
<p>For instance, even though <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">every number can be represented as a sum of four squares</a>, the greedy algorithm uses five for <script type="math/tex">23 = 16 + 4 + 1 + 1 + 1</script>. And similarly, for prime-number coinage (plus a one-cent coin, again not greed-optimal), the hard-to-represent values are the <a href="https://en.wikipedia.org/wiki/Pillai_sequence">Pillai sequence</a></p>
<p style="text-align:center"><script type="math/tex">1, 4, 27, 1354, 401429925999155061,\dots</script> (<a href="https://oeis.org/A066352">OEIS A066352</a>).</p>
<p>To find the next term in the sequence, we need to find the first <a href="https://en.wikipedia.org/wiki/Prime_gap">prime gap</a> bigger than 401429925999155061. As the OEIS entry states, reaching a gap this big is likely to require hundreds of millions of digits. So in practice, all reasonable amounts of change require only four coins or bills in the prime number coinage system, even when we make change greedily. <a href="https://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach’s conjecture</a> suggests, though, that a more clever change-making strategy would need only three coins for every value. So greed is not always good.</p>
<p style="text-align:center"><img src="/blog/assets/2017/RembrandtDriving.jpg" alt="Christ Driving the Money Changers from the Temple, Rembrandt, 1626" /></p>
<p style="text-align:center;font-size:75%"><em><a href="https://commons.wikimedia.org/wiki/File:Rembrandt_Christ_Driving_the_Money_Changers_from_the_Temple.jpg">Christ Driving the Money Changers from the Temple</a></em>, <a href="https://en.wikipedia.org/wiki/Rembrandt">Rembrandt</a>, 1626</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/ADXAhhMNXT2">G+</a>)</p>David EppsteinI think most people make change for amounts of money using a greedy algorithm: repeatedly start with an empty pile of money, and repeatedly add the largest-valued coin or bill that keeps the value of the pile at or under the desired total, until the total is reached. This can use more coins than necessary, though, for exotic coinage systems, or even some less-exotic ones like US coins without the nickels.