0xDE
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0xDE - LiveJournal.comThu, 24 Jul 2014 00:50:58 GMTLiveJournal / LiveJournal.com110111107784841personalhttp://l-userpic.livejournal.com/32934265/77848410xDE
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100100http://11011110.livejournal.com/291980.htmlThu, 24 Jul 2014 00:50:58 GMTBig grids in outerplanar strict confluent graphs
http://11011110.livejournal.com/291980.html
I was wondering whether the <a href="http://arxiv.org/abs/1308.6824">outerplanar strict confluent drawings</a> I studied in a Graph Drawing paper last year had underlying diagrams whose <a href="https://en.wikipedia.org/wiki/Treewidth">treewidth</a> is bounded, similarly to the treewidth bound for the usual <a href="https://en.wikipedia.org/wiki/Outerplanar_graph">outerplanar graphs</a>. The confluent graphs themselves can't have low treewidth, because they include large <a href="https://en.wikipedia.org/wiki/Complete_bipartite_graph">complete bipartite graphs</a>, but I was hoping that a treewidth bound for the diagram could be used to prove that the graphs themselves have low <a href="https://en.wikipedia.org/wiki/Clique-width">clique-width</a>. Sadly, it turns out not to be true.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/hyperbolic-outerconfluent.png"></div><br /><br />The drawing above, shown with two levels of cells surrounding the central pentagon, can be extended to any number of levels. Besides the three shapes of faces shown here (with three, four, and five sharp points) there's one more possible shape, with three sharp points that are not consecutive, that would take a couple more levels to complete — the five points on the boundary with three incoming edges would each form the base of one of these shapes — but that's the only other thing that can happen. The drawing is based on the <a href="https://en.wikipedia.org/wiki/Order-5_pentagonal_tiling">order-5 pentagonal tiling</a> of the hyperbolic plane, and consists of five-sided regions with five regions meeting at each confluent junction. By the theory from my GD paper, it's not possible to find a different drawing with the same vertices in the same order in which some junctions have been merged, reducing the treewidth of the drawing. And because it can be extended to arbitrarily large patches of the pentagonal tiling (analogous to arbitrarily large grids in the Euclidean plane) it has unbounded treewidth.<br /><br />It doesn't seem to work to do this directly with the square grid because some confluent junctions will have three connections in one direction and one in the other, allowing them to be merged with a neighboring junction. Using a hyperbolic tiling pattern allows all junctions to have at least two connections in each direction.<a name='cutid1-end'></a>http://11011110.livejournal.com/291980.htmlconfluencegraph drawingpublic0http://11011110.livejournal.com/291634.htmlMon, 21 Jul 2014 23:54:27 GMTUsing finite automata to draw graphs
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The diagram below describes a finite state machine that takes as input a description of an <a href="https://en.wikipedia.org/wiki/Indifference_graph">indifference graph</a>, and produces as output a <a href="https://en.wikipedia.org/wiki/1-planar_graph">1-planar drawing</a> of it (that is, a drawing with each edge crossed at most once).<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/1planar-unit-interval.png"></div><br /><br />Indifference graphs are the graphs that can be constructed by the following process. Initialize an active set of vertices to be the empty set, and then perform a sequence of steps of two types: either add a new vertex to the active set and make it adjacent to all previous active vertices, or inactivate a vertex (removing it from the active set but not from the graph). Thus, they can be represented by a sequence of binary values that specify whether the next step is to add or inactivate a vertex. These values are the input to the finite state machine.<br /><br />In <a href="http://arxiv.org/abs/1304.5591">a paper at WADS last year with Bannister and Cabello</a>, we showed <a href="http://11011110.livejournal.com/267767.html">how to test 1-planarity for several special classes of graphs</a>, but not for indifference graphs. Some of our algorithms involved proving the existence of a finite set of forbidden configurations, and that idea works here, too: an indifference graph turns out to be 1-planar if and only if, for every K<sub>6</sub> subgraph, the first three vertices of the subgraph (in the activation order) have no later neighbors outside the subgraph, and the last three vertices have no other earlier neighbors. K<sub>6</sub> is 1-planar, but it has essentially only one drawing (modulo permutation of the vertices), and any example of this configuration would have a seventh vertex connected to four of the K<sub>6</sub> vertices, something that's not possible in a 1-planar drawing.<br /><br />At one level, the state diagram above can be viewed as a diagram for detecting this forbidden configuration. Every right-going transition is one that adds an active vertex, and every left-going transition is one that removes an active vertex. If a transition of either type does not exist in the diagram, it means that a step of that type will lead to an inescapable failure state. But the only missing transitions are the ones that would create a six-vertex active set by a sequence of transitions that does not end in three consecutive right arrows (creating a K<sub>6</sub> in which one of the last three vertices has an earlier neighbor) or the ones that would exit a six-vertex active set by a sequence of transitions that does not begin with three consecutive left arrows (creating a K<sub>6</sub> in which one of the first three vertices has a later neighbor). So, this automaton recognizes only the graphs that have no forbidden configuration.<br /><br />On another level, the drawings within each state of the diagram show how to use this finite state machine to construct a drawing. Each state is labeled with a drawing of its active vertices, possibly with a yellow region that represents earlier inactive parts of the drawing that can no longer be modified. The numbers on the vertices give the order in which they were activated. For each left transition, it is always possible to remove the oldest active vertex from the drawing and replace the parts of the drawing surrounding it by a yellow region to create a drawing that matches the new state. Similarly, for each right transition, it is always possible to add one more active vertex to the drawing, connect it to the other active vertices, and then simplify some parts of the drawing to yellow regions, again creating a drawing that matches the new state. Therefore, every graph that can be recognized by this state diagram has a 1-planar drawing.<br /><br />Since the machine described by the diagram finds a drawing for a given indifference graph if and only if the graph has no forbidden configurations, it follows that these forbidden configurations are the only ones we need to describe the 1-planar graphs and that this machine correctly finds a 1-planar drawing for every indifference graph that has one. This same technique doesn't always generalize: A result from my WADS paper that it's NP-complete to test 1-planarity for graphs of bounded bandwidth shows that, even when a class of graphs can be represented by strings of symbols from a finite alphabet it's not always going to be possible to find a finite state machine to test 1-planarity. But it would be interesting to find more graph classes for which the same simple technique works.<a name='cutid1-end'></a>http://11011110.livejournal.com/291634.htmlgraph drawingpublic0http://11011110.livejournal.com/291361.htmlSat, 12 Jul 2014 03:25:09 GMTFour preprints
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I noticed that there was a higher-than-usual density of arxiv preprints among the web pages I'd been bookmarking lately, so I thought maybe I'd share. The first one, especially, is very timely:<br /><br /><b>From the "Brazuca" ball to Octahedral Fullerenes: Their Construction and Classification</b>, Yuan-Jia Fan, Bih-Yaw Jin, <a href="http://arxiv.org/abs/1406.7058">arXiv:1406.7058</a>, <a href="https://medium.com/the-physics-arxiv-blog/mathematicians-solve-the-topological-mystery-behind-the-brazuca-world-cup-football-2e11ab1f4391">via</a>. The classical pentagon and hexagon soccer ball pattern (introduced for the 1970 World Cup) later became even more famous as the structure of the <a href="https://en.wikipedia.org/wiki/Buckminsterfullerene">buckminsterfullerene</a> Carbon-60 molecule, from which the fullerene graphs (planar graphs in which all faces are pentagons or hexagons) took their name. Another soccer ball pattern, used in the 2006 world cup, is topologically a truncated octahedron but with distorted face shapes that reduce its symmetry to tetrahedral; there also exist fullerenes with tetrahedral symmetry. And there's a new soccer ball pattern for the Brazil world cup, with the topology of the cube; the "via" article says that it has octahedral symmetry but I'm not convinced, because it doesn't seem to have the reflection symmetries that octahedra should have. Nevertheless Fan and Jin asked: are there fullerenes with octahedral symmetry? The positive answer comes from a nice construction involving cutting equilateral triangles out of the hexagonal tiling and then gluing them together to make a finite polyhedron.<a name='cutid1-end'></a><br /><br /><b>The Shortest Path to Happiness: Recommending Beautiful, Quiet, and Happy Routes in the City</b>, Daniele Quercia, Rossano Schifanella, Luca Maria Aiello, <a href="http://arxiv.org/abs/1407.1031">arXiv:1407.1031</a>, <a href="http://gizmodo.com/yahoos-developing-a-map-algorithm-to-find-the-most-beau-1602266077">via</a>. Suppose you want an online map service to give you a walking tour of a city. You probably wouldn't want the shortest path from one part to another, but rather the nicest path. Changing how you weight the edges of the underlying graph to prioritize them differently is not so difficult (and I wrote <a href="http://arxiv.org/abs/cs.DS/9907001">a paper</a> long ago on how you might adjust these weights to fit different users' preferences) but the harder part is gathering the data to measure what it means for a route to be nice. These authors approach the problem with online photo databases, in one case crowdsourcing the problem of quantifying niceness and in another attempting to use the image tags to determine it automatically. The analysis of how well it worked seemed very anecdotal and handwavy to me but maybe that's the nature of the subject.<a name='cutid2-end'></a><br /><br /><b>The Convex Configurations of "Sei Shonagon Chie no Ita" and Other Dissection Puzzles</b>, Eli Fox-Epstein, Ryuhei Uehara, <a href="http://arxiv.org/abs/1407.1923">arXiv:1407.1923</a>. Given a puzzle like the <a href="https://en.wikipedia.org/wiki/Tangram">tangram</a> (a square subdivided into seven convex pieces) how many ways are there of rearranging it into convex shapes? I asked a similar question (with different shapes) in <a href="http://www.ics.uci.edu/~eppstein/pubs/Epp-COMB-01.pdf">one of my old talks</a> but the tangram question (and its answer) have been known for much longer, <a href="http://www.jstor.org/stable/2303340">since at least 1942</a>. This paper solves the same question for some more complex puzzles of a similar type. Like the 1942 tangram paper, the new preprint uses the observation that there is a smaller tile into which the puzzle pieces can all be subdivided, such that any solution is a subset of a regular tesselation of the plane by this tile. I don't know of a general algorithm for solving this kind of problem efficiently; perhaps there's something interesting to be done in that direction.<a name='cutid3-end'></a><br /><br /><b>Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts</b>, Radu Curticapean, Dániel Marx, <a href="http://arxiv.org/abs/1407.2929">arXiv:1407.2929</a>. Ok, unlike the other ones this is not recreational mathematics. But it still interested me. The problem concerns the complexity of counting copies of subgraphs in larger graphs, something that seems to be quite topical lately. If the subgraph has a fixed number <i>k</i> of vertices and the larger graph has <i>n</i> vertices, then there's an obvious <i>O</i>(<i>n</i><sup><i>k</i></sup>) algorithm: just try all <i>k</i>-tuples of vertices, and it's not hard to extend this to the case where the subgraph has a subset of <i>k</i> vertices that together cover all of its edges. As Curticapean and Marx show, these are the only cases in which one gets a polynomial time algorithm (assuming standard complexity-theoretic conjectures): counting subgraphs from a given class does not have a fixed-parameter tractable algorithm unless the vertex cover number is bounded.<a name='cutid4-end'></a>http://11011110.livejournal.com/291361.htmlsubgraph isomorphismpaperspublic3http://11011110.livejournal.com/291080.htmlSun, 06 Jul 2014 06:06:57 GMTBlack Phoebe
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For the past month or so, until this weekend when they moved out, we've had some squatters on our front porch: a family of <a href="https://en.wikipedia.org/wiki/Black_phoebe">Black Phoebes</a>. They conveniently set up their nest in clear sight of a window over the front door, through which I could aim the camera. <br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/phoebe/3-m.jpg" border="2" style="border-color:black;" /></div>http://11011110.livejournal.com/291080.htmlphotographypublic0http://11011110.livejournal.com/290949.htmlWed, 02 Jul 2014 16:38:04 GMTSeth Teller
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Via <a href="http://newsoffice.mit.edu/2014/professor-seth-teller-dies">MIT's news office</a> I learn that Seth Teller has died, at the young age of 50. Seth primarily worked in robotics and vision, but was also a regular participant at the Symposium on Computational Geometry. For more about his many accomplishments, read the MIT story.<br /><br />ETA: <a href="http://www.scottaaronson.com/blog/">Scott Aaronson has more</a>http://11011110.livejournal.com/290949.htmlcomputational geometrypublic1http://11011110.livejournal.com/290793.htmlMon, 30 Jun 2014 05:21:11 GMTBook:Graph Drawing
http://11011110.livejournal.com/290793.html
One of Wikipedia's less well-known features is its Book: namespace. The things there are called books, and they could be printed on paper and bound into a book if you're one of those rare Wikipedia users who doesn't use a computer to read things, but really they're curated collections of links to Wikipedia articles. I've made two of them before, <a href="https://en.wikipedia.org/wiki/Book:Graph_Algorithms">Book:Graph Algorithms</a> and <a href="https://en.wikipedia.org/wiki/Book:Fundamental_Data_Structures">Book:Fundamental Data Structures</a>, which I have used for the readings in my graduate classes on those topics because I wasn't satisfied with the textbooks on those subjects. This week I put together a third one, <a href="https://en.wikipedia.org/wiki/Book:Graph_Drawing">Book:Graph Drawing</a>.<br /><br />It's not complete (what on Wikipedia is?), and the writing quality and depth of coverage are as variable as always, but there are about 100 topics there and I hope that collecting them in this way proves useful. I've listed a few more things that I think should be added but don't yet have their own Wikipedia articles on <a href="https://en.wikipedia.org/wiki/Book_talk:Graph_Drawing">the talk page</a>, but if you see something else missing then please let me know or, even better, add it.http://11011110.livejournal.com/290793.htmlgraph drawingwikipediapublic3http://11011110.livejournal.com/290437.htmlFri, 27 Jun 2014 21:18:02 GMTThe future of SoCG
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<a href="http://makingsocg.wordpress.com/2014/06/24/voting-is-open/">Voting opened this week</a> for members of the compgeom-announce mailing list, on whether the annual Symposium on Computational Geometry should leave ACM, as the <a href="http://computationalcomplexity.org/forum/open-letter/">Conference on Computational Complexity has recently done from IEEE</a>.<br /><br />There's a lot more opinion on both sides of the issue, and arguments both for staying with ACM and for leaving, in a series of postings at <a href="http://makingsocg.wordpress.com/">MakingSoCG</a>. If you're a compgeom-announce member, please inform yourself and then make your own opinion known through the vote. Past votes on the same issue had unconvincing results due to low turnout; we don't want the same problem to happen again.http://11011110.livejournal.com/290437.htmlcomputational geometryconferencespublic0http://11011110.livejournal.com/290247.htmlFri, 27 Jun 2014 05:02:24 GMTDobby is a free elf
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It must be graduation-and-moving-out time again. Seen this morning, in the parking lot of a UCI apartment complex for postdocs and visiting researchers:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/dobby/Dobby-m.jpg" border="2" style="border-color:black;" /></div>http://11011110.livejournal.com/290247.htmlcarsharry potteruciphotographypublic0http://11011110.livejournal.com/289956.htmlWed, 04 Jun 2014 05:33:31 GMTReverse perspective
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While cleaning up some badly-sourced articles on Wikipedia, I ran across the one on <a href="https://en.wikipedia.org/wiki/Reverse_perspective">reverse perspective</a>, a strange drawing style in which nearby objects are shown as small and far-away objects are shown as big. Despite the strangeness, it's mathematically consistent: it's what you get when you put objects between the perspective point (your eye in conventional perspective) and the viewing plane (a window through which you're looking at the world). Here's a nice example:<br /><br /><div align="center"><lj-embed id="44" /></div><br /><br />Although it's been used a lot in art (sometimes deliberately, for effect, sometimes naively, and sometimes for reasons lost to history and hotly debated), I don't know of any attempts to use it for visualization. It has some effects that might be useful, notably the ability to see more sides of an object at once than is possible in a conventional view.http://11011110.livejournal.com/289956.htmlinformation visualizationwikipediageometryartpublic0http://11011110.livejournal.com/289592.htmlSun, 25 May 2014 18:59:57 GMTPADS updated to Python 3
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<a href="http://www.ics.uci.edu/~eppstein/PADS/">PADS</a>, my collection of Python algorithm implementations, has finally been updated to work under both Python 2 and Python 3. Most, but not all, of its modules have been tested under both version of Python.<br /><br />Most of the code was already pretty close to working. The most frequent changes I needed to do: use next(iterator) instead of iterator.next(); avoid xrange, iteritems, and itervalues; use list(dict.items) for loops that change the dictionary; wrap arguments to print in parens.http://11011110.livejournal.com/289592.htmlpythonpublic2http://11011110.livejournal.com/289432.htmlTue, 20 May 2014 04:29:37 GMTCongratulations to Drs. Parrish and Pszona
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It must be that time of year...Friday and today I took part in two more thesis defenses, for Brian Parrish and Paweł Pszona.<br /><br />Brian is a mechanical engineering student, supervised by fellow Wikipedia editor <a href="https://en.wikipedia.org/wiki/User:Prof_McCarthy">Mike McCarthy</a>. His thesis research was to develop a system that can automatically analyze the configuration space of an eight-bar linkage and determine whether it can smoothly move between a desired set of poses. Somehow I got involved because setting up a system of equations describing the motion of these linkages involves some interesting graph algorithm problems. He's been working at Northrop Grumman, who supported him through his doctorate, and I imagine will continue working for them afterward.<br /><br />Paweł is a theoretical computer scientist, supervised by Mike Goodrich. The topics he included in his thesis were external-memory approximation of graph <a href="https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)">degeneracy</a>, applications of <a href="http://Order-maintenance problem">list labeling</a> in the visualization of dynamic graphs, and three-dimensional <a href="https://en.wikipedia.org/wiki/Arc_diagram">arc diagrams</a>. For the third of these topics, he brought along some props, 3d prints of his graph visualizations, which I think make the graph structure much clearer than their 2d projections. He has also done some interesting work with Goodrich (not part of his thesis) on <a href="http://arxiv.org/abs/1306.3000">making parametric search practical</a>, and very recently published another paper with me and others on <a href="http://arxiv.org/abs/1404.0286">cuckoo hashing on storage devices with limited rewrite capacity</a>. After he finishes he will be joining navigation-system company <a href="http://www.tomtom.com">TomTom</a> in Berlin.<br /><br />Congratulations, both of you!http://11011110.livejournal.com/289432.htmlucialgorithmspublic0http://11011110.livejournal.com/289243.htmlMon, 19 May 2014 03:13:28 GMTParking at The Lab
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<div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/labparking/LabParkingLot-m.jpg" border="2" style="border-color:black;" /></div><br /><br />Despite the name, it's not a university facility; it's a nearby mall with a nice Cuban restaurant in it. The stripy building is also a restaurant, but not one I've tried.http://11011110.livejournal.com/289243.htmlcarsphotographypublic0http://11011110.livejournal.com/288870.htmlSun, 11 May 2014 22:07:04 GMTCubic 1-planarity
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Here's a graph drawing problem I don't know how to solve: find the smallest <a href="https://en.wikipedia.org/wiki/Cubic_graph">3-regular graph</a> that is not <a href="https://en.wikipedia.org/wiki/1-planar_graph">1-planar</a>.<br /><br />The examples I tried with up to 24 vertices are 1-planar. Below are the 20-vertex <a href="https://en.wikipedia.org/wiki/Desargues_graph">Desargues graph</a>,<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/Desargues_graph_1planar.png"></div><br /><br />the 24-vertex <a href="https://en.wikipedia.org/wiki/McGee_graph">McGee graph</a> (7-cage),<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/McGee_graph_1planar.png"></div><br /><br />and the 24-vertex <a href="https://en.wikipedia.org/wiki/Nauru_graph">Nauru graph</a>:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/Nauru_graph_1planar.png"></div><br /><br />(I'm not requiring the edges to be straight line segments; they're just drawn that way.)<br /><br />On the other hand, for the 28-vertex <a href="https://en.wikipedia.org/wiki/Coxeter_graph">Coxeter graph</a> and 30-vertex <a href="https://en.wikipedia.org/wiki/Tutte%E2%80%93Coxeter_graph">Tutte–Coxeter graph</a> (8-cage) I have been unable to find a 1-planar drawing. So my guess is that the answer is somewhere in this range.<br /><br />The fastest method I know of for testing whether graphs are 1-planar algorithmically (try all pairings of edges, for each pair of edges insert a dummy node to represent their crossing, and test planarity of the result) is going to max out at 14 or so vertices, I think, not good enough to solve the problem. On the other hand, lately there's been some progress on solving other crossing number problems exactly by formulating them as logic problems using the Hanani–Tutte theorem and throwing SAT solvers at them. Maybe something like that can work here, too.<a name='cutid1-end'></a>http://11011110.livejournal.com/288870.htmlunsolvedgraph drawingpublic4http://11011110.livejournal.com/288589.htmlSat, 10 May 2014 05:46:04 GMTCongratulations, Dr. Simons!
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Today, Joe Simons (a joint advisee of Mike Goodrich and me) passed his thesis defense. Most of <a href="http://www.ics.uci.edu/~eppstein/pubs/a-simons.html">my own research with Joe</a> has been on graph drawing, but his dissertation concerned several results on different models of time in dynamic geometric data structures, including the <a href="http://arxiv.org/abs/1109.0312">retroactive model</a> (in which updates you make to past versions of the timeline have effects that propagate through other more recent updates), the windowed model (in which you have to aggregate the events that happened within a query window of time into a larger structure), and the <a href="http://arxiv.org/abs/1204.4714">local-update model</a> (in which objects can move, but only proportionally to their size). Joe has accepted a position at Google starting this summer.http://11011110.livejournal.com/288589.htmlcomputational geometryucipublic0http://11011110.livejournal.com/288490.htmlFri, 09 May 2014 06:57:28 GMTNew disjoint paths algorithm
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Via the <a href="http://processalgebra.blogspot.com/2014/05/best-paper-awards-at-icalp-2014.html">Process Algebra Diary</a>, I learned of a recent paper by Andreas Björklund and Thore Husfeldt, <a href="http://thorehusfeldt.files.wordpress.com/2010/08/spdp-e5d5661.pdf">Shortest Two Disjoint Paths in Polynomial Time</a>, which just won the Track A best paper award at ICALP. It looks like a very interesting paper; I can see why it won the award.<br /><br />Here's the problem it solves: you're given a graph with two pairs of terminals <i>s</i><sub>1</sub>–<i>t</i><sub>1</sub> and <i>s</i><sub>2</sub>–<i>t</i><sub>2</sub>. The goal is to find disjoint paths connecting the two pairs of terminals with minimum total cost. It almost sounds like the problem solved by <a href="https://en.wikipedia.org/wiki/Suurballe's_algorithm">Suurballe's algorithm</a>, but that one allows either <i>s</i> to be connected to either <i>t</i>, while here we have to connect the correct pairs. The specific version of this problem they solve is for undirected and unweighted graphs, with disjointness meaning either that the paths share no vertices or that they can share vertices but not edges. The unweighted part could easily be modified to paths weighted by small integers. Directed or real-weighted versions would also be natural but I don't know what's known about them and I don't think this paper's ideas are likely to work for them.<br /><br />My understanding of this all is still a bit vague, but here's a summary.<br /><br />The techniques are algebraic rather than combinatorial, meaning that rather than working with paths directly it works with matrices and polynomials and things like that. This is a bit unusual for graph algorithms but not unprecedented; for instance there are standard maximum matching algorithms that work this way. The number of perfect matchings is a matrix permanent, and permanents are difficult (#P-complete) to compute. But if there's only one matching, the permanent is the same as the determinant: both are just the number one. By using random edge weights from a suitable range of integers (large enough to make it work, small enough not to blow up the computation time) one can <a href="https://en.wikipedia.org/wiki/Isolation_lemma">"isolate"</a> a single matching with smaller weight than the others, compute a determinant of a matrix constructed from the weighted graph, and use it to find the matching you isolated. The coefficients of the matrix are powers of two and the lowest nonzero power of two in the binary representation of the determinant can be used to read off the matching.<br /><br />The new algorithm is similar in flavor, but trickier, and is inspired by Valiant's work on <a href="https://en.wikipedia.org/wiki/Holographic_algorithm">holographic algorithms</a>. Again, it uses permanents. You can compute them modulo 2 because in that case it's the same as the determinant, and what the authors show is that this can be extended in two ways, simultaneously: to modulo 4 instead of modulo 2, and to matrices whose coefficients are polynomials mod 4 instead of just numbers. The short summary of this part of the algorithm is that it performs a sequence of pivots to make one column of the matrix even, computes the permanent of a minor of the matrix eliminating the even column, and then does some stuff to get from that result to the permanent of the whole matrix. "Some stuff" involves using permanents of matrices of modulo 2-polynomials in order to compute some correction terms to add to the subpermanent.<br /><br />With these fancy modular permanents in hand, a disjoint paths problem that has a single optimal solution can be found in a very similar way to the matching solution: cook up a matrix of mod-4 polynomials, such that the lowest-degree nonzero coefficient in its permanent can be used to read off the answer. In the general case, when there might be more than one optimal solution, a random weighting (again, from a range of integers that's not too big and not too small) isolates one of them and allows the rest of the algorithm to work.<a name='cutid1-end'></a>http://11011110.livejournal.com/288490.htmlgraph algorithmspublic3http://11011110.livejournal.com/288042.htmlMon, 28 Apr 2014 05:22:06 GMTBanksy?
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I found this parachute rat in Costa Mesa today. I have no idea whether it's authentic — the lack of detail kind of suggests it isn't, but given the history of the artist in question, how meaningful is authenticity anyway?<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/parachuterat/ParachuteRat-m.jpg" border="2" style="border-color:black;" /></div><br /><br />Not coincidentally, but perhaps ironically given the subject matter, <a href="http://11011110.livejournal.com/268225.html">almost exactly a year ago</a> I recommended a Korean time travel movie, <i>Young Gun In The Time</i>, which sadly still seems very difficult to find. This year's time travel movie recommendation is for an Australian romance, <i>The Infinite Man</i>. Tightly plotted, quite stage-play-like in its very sparse cast and setting, darkly funny, and in some ways reminiscent of Rashomon in the way it keeps bringing in unexpectedly different viewpoints to the same repeated scenes. I enjoyed it. There's still <a href="http://newportbeach.festivalgenius.com/2013/films/theinfiniteman_hughsullivan_newportbeach2014">still another chance to see it this coming Wednesday</a>, if you're local, interested, and lacking your own time machine.http://11011110.livejournal.com/288042.htmlgraffitiphotographypublic0http://11011110.livejournal.com/287956.htmlMon, 21 Apr 2014 18:33:47 GMTIndifference graphs and their construction
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I just added a new article to Wikipedia on <a href="https://en.wikipedia.org/wiki/Indifference_graph">indifference graphs</a> (also known as unit interval graphs or proper interval graphs): the graphs formed from sets of points on the real line by connecting every two points whose distance is less than one.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/Indifference-graph.png"></div><br /><br />There are many papers on algorithms for going from the graph to a geometric representation in linear time. The following method for the reverse problem, going from the set of points (or equivalently unit intervals) to its graph must be known, possibly in the context of its generalization to higher dimensional <a href="https://en.wikipedia.org/wiki/Unit_disk_graph">unit disk graphs</a>, but I don't know a good reference for it.<br /><br />Given a set of real numbers:<ol><li>Round each one down to the nearest smaller integer.</li><li>Use a hash table to collect numbers that round to the same integer.</li><li>For each number <i>x</i>, use the hash table to find each other number <i>y</i> whose rounded value is within one of the rounded value of <i>x</i>.</li><li>Create an edge in the graph for every pair (<i>x</i>,<i>y</i>) found in this way whose distance is at most one.</li></ol>Each pair (<i>x</i>,<i>y</i>) can be charged against the hash table entry of <i>x</i> or <i>y</i> that has the larger number of inputs mapped to it. In this way, each hash table entry gets a charge proportional to the square of its number of values. But on the other hand every pair of inputs that map to the same hash table entry form an edge, so the number of edges is at least proportional to the sum of the squares of the hash table entry sizes. Thus, the total work is at most proportional to the total output size.<a name='cutid1-end'></a><br /><br />Update, April 22: It appears that the correct reference for this is Bentley, Jon L.; Stanat, Donald F.; Williams, E. Hollins, Jr.<br />The complexity of finding fixed-radius near neighbors. Information Processing Lett. 6 (1977), no. 6, 209–212.http://11011110.livejournal.com/287956.htmlgraph algorithmswikipediagraph theorypublic6http://11011110.livejournal.com/287741.htmlSat, 19 Apr 2014 21:07:51 GMTStructures in solution spaces
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As <a href="http://11011110.livejournal.com/284935.html">I promised earlier</a>, here is the video for my talk on "structures in solution spaces" at the <a href="http://11011110.livejournal.com/285302.html">Conference on Meaningfulness and Learning Spaces</a> last February.<br /><br /><div align="center"><lj-embed id="39" /></div><br /><br />It was a wide-ranging talk, about learning spaces, distributive lattices and Birkhoff's representation theorem for them, rectangular cartograms, antimatroids, the 1/3-2/3 conjecture for partial orders and antimatroids, partial cubes, and flip distance in binary trees and point sets. It was also about an hour long, so don't watch unless you have the time. For those with shorter attention spans, I've also put up a pdf file of <a href="https://www.ics.uci.edu/~eppstein/pubs/Epp-IMBS-14.pdf">my talk slides</a>.<br /><br /><a href="https://www.youtube.com/playlist?list=PLQw7KTnzkpXcxHV9ehbYiB_DjQM1Qblnj">The rest of the talks from the conference are also online</a>. For those who like me are interested in discrete mathematics and discrete algorithms, Fred Roberts' talk on when the output of an algorithm is meaningful and Jean-Paul Doignon's talk on polyhedral combinatorics might be particularly interesting.http://11011110.livejournal.com/287741.htmlmedia theorytalkspublic0http://11011110.livejournal.com/287439.htmlMon, 14 Apr 2014 06:50:49 GMTFrom when even the cars had moustaches
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Every year about this time, my campus holds a spring celebration, with all the student organizations setting up pavilions in the park to sell food and advertise for new members, with the creative anachronists bashing each other with padded swords, and with a car show. Why a car show? I don't know, but I always enjoy seeing the variety of shapes and colors compared to today's mostly-the-same boxes. My favorite this year was a 1950 Chevy Fleetline, found rusting in a swamp in Tenessee; after a lot of work restoring it, its owner decided to cover it in clear-coat, showing off the patina on its metal, rather than just painting it. And then drove it cross country, towing a trailer full of his stuff, to UCI where his girlfriend is a grad student.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/wayzgoose14/Fleetline1-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><a href="https://www.ics.uci.edu/~eppstein/pix/wayzgoose14/Fleetline2.html">The moustache is more visible in this one</a>. And <a href="https://www.ics.uci.edu/~eppstein/pix/wayzgoose14/index.html">here are the rest of the photos</a>.http://11011110.livejournal.com/287439.htmlcarsuciphotographypublic1http://11011110.livejournal.com/287079.htmlSun, 13 Apr 2014 05:07:36 GMTCongratulations to Tetsuo Asano
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<a href="http://11011110.livejournal.com/274087.html">When I last saw</a> <a href="https://en.wikipedia.org/wiki/Tetsuo_Asano">Tetsuo Asano</a>, he was giving a research talk at WADS, and openly worrying that it might be his last one. We all thought it was because of mandatory retirement (still legal in Japan). But, it turns out, no. Instead, <a href="http://www.jaist.ac.jp/profiles/info_e.php?profile_id=00033">he's the new president of JAIST</a>. Congratulations, Tetsuo!<br /><br />I'll probably miss SoCG, in Kyoto this year, but for those who will be going, there will be an associated <a href="http://www.jaist.ac.jp/~otachi/tetsuo65/">workshop in honor of Asano's 65th birthday</a>.http://11011110.livejournal.com/287079.htmlcomputational geometryacademiapublic0http://11011110.livejournal.com/286767.htmlSat, 12 Apr 2014 00:42:44 GMTUsing complete binary trees to prove the power of two choices
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The power of two choices in load balancing is well known: If one throws <i>n</i> balls independently at a similar number of bins (as in hash chaining), some bin will typically have Θ(log <i>n</i>/log log <i>n</i>) balls in it, but if one draws two random bins for each ball, and places the ball greedily into the less-full of these two bins, the maximum load will be much smaller, Θ(log log <i>n</i>). And if one clairvoyantly chooses which of the two bins to place each ball into (or uses cuckoo hashing to move the balls around between their two bins as later balls come in) it is very likely that one can achieve only a constant load.<br /><br />The log-log result is originally by Azar, Broder, Karlin, and Upfal, and is well explained in a survey by Mitzenmacher, Richa, and Sitaraman, "<a href="http://people.cs.umass.edu/~ramesh/Site/PUBLICATIONS_files/MRS01.pdf">The power of two random choices: A survey of techniques and results</a>", which includes three different proof methods. Here's a fourth, which is related to but I think different from witness trees.<br /><br /><blockquote>We suppose that the balls are thrown into bins one-by-one, and consider two graphs defined from this random process. The first graph, G<sub>1</sub>, has the bins as its vertices; each ball defines an edge in G<sub>1</sub> whose endpoints are the two choices for that ball. As long as the number of bins is at least 2 + ε times the number of balls (for some ε > 0) we know from standard results on random graphs that with high probability every connected component of G<sub>1</sub> has <i>O</i>(log <i>n</i>) vertices and is either a tree or a pseudotree (tree plus one edge).<br /><br />The second graph, G<sub>2</sub>, is a directed graph that instead has the balls as its vertices. each ball in G<sub>2</sub> has at most two outgoing neighbors: the most recent balls to be previously placed in its two bins. When a component of G<sub>1</sub> is a tree, so is the corresponding component of G<sub>2</sub>, with each vertex of G<sub>1</sub> (a bin) expanded into a path in G<sub>2</sub> (the balls that end up being placed in this bin). Similarly, when a component of G<sub>1</sub> is a pseudotree, so is the corresponding component of G<sub>2</sub>. And since each component of G<sub>1</sub> has at most the same number of edges as it has vertices, and these edges correspond to the vertices of G<sub>2</sub>, the components of G<sub>2</sub> also have <i>O</i>(log <i>n</i>) vertices with high probability.<br /><br />Now, suppose that a ball is the <i>i</i>th to be added to its bin, and look at its two neighbors. When <i>i</i> > 1, each of these must exist, and must be at least the (<i>i</i> − 1)st ball to be added to its bin, and so on. Thus, if this ball is part of a tree component of G<sub>2</sub>, it is the root of a complete binary tree of height <i>i</i> within that component. If instead it is part of a pseudotree component, we can remove one edge from the component to turn it into a tree and get a complete binary tree missing at most one branch. So in either case the component of G<sub>2</sub> containing this ball contains a subtree with at least 2<sup><i>i</i> − 1</sup> vertices in it.<br /><br />In order for the number of nodes in the subtree, 2<sup><i>i</i> − 1</sup>, to be no larger than the number of nodes in the whole component, <i>O</i>(log <i>n</i>), it must be the case that <i>i</i> = <i>O</i>(log log <i>n</i>), QED.</blockquote><br /><br />One drawback of this argument, relative to the other proofs of the same result, is that it seems to require stronger assumptions on the numbers of balls and bins: it stops working when the number of bins drops below twice the number of balls. On the other hand, I think it's an easy way to teach the power of two choices, and relates well to other important theory concerning random graphs.<br /><br />It also seems to be a versatile argument that can be used for other related problems. The context here is a new paper of mine with Goodrich, Mitzenmacher, and Pszona, "Wear minimization for cuckoo hashing: how not to throw a lot of eggs into one basket", <a href="http://arxiv.org/abs/1404.0286">arXiv:1404.0286</a>, to appear at SEA 2014. In it, we used the same type of argument to show that a variant of cuckoo hashing, with three choices instead of two, avoids making large numbers of changes to any of its memory cells, a property useful for certain memory technologies. The paper also includes experiments with an implementation of this variant showing that it works well in practice.<a name='cutid1-end'></a>http://11011110.livejournal.com/286767.htmlhashingalgorithmspaperspublic8http://11011110.livejournal.com/286633.htmlSat, 05 Apr 2014 22:18:09 GMTUpgrade to Mavericks
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I had been running Snow Leopard (OS X 10.6) on my Macs, because I wasn't convinced that later versions were much of an improvement in usability. But Apple has apparently stopped providing security updates for a version as old as mine, so this week I updated to Mavericks. So far I've encountered only minor glitches: the update lost a couple of configuration settings (the monitor arrangement for my two-screen desktop and the preferred browser on my laptop), I had to reinstall git and update python, and some very old PowerPC software that I had forgotten I even had either stopped working or became more obviously flagged as non-working.<br /><br />In honor of the successful upgrade, and Apple's new naming theme, here's some big-wave action at a closer beach (about five miles as the crow flies from my house).<br /><br /><div align="center"><lj-embed id="35" /></div>http://11011110.livejournal.com/286633.htmltoolspublic4http://11011110.livejournal.com/286355.htmlSat, 05 Apr 2014 07:18:33 GMTGraphs with many cycles and doubled cycle minors
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This one isn't my problem – it came from MathOverflow – but it's frustrating me that I haven't been able to solve it, so I hope by reposting it here I can get some more attention to it and maybe even a solution.<br /><br />The question is: <a href="http://mathoverflow.net/questions/161006/do-graphs-with-large-number-of-cycles-always-contain-large-necklace-minor">let <i>F</i><sub>k</sub> be the family of graphs having no minor in the form of a cycle with doubled edges, of length <i>k</i>. Do the graphs in this family have a polynomial number of cycles (with the exponent of the polynomial depending on <i>k</i>, ideally linear in <i>k</i>)?</a><br /><br />Here are some examples of graphs for which the number of cycles is large relative to the size of the largest doubled cycle minor:<br /><br />A biconnected graph with no length-3 doubled cycle minor must be either a subdivision of <i>K</i><sub>4</sub> or a series-parallel graph in which one of the two arguments to each series composition is a path. All such graphs have O(<i>n</i><sup><i>2</i></sup>) cycles.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/bicycle-minor/3-bicycle-free.png"></div><br /><br />In outerplanar graphs, the length <i>k</i> of a doubled cycle minor is the number of leaves of the dual tree. Each cycle can be formed by pruning some branches of the tree, and there are at most <i>k</i> branches to prune, so there are O(<i>n</i><sup><i>k</i></sup>) cycles in the outerplanar graphs with no doubled <i>k</i>-cycle minor. For instance, the outerplanar graph below has four leaves.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/bicycle-minor/outerplanar.png"></div><br /><br />The complete bipartite graph <i>K</i><sub><i>k</i>,<i>n</i></sub> (for <i>k</i> ≤ 2<i>n</i></sub>) has maximum doubled cycle minor length <i>k</i> and O(<i>n</i><sup><i>k</i></sup>) cycles.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/bicycle-minor/bipartite.png"></div><br /><br />In complete graphs, the length of a doubled cycle minor is approximately 2/3 the number of vertices. (Each edge contraction can double only two of the edges of a cycle.) The number of cycles is factorial in the number of vertices, so the exponent of the number of cycles (as a polynomial in <i>n</i>) is also linear in <i>n</i>.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/bicycle-minor/complete2bicycle.png"></div><br /><br />Form a tree with height log<sub>2</sub> <i>k</i> and branching factor <i>b</i> (possibly much larger than <i>k</i>). Connect each node to all its ancestors and connect the set the children of each node by a spanning tree. If we didn't include the spanning trees, the result would have low tree-depth, causing all of its cycles to have length at most <i>k</i>, from which it follows that there are O(<i>n</i><sup><i>k</i></sup>) cycles and maximum doubled cycle length <i>k</i>. The spanning trees don't significantly increase the number of cycles (you can use at most one path per spanning tree in a cycle, and each cycle touches at most <i>k</i> of these trees) and for the same reason doesn't significantly increase the length of the longest doubled cycle minor.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/bicycle-minor/tree-depth.png"></div><br /><br />Somehow I have the feeling that the last example, based on tree-depth, is on the right track. For minor-closed graph families, having no large cycle minor is the same as each biconnected component having bounded tree-depth, so having no large doubled cycle minor should be the same as having some structure like tree-depth but a little more complicated. But I can't prove anything of the sort.<br /><a name='cutid1-end'></a><br /><br />ETA 2014-04-10: Solved, negatively, by a <a href="http://11011110.livejournal.com/259176.html">planar bipartite permutation graph</a>.http://11011110.livejournal.com/286355.htmlgraph theorypublic2http://11011110.livejournal.com/286162.htmlSat, 29 Mar 2014 06:07:03 GMTGreetings from Barbados
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I'm back now, but this beach house in Barbados was my home for the last week, as I attended the 29th Bellairs Winter Workshop on Computational Geometry (my first time there).<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs/Seabourne-m.jpg" border="2" style="border-color:black;" /></div><br /><br />The format of the workshop is very much aimed at making new research, not just sharing what the participants have done elsewhere (as many other workshops and conferences do). We met as a group twice each day for three-hour group sessions, one in the morning and another in the evening, with afternoons as free time. The sessions were held outdoors at a set of picnic tables, and participants organized themselves into smaller groups of from two or three to half a dozen or so people, to discuss different open problems that had been prepared in advance (most of which saw significant progress by the end of the workshop). Preliminary results were announced to the whole group at most sessions, so even if you worked on a subset of the problems you had a sense of what everyone else was doing. And moving from one group to another was encouraged: you didn't have to choose one problem to work on and stick to it even when you were getting nowhere.<br /><br />For me, the long afternoon break gave me time to think about the problems on my own (while relaxing in a deck chair with a beach view and catching up on the internet) so I could come back to the discussions with fresh ideas. I heard several people remark that having two intensive sessions so far apart made it feel like we were getting in two days of work for every day of the workshop. The hope of being invited back to such a pleasant place also provided some motivation to work hard. And group social activities such as our meals together and the daily meet-up on the beach at sunset to watch for the green flash (sadly, not in evidence this year) helped build cameraderie and made it easier to join in on research discussions with groups of people you might not already know.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs/Sunset-m.jpg" border="2" style="border-color:black;" /></div><br /><br />I'm not yet sure in what form and when the results of this all will appear (so I'm being intentionally vague about exactly what we discussed) but expect to see some new names in the co-authors of some of my future papers.<a name='cutid1-end'></a>http://11011110.livejournal.com/286162.htmlconferencesphotographypublic0http://11011110.livejournal.com/285735.htmlThu, 20 Mar 2014 05:10:58 GMTCages
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<p>The <a href="https://en.wikipedia.org/wiki/Cage_(graph_theory)">cages</a> are a class of graphs that I think deserve to be more widely known, because they have a lot of interesting properties that make them useful as counterexamples in graph algorithms and graph theory. They're hard to construct and we don't have a lot of explicit descriptions of them, but that's not so important when you're using one as a counterexample.</p>
<p>First, what is a cage? The cages are parameterized by two numbers, <i>r</i> and <i>g</i>. An (<i>r</i>,<i>g</i>)-cage is a graph that:
<ul>
<li>is <i>r</i>-regular: each vertex is incident to exactly <i>r</i> edges,</li>
<li>has girth <i>g</i>: each cycle has at least <i>g</i> edges in it, and</li>
<li>is as small as possible: no graph with fewer vertices is also <i>r</i>-regular with girth <i>g</i>.</li>
</ul>
An example is the <a href="https://en.wikipedia.org/wiki/Petersen_graph">Petersen graph</a>, which is the unique smallest 3-regular graph with girth 5 and therefore is also known as the (3,5)-cage. Another, less well known, is the <a href="https://en.wikipedia.org/wiki/Robertson_graph">Robertson (4,5)-cage</a>. Challenge: find a better layout than Robertson's, below (which I took from Wikipedia and at least shows some hints at symmetry), or than the tangled circular layouts also shown in the Wikipedia article.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/Robertson-cage.png"></p>
<p>I don't think it's obvious that cages exist, for all combinations of <i>r</i> and <i>g</i>, but they do. They have to be big: A simple argument based on the fact that a breadth first search of depth up to half the girth can't find any cycles shows that these graphs must have Ω((<i>r</i> − 1)<sup><i>g</i>/2 − O(1)</sup>) vertices, exponential in <i>g</i> even when <i>r</i> is bounded. Turning this around, if <i>n</i> is the number of vertices in such a graph, <i>g</i> can be at most 2 log<sub><i>r</i> − 1</sub> <i>n</i> + O(1), only logarithmically small.</p>
<p>Bollobás and Szemerédi ["Girth of sparse graphs", J. Graph. Th. 2002] say that "the first good upper bound" on the size of a cage was by Erdős and Sachs in 1963. I don't know how they proved it (it's in German), but my first guess would be a form of the probabilistic method: <a href="http://arxiv.org/abs/cs.DM/9907002">random regular graphs have only a small number of short cycles</a>, so probably there's some way of getting rid of them and leaving a remaining regular graph that has none at all. But as far as I know the best bounds are not probabilistic, but instead use the method of Ramanujan graphs, by Lubotzky, Phillips, and Sarnak [Combinatorica 1988].
A Ramanujan graph is a graph whose second eigenvalue is nearly as large as possible, which in turn implies that it is a good <a href="https://en.wikipedia.org/wiki/Expander_graph">expander</a>.
In their paper, Lubotzky et al use an algebraic construction to find a family of graphs with <i>g</i> = (4/3 − o(1))log<sub><i>r</i> − 1</sub> <i>n</i> and they show that having such a high girth implies that the graph is a Ramanujan graph.</p>
<p>So we know that the girth of a cage is logarithmic in its size. But is the right constant on the logarithm 2 or 4/3? Bollobás and Szemerédi say that it should be 2, and I don't have any reason to doubt it myself. If so, the graphs of Lubotzky et al are not themselves cages, but at least they give us an upper bound on how big cages need to be.</p>
<p>Beyond having high expansion, some of the other properties of cages involve high local symmetry, dense minors and high treewidth, and bad surface embeddings:</p>
<ul>
<li><p>Some of the known small cages are highly symmetric graphs, but there's no particular reason to believe that continues for larger sizes. However, they are locally symmetric: the neighborhood of every vertex looks like every other vertex, out to a radius of half the girth, just a regular tree with no cycles to be seen.</p></li>
<li><p>For fixed values of <i>r</i>, a cage is a sparse graph. However, the cages necessarily have much denser minors. To see this, take an arbitrary spanning tree of a cage, and choose a set of equally spaced levels of the tree spaced about <i>g</i>/4 levels apart, such that there are about 4<i>n</i>/<i>g</i> vertices in the chosen levels. Contract the tree edges connecting each node with its nearest ancestor in a chosen level (or the root if no such ancestor exists). Because the graph has high girth, these contractions won't produce any self-loops or doubled edges, giving a minor that still has almost as many edges as it did before but with a number of vertices that has been reduced by a logarithmic factor. Even if you delete all but (1 + ε)<i>n</i> of their edges, the same argument shows that the remaining graph will still have a dense minor. And because of their high expansion, cages also do not obey anything like the <a href="https://en.wikipedia.org/wiki/Planar_separator_theorem">planar separator theorem</a>, and instead have linear treewidth.</p></li>
<li><p>If a graph is embedded on a surface, <a href="https://www.ics.uci.edu/~eppstein/junkyard/euler/">Euler's formula</a> tells us the genus of the surface as a function of its number of vertices, edges, and faces. For a cage, the number of faces must be very small (each edges is in two faces but each face has logarithmically many edges), from which it follows that the genus must be linear. In fact, the minimum genus for these graphs is very close to the maximum genus of any regular graph with the same degree. They also can't be embedded nicely in the plane: Leighton proved that for bounded degree graphs, the number of crossings in any planar drawing is proportional to the square of the bisection width. Because they are expanders, cages have linear bisection width and quadratic <a href="https://en.wikipedia.org/wiki/Crossing_number_(graph_theory)">crossing number</a>.</p></li>
</ul><a name='cutid1-end'></a>http://11011110.livejournal.com/285735.htmlgraph theorypublic7