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Oxymoronic

Fri, 11 Dec 2009 02:06:58 GMT

Unintentially humorous email subject line of the day: "Impact Your Success With Effective Communication". I don't see why I should trust someone who uses impact as a verb in that context to tell me anything about communication. ~~Physician~~ Dean of Continuing Education, heal thyself.

STACS acceptances

Mon, 07 Dec 2009 08:36:14 GMT

Via polylogblog I learn that the list of accepted papers at STACS has become available.

Three graph algorithm titles caught my attention enough to get me to track down their online versions, though I haven't had time to do much more than scan their abstracts:

Finding Induced Subgraphs via Minimal Triangulations (arXiv:0909.5278) by Fomin and Villanger. This turns out to be about finding moderately-exponential algorithms that find subgraphs of bounded treewidth that are as large as possible and to solve subgraph isomorphism problems when the subgraph to be found is large but of bounded treewidth.

Two-phase algorithms for the parametric shortest path problem by Chakraborty, Fischer, Lachish, and Yuster. Parametric optimization problems involve an input graph with weights that vary (linearly or more complicatedly) with some parameter; one must solve an optimization problem like shortest paths for particular parameter values. In the version studied here, one doesn't want all the different shortest paths for all different parameter values (there can be superpolynomially many different paths even when the edge weights vary linearly); rather, we'd like to compute the shortest path for some small set of parameter values, more quickly than the O(mn) per value that it would take to run Bellman-Ford separately for each value. (One can't use Dijkstra because some edges may be negative.) The authors show that, with polynomial preprocessing, shortest paths for each parameter value may be found in O(m + n log n) time, as fast as Dijkstra.

Planar Subgraph Isomorphism Revisited (arXiv:0909.4692) by Dorn. This is a followup to one of my papers, which provides a completely impractical algorithm for finding any k-vertex subgraph (for fixed k) in a larger planar graph in linear time. The algorithm described here has been improved: it's now exponential in k rather than k log k. I'm still not convinced that it's at all close to practical, but it's a step in the right direction.

Paper permutohedron

Fri, 04 Dec 2009 17:16:53 GMT

Make your own paper orthogonal permutohedron! Here's how. Print out the pattern, cut the solid lines, valley fold the dashed lines, and mountain fold the dash-dot lines. (I used a box-cutter both to make the cuts and to prescore the folds.) Here's what it looks like when you're done:

See also this fractal staircase, another orthogonal polyhedron constructed in a similar way.

Orthogonal polyhedra

Fri, 04 Dec 2009 01:17:31 GMT

Three quick puzzles concerning the polycube depicted below: (1) Cut it into two pieces that can be reassembled into a cube. (2) Fill space with translated, rotated, and reflected copies of it. (3) Fill space with translated and rotated (but not reflected) copies of it.

This polyhedron is an axis-parallel permutohedron: its 24 vertices can be labeled with the 24 permutations on four objects, in such a way that each edge connects two permutations that are related by a swap of two adjacent values. My new paper with Elena Mumford, Steinitz Theorems for Orthogonal Polyhedra (arXiv:0912.0537), aims to provide a complete classification of orthogonal polyhedra such as this one in the same way that Steinitz's theorem provides a complete classification of convex polyhedra.

Convex polyhedra have graphs that are exactly the planar 3-connected graphs, from which we know that there is (up to combinatorial equivalence) only one four-face polyhedron (the tetrahedron), two five-face polyhedra (the triangular prism and the square pyramid), etc. The "simple orthogonal polyhedra" that we study in an analogous way have axis-parallel faces that are simple polygons, three perpendicular edges at each vertex, and the topology of the sphere. As we show, these polyhedra are exactly described by the planar bipartite 3-regular graphs with a weaker connectivity condition than Steinitz's: removing any two vertices splits the graph into at most two connected components (counting an edge between the two removed vertices as a component). From this graph-theoretic characterization it follows that there is exactly one six-face orthogonal polyhedron (the cube), one eight-face orthogonal polyhedron (an L-shaped hexagonal prism), etc.

The key building blocks for our classification are a more restricted class of orthogonal polyhedra: ones that, like the permutohedron above, can be viewed from a point of view from which only a single vertex is hidden behind the other faces. As we show, a planar graph forms the set of vertices and edges of a simple orthogonal polyhedron with one hidden vertex if and only if its dual graph has a structure like the one in the drawing below: a cycle or set of cycles that passes through each interior vertex and that uses exactly one edge from every white interior triangle. Geometrically, the cycles separate the mountain folds from the valley folds of the polyhedron. If the dual graph is 4-connected, then it always has a cycle cover of this type, and the other polyhedra of our classification can be constructed by gluing together polyhedra for dual-4-connected subgraphs.

In my previous paper on bendless orthogonal drawing, I showed how to represent the permutohedron (and many other graphs) as a system of axis-parallel line segments in 3d, so that three mutually perpendicular line segments meet at each vertex. But in that paper, the line segments were allowed to cross each other, and the drawings I found were in general very far from being polyhedral. One of the results in the new paper is that, whenever a planar graph has a bendless drawing of this type, then it is the graph of a polyhedron: all crossings and other forms of self-intersections can be eliminated.

Coincidentally, today's arXiv papers also include another one related to Steinitz's theorem (the real one, about convex polyhedra): On the number of spanning trees a planar graph can have (arXiv:0912.0712) by Kevin Buchin and André Schulz. Buchin and Schulz show that, in order to realize a 3d convex polyhedron using integer coordinates, one needs only O(n) bits per coordinate, improving an exponential bound from Steinitz' original paper and several other previous polynomial bounds.

Fake conference scams

Fri, 27 Nov 2009 22:04:46 GMT

Old and busted: fake conferences that scam money out of academics using high registration fees, accept everything rather than subjecting submissions to any sort of peer review, but actually exist: you can attend them, meet the other dupes, and get a proceedings.

The new hot trend: fake conferences that don't exist, come with (nonexistent) paid travel expenses to invited speakers and, presumably, scam money out of academics by getting them to send their banking data to the organizers.

Via BoingBoing.

Not drowning but waving

Wed, 25 Nov 2009 08:38:37 GMT

When I'm in the heat of coming up with new ideas on a research project I have a bad habit of sending an email to my collaborators and then, a minute or an hour or a day later, another and another, correcting or contradicting or elaborating on the previous ones, until it can be difficult to figure out which parts of what I sent were important or bogus or relevant. Wouldn't it be better to have a system that's sort of like email, in that it allows one to send messages that only one's trusted collaborators sees, but sort of like a wiki, in that everything can be edited after the fact by anyone else (with the old versions still viewable)? And while we're at it, why not make this system sort of like a threaded web forum where one can share a whole long thread with additional people long after it starts, and sort of like an outliner that lets you build a hierarchical structure for your ideas, and maybe also sort of like a chat system where one can see what everyone else in the same thread is typing as they type it? And even sort of like LaTeX^* in that it knows how to format equations and not just the usual text and pictures?

That's what Google Wave is.

I've been using Wave for all of a week now for one of my papers and I already regret the times I lapsed back into my old bad habits to work on it by email. I'm not giving up on email any time soon but for some situations there are other tools that are better, and I think this is one of those situations. Wave is in beta (justifiably so: the server is...robust) and still invitation-only. But apparently a week is long enough for me to be an old hand^** and start inviting a few other people to join me, so if you want an invite and somehow haven't managed to score one already elsewhere then feel free to ask here.

^*Ok, the LaTeX part isn't there by default. It's an extra gadget that you have to add on, but it's easy to do so. I haven't used it much, though: most of the time when I need math it's just for simple formulas like O(n²) that are easy enough to type inline.

^**If I'm an old hand, then Suresh and Lance are ancient and decrepit and over-the-hill.

~~Update: all eight of my initial set of invitations are used now but bhael has more; see the comments.~~

Update update: I have invites again.

Growth and Decay in Life-Like Cellular Automata

Tue, 17 Nov 2009 03:44:21 GMT

If I haven't been posting much here in the last couple of weeks, it's because too much of my writing energy has been going into writing actual papers. Here's one of them: Growth and Decay in Life-Like Cellular Automata, arXiv:0911.2890. The aim of the paper is to predict which two-dimensional cellular automaton rules will behave similarly to Conway's Game of Life. But a large part of the difficulty in making this sort of prediction is finding the right definition of "similarly".

A standard method for classifying automata, due to Wolfram, is to look at the behavior of the automaton on random initial conditions. Very roughly speaking, four outcomes are possible: the field can go from random to uniform, it can develop scattered still life and oscillator patterns but without any persistence of long-term complex behavior, it can remain random, or something else can happen. That "something else" supposedly describes Life, although really it seems that Life develops scattered still lifes and oscillators but just does it more slowly than some other rules.

From my point of view, I'm not so concerned with random inputs. There are interesting things to say about Life's behavior on random initial conditions, but it's the non-random inputs that more fascinate me, the ones with some structure to them like the caterpillar and metapixel that are far too big to be found in practice by a random search (despite the theoretical truth that every possible pattern happens in a large enough random field). From this point of view Wolfram's classification is unsatisfying: these large structured patterns with interesting behavior can be found in many different rules that have very different behavior on random inputs. It's also overly subjective (why is Life Class IV and not Class II?) and unpredictive — it doesn't tell you in advance how a rule is going to behave, it only serves to describe the behavior you've already observed.

So a large part of the new paper describes a very crude classification that despite its crudity works better for my purposes: automata are grouped into four classes according to whether they support patterns that can grow beyond any bounding box enclosing them, and whether they support patterns that can die out completely. The interesting rules are likely to be the ones for which both answers are yes, and we can classify the vast majority of rules without having to look at the individual behavior of each one.

After defining this classification and using it to map out the rule space, the rest of the paper catalogues a collection of rules that support the complex patterns I'm looking for. One standout here is B013468/S02, which supports many small spaceships and oscillators and has a glider gun much smaller than the one in Life. It's also interesting from the Wolfram random-start point of view: when seeded with a random field (best with 46% live cells and 54% dead ones), it develops into large regions of cells alternating at each step between being alive and dead, each crisscrossed by large numbers of gliders, puffers, and other patterns sent out from the chaotic boundaries between regions of different phases. Because these boundaries gradually lose their curvature over long time scales, the number of steps before this system stabilizes to scattered oscillators is much higher than it is in Life.

(See also)

Boo

Mon, 02 Nov 2009 07:16:02 GMT

My photos from Halloween are now online. Pumpkin picking, carved pumpkins, my son and his friend in costume, that sort of thing.

Slopes of intersecting line segments

Tue, 27 Oct 2009 01:12:10 GMT

As a follow-up to the problem of coloring triangle-free line segment intersection graphs that I posted the other day, here's another line segment intersection representation of the Grötzsch graph. This time I drew it with only four different slopes of line segments, making it obvious that it can be 4-colored (just use a different color for each slope). And conversely the fact that this graph requires four colors implies that I couldn't have drawn it with fewer than four slopes. Is there a bound on the number of distinct slopes needed to realize all triangle-free line segment intersection graphs? It seems likely to be hard to prove that this number is bounded, if it is, since that would also solve the coloring problem. But maybe it's easy to come up with a family of triangle-free line segment intersection graphs that require an unbounded number of slopes?

A natural candidate for an (infinite) line segment intersection graph requiring an unbounded number of slopes is the order-4 pentagonal tiling of the hyperbolic plane, the Klein model of which forms a triangle-free line segment arrangement in the Euclidean plane. But I don't see an easy way to get a handle on how many slopes it needs.

Another case of plagiarism

Mon, 26 Oct 2009 21:21:05 GMT

Via Mike Trick's twitter feed I learn of a case of plagiarism published on SIAM's web site. Most of the papers that were plagiarized seem to be in operations research, but one of them is computational geometry: one of the victimized authors is Godfried Toussaint, whose paper (copied by the plagiarists) concerns a one-dimensional geometric matching problem for testing how similar two point sets are that is somewhat related to the calculation of Haussdorff distance.

To me the most interesting part of the story is the lack of response from the journals and universities associated with the plagiarists. Do they hope to avoid publicity by keeping the story quiet? But by failing to respond they have prompted SIAM to take this public step, and made themselves look like collaborators in the plagiarism.

Four open from IPAM

Sat, 24 Oct 2009 21:01:54 GMT

I spent the last week visiting the Institute for Pure and Applied Mathematics at UCLA, for a workshop on combinatorial geometry. Rather than post here about the many exciting new results I heard about (for which see the conference program) I thought I'd describe a few open problems that came up in some of the talks.

Jacob Fox began the conference by describing his work with János Pach on separator theorems for string graphs. He mentioned a problem that he attributed to Erdős: suppose that A is a collection of line segments in the plane, no three of which intersect pairwise. How many colors are needed, in the worst case, to color the segments of A so that no two crossing segments have the same color. In other words, what is the chromatic number of a triangle-free intersection graph of line segments? It is not even known whether this number has a finite bound. For instance, the line segments below represent the Grötzsch graph, a triangle-free graph requiring four colors.

Rom Pinchasi spoke about "some unrelated problems", one of which concerned blocking visibilities among sets of points in the plane. If one is given n blue points, how many additional red points must one place so that there is a red point between every pair of blue points? If the points are in sufficiently general position (so that any red point can block at most two pairs of blue points) then a quadratic number of blockers is needed, but it seems to be the case that, even if one carefully arranges the blue points so that they are easily blocked, a superlinear number of blockers may be needed. Specifically, Pinchasi conjectured that if no three blue points are collinear, they can't be blocked with fewer than Ω(n log n) red points. He also considered a version of the problem where the blue points may have collinearities and in which, on each line determined by two or more blue points, one needs only to place a red point somewhere between the two extreme blue points. For this variation, he proved that at least n/2 blockers are necessary (unless all blue points are on one line). However, the arrangement below (one of two examples of point sets with few ordinary lines) was the only one he could find in which fewer than n were sufficient.

Although Richard Ehrenborg's talk was about something else, he began with another unrelated problem, concerning spanning trees in bipartite graphs. He defined a class of graphs that he called Ferrers graphs, a bipartite variation of threshold graphs, as follows. From any Ferrers diagram with x rows and y columns, we form a bipartite graph with x vertices u_i on one side of the bipartition and y vertices v_i on the other side, where u_i is connected to v_j if there's a dot or square in the ith row and jth column of the diagram; I've drawn an example below. For these graphs, there's a very simple formula for the number of spanning trees: multiply together all the degrees of all the vertices, and divide this product by xy. Ehrenborg conjectures that the same formula (the product of the degrees divided by the numbers of vertices on each side of the bipartition) is an upper bound for the number of spanning trees in any bipartite graph.

József Solymosi finished off the conference by a talk about the famous and still unsolved problem of how many pairs of points can be at unit distance among n points in the plane, or equivalently how many edges can be in a unit distance graph. He asked about a related problem, progress on which would lead to progress in the original graph: given n unit circles in the plane, how many points can there be that are crossed by three or more of the circles? One can get a superlinear lower bound by placing circle centers on a grid with a carefully chosen spacing. For instance, in the grid of circles shown below, the radius of the circles is 5/2 times the grid spacing, and the points of triple intersection (some of which are marked in green) form a diagonal grid twice as dense as the grid of circle centers. However, for upper bounds, nothing better than the trivial O(n²) is known.

Commutative diagrams

Wed, 21 Oct 2009 18:22:54 GMT

Aesthetics of commutative diagrams. An interesting discussion of graph drawing in the context of graphs that represent mathematical formulas. Does one lay them out in a nice grid pattern which makes the formulas easier to read, or does one choose a less uniform vertex placement in which the global shape of the diagram (its outside face) conveys the intended meaning of the diagram?

Curves in graph drawing

Thu, 15 Oct 2009 03:17:31 GMT

The graph drawn above is the planar dual to the Herschel graph. The Herschel graph has nine quadrilateral faces, so its dual is a 9-vertex 4-regular planar graph, and therefore can be drawn as the arrangement graph of a system of smooth curves, or as in this case as the self-crossings of a single smooth curve.

Much of the research in graph drawing uses a drawing style in which edges are drawn as straight line segments, or as sequences of straight line segments articulated by "bends", corners where two line segments meet. But I think that, in many cases, smooth curves can create a more interesting aesthetic effect, that the extra freedom inherent in drawing curves makes it possible to spread out the features of the drawing more evenly, and that smooth curves are likely to be easier for the eye to follow than sharp bends.

No doubt Mark Lombardi would agree.

Paths and edge-colorings in hypercubes

Fri, 09 Oct 2009 00:39:41 GMT

By way of clearing some space off my whiteboard, here's a drawing that's been taking up space in the middle of it for too long:

The context is a problem of Norine on edge-antipodal colorings of cubes: is it the case that, whenever the edges of a hypercube are two-colored in such a way that edges directly opposite each other have opposite colors, then some pair of opposite vertices are connected by a path of edges of only one color?

I was looking at a stronger variation of the problem: if a cube's edges are two-colored arbitrarily, is there always either a monochromatic antipodal path or a monochromatic zone (a family of parallel edges, all given the same color)? If true this would answer Norine's problem, because an antipodally colored cube can't have a monochromatic zone. But it's not true, and the drawing shows why not. It's a six-dimensional hypercube (not antipodally colored), in which there are no monochromatic zones (each class of parallel edges has both colors). But there are also no monochromatic antipodal paths: an antipodal path would have to project to an antipodal path in a little three-dimensional cube as well as one in the big three-dimensional cube, but in the little cube the only antipodal paths are brown and in the big one they are only purple.

There's another variation that I've also thought about, without much success: given an arbitrily two-edge-colored hypercube, must there always exist a path between two opposite vertices in which there exists at most one pair of differently-colored consecutive edges? Norine's problem and this problem don't require the paths to be shortest paths but that would also be interesting.

Updated bibtex style for easy hyperlinking of articles

Thu, 01 Oct 2009 22:59:30 GMT

In conjunction with Bill Gasarch's call for including as many links as possible in the bibliographies of our papers (and for us to take some other more important steps towards open access; see also here) I've updated a BibTeX style file, abuser.bst, that I've been using in conjunction with pdflatex and the ~~url package~~ hyperref package in order to make hyperlinks in my papers' bibliographies.

Basically, you use this package very much like the standard BibTeX abbrv style, with the following changes:

For book series, such as LNCS, it is preferable to use number={nnnn} rather than volume={nnnn} to indicate the number of the book within the series; this frees the volume to indicate the volume of a multivolume work (which might exist within a differently numbered series). The formatting is also a little more compact; e.g. series={LNCS}, number={1234} produces "LNCS 1234" rather than "vol. 1234, other information, LNCS".

url={...} includes the given url within the formatted reference, as a hyperlink for TeX systems that support hyperlinking.

eprint={...} includes the arxiv paper with the given eprint number, as a hyperlink for TeX systems that support hyperlinking.

doi={...} includes the given doi, as a hyperlink for TeX systems that support hyperlinking. If you're not familiar with the doi system, it's a way to provide links to the publisher's web site for journal and conference papers that is supposedly more permanent than just using a url (as publishers often change their url schema but are not supposed to change their dois). So this doesn't do anything about the open access issue but does allow easy hyperlinking of online published content. The doi is usually included somewhere on the publisher's web page for an article but can also be looked up using crossref.org.

There's a note at the top of the file about Dutch names being sorted correctly (that is, "van Kreveld" should be sorted under "K"), but I no longer remember what I did to achieve this.

If you need something other than abbrv, you're on your own, but I hope this is helpful to at least a few other people than myself.

Dublin and Chicago photos

Wed, 30 Sep 2009 05:46:53 GMT

I just finished uploading five new sets of photos:

Mathematical sculpture at Trinity College Dublin. Or sort-of mathematical: the artist may have intended one of these sculptures to model DNA, but to a topologist it looks like a torus link. Apparently every year the Hamilton Workshop on Geometry and Topology uses one of these as a logo, but they're running out: soon they'll have to go with the kitschy sculpture of a buxom Molly Malone outside the college gate.

Dublin. Or the parts of it that I saw outside of Trinity and Guinness. With my usual complement of graffiti.

The Guinness Storehouse. Supposedly Ireland's biggest tourist attraction. I didn't take many photos inside, although I found the tour quite interesting; most of the photos are of the 360-degree view of the Guinness brewery that one gets from the bar at the top of the tour.

Chicago. The weather was not really conducive to photography but I took a couple of shots anyway.

Graph Drawing 2009. If you didn't go, now's your chance to see what you missed. If you did go, you can see how many photos you and your friends are in. I hope none of the ones I kept are too embarrassing or unflattering. ETA: The GD09 web site now also links to another set of photos by Pranava Jha.

Some statistics about this year's Graph Drawing program

Sat, 26 Sep 2009 03:14:43 GMT

After two conferences I'm too tired to successfully post anything serious. So, since Graph Drawing lacks a public business meeting at which to present these, here instead are some statistics on acceptances and rejections at the conference.

It was a bad year to have only one co-author: (Here and in the other chart below, dark blue represents accepted papers and light blue the rejected ones.)

Here are the most popular title words (excluding stop words), sorted by the acceptance rate of the papers they appeared in. Somehow it pleases me that the two most frequent were "graph" and "drawing". Also, papers on optimal algorithms for orthogonal grid labeling were unusually unpopular this year.

And finally, what would Graph Drawing be without a drawing of a graph? This one has a vertex for each accepted paper and an edge between two papers that share a co-author. Somehow this looks very different from an equally sparse Erdős–Rényi random graph: the fact that it's built out of cliques (sets of papers with a common co-author) seems very visible.

Crossing resolution of bounded-degree graphs

Thu, 24 Sep 2009 05:08:09 GMT

Two of the papers presented today at Graph Drawing (and one earlier at WADS) concerned drawings in which edges that cross are required to do so at a high angle. One of today's papers ("On the perspectives opened by right angle crossing drawings", by Angelini et al) included some results on bounded degree graphs, showing that graphs with sufficiently low degree bounds could be given drawings with right-angled crossings and few bends per edge. The other paper ("Area, curve complexity, and crossing resolution of non-planar graph drawings", by Di Giacomo et al) generalized the problem, looking not just at right angled drawings but at drawings in which all crossings have an angle bounded away from zero; the "crossing resolution" of the title is the minimum angle of any crossing. So what if we put both of these together and look at the crossing resolution of bounded-degree graphs?

As I already pointed out a year ago, an algorithm from a paper with Duncan and Kobourov in SoCG 2004 already achieves good crossing angles with no bends when the degree is very small (at most three). This algorithm draws the graph with its vertices on a grid and its edges in two planar layers such that one layer of the edges connects pairs of vertices on adjacent grid columns (causing them to be close to vertical) and the other layer connects pairs of vertices on adjacent grid rows (causing them to be close to horizontal). Two edges can cross only if all four of their endpoints lie on distinct rows and columns of the grid, ruling out what would otherwise be some bad cases. So the worst case appears to be one in which a slope-3 and a slope-1/3 edge cross each other, with angle π/2 − 2tan⁻¹(1/3), approximately 53 degrees.

A limit to the vertex degree bounds that force bendless drawings with high crossing resolution to exist is provided by a paper by Barát, Matoušek, and Wood that proves that graphs with degree nine may have unbounded geometric thickness. A drawing in which all crossings have angles bounded away from zero has bounded geometric thickness: if one partitions the edges of the graph into a constant number of subsets within which all slopes are close to each other, each subset must form a planar graph. So, since degree-9 graphs don't have bounded geometric thickness, they also don't have bounded crossing resolution.

For bendless drawings of graphs with degree from four to eight, it seems that it is still unknown how the crossing resolution may depend on the degree. And even for degree three, perhaps better crossing resolution may be achieved by some other algorithm.

ETA: János Pach points out that the same relation between geometric thickness and crossing resolution allows one to prove an O(1/n) bound on the crossing resolution of n-vertex complete graphs from known results on geometric thickness of complete graphs, improving a logarithmic bound from the Di Giacomo et al paper and an O(1/√n) bound obtained from a result of Aronov et al. that any n points in general position have Ω(√n) pairwise crossing line segments.

Report from the Hamilton Workshop

Sat, 19 Sep 2009 20:37:24 GMT

I've been visiting Dublin, Ireland (my first time here) for the Fifth William Rowan Hamilton Geometry and Topology Workshop at the Hamilton Mathematics Institute of Trinity College, where I spoke about hyperconvex metric spaces, finding optimal stars, orthogonal convex hulls, and embedding into the Manhattan plane.

It was a good conference, but a bit like stepping into a parallel world where everyone's evil twin does low-dimensional topology and geometric group theory: the workshop theme had the phrase "Computational Geometry" in it, but it's an Other Computational Geometry that really mostly means low-dimensional topology and geometric group theory; there's an Other David Epstein (spelled wrong) who does etc and was here last year; and there's an Other Kevin Wortman (not spelled wrong) who does etc and wasn't here, but whose name caused some confusion when I referred to the work of my former student Kevin Wortman. Still, my interests and those of the Other Computational Geometers have enough overlap that I think we all got something useful out of the experience.

I'm not going to recount all of the talks (especially because I didn't understand them all) but here are a few that caught my attention.

Nathan Dunfield has posted his slides on “Practical Solutions to Hard Problems in 3-Dimensional Topology.” Hard in this context means, actually, as 3d topology goes, pretty easy: problems such as finding incompressible surfaces in manifolds that can be solved in merely exponential time by triangulating the manifold (that is, representing it as a complex of glued-together tetrahedra), observing that the thing you want can be represented as a normal surface, describing all normal surfaces as the integer points in an unbounded high-dimensional convex polytope, and looking at the boundary rays of the polytope. Despite the complication this all works pretty well (he said) when the number of tetrahedra is moderate (say 30 or 40), but these methods hit a wall somewhere around 50 tetrahedra and he needed more like 120 of them to define some examples of manifolds coming from a number-theoretic construction. So he found some heuristics and tricks that allowed him to guess the surfaces he was looking for without the exhaustive searching; there seems to be little hope of guaranteeing that these tricks work well in general, but they worked well enough for his examples.

János Pach was there as well and gave a talk I'd mostly seen before (but I think the rest of the audience hadn't) on bounding the numbers of edges in a graph when you know that it has a nice enough drawing (say, there is a constant bound on the size of the largest mutually crossing set of edges).

Robert Meyerhoff spoke on “Computer-Aided Analysis of Hyperbolic 3-Manifolds.” The use of the computer in this work is very quantitative and numerical. For instance, as part of a different calculation he needed to show that (with, it turns out, six or seven exceptions) if one expands a toroidal tube around the shortest non-contractable curve in a compact hyperbolic 3-manifold, one can keep expanding until the tube radius is at least log(3)/2 before it runs into itself and can't expand any more. The solution technique involves finding several numerical parameters of the manifold (such as the length of the non-contractable curve, the shortest distance between two copies of the curve in the universal cover, etc) and using a computer to map out blocks in the parameter space where a radius lower than log(3)/2 would lead to a contradiction (e.g. some other curve that is shorter than the supposedly shortest one). The output of this search (a partition of parameter space into blocks and data describing of what goes wrong within each block) forms what is in principle a rigorous proof; a program that verifies the proof can be much simpler than one that searches for it, and the level of rigor resulting from this sort of search seems to be quite acceptable to the low dimensional topologists.

Tim Riley had a fun talk that involved asymptotic analysis of quickly growing functions of a type that was familiar to me from theoretical CS. Specifically, he looked at the following process one could play on a sequence of finitely many non-negative numbers: remove the first number from the sequence, and replace each remaining positive number i in the sequence by the pair of numbers i, i − 1. For instance, the sequence 3,3,3 would become 3,2,3,2, then 2,1,3,2,2,1, then 1,0,3,2,2,1,2,1,1,0, then 0,3,2,2,1,2,1,1,0,2,1,1,0,1,0,0, etc. As he showed, any sequence eventually becomes empty after a finite number of steps like this, but a sequence of n i's takes roughly A(i,n) steps where A is the Ackermann function. He used this rapid growth to define groups with unexpected properties.

There were also several talks presented entirely on the chalkboard, something I still see occasionally in mathematics department seminars but not at the other conferences I go to. I found it quite refreshing, but I think it would only work for hour-length talks, not for the 20-minute ones we more frequently get.

Not the Nauru graph

Fri, 18 Sep 2009 07:59:00 GMT

The orthogonal polyhedron shown below has an interesting combination of properties: it is orthogonally convex (any axis-parallel line intersects it in a point, an interval, or the empty set) but not simply-connected.

My first thought on completing this drawing was: its skeleton is a highly symmetric cubic torus graph with 24 vertices...must be the Nauru graph! But it isn't. It's vertex-transitive but, unlike the Nauru graph, not edge-transitive: the edges that wrap the short way around the torus each belong to only two six-cycles (the two faces on either side of the edge) but the edges that wrap the long way around belong to three (the two faces and their equatorial cycle).

More Banff photos

Sun, 13 Sep 2009 19:06:49 GMT

As promised, I've finally put up Diana's photos from Banff, complete with grizzle bear.

Much of the delay was due to a chain of reinstallations of software and libraries I needed to perform to get my web photo gallery software working again after upgrading to Snow Leopard. Why doesn't software just work? (Asks the computer scientist, who should know better.)

Another theory blog

Fri, 11 Sep 2009 16:49:50 GMT

To go along with her newly minted assistant professorship at Oregon State University, Cora Borradaile has started a new blog, Silent Glen Speaks. So far there's only an introductory post (and, on the RSS feed, some abstracts of older papers) but I'll be watching for more. Welcome to the blogosphere, Cora!

Manhattan embedding, paired approximation, and stragglers

Fri, 11 Sep 2009 00:58:05 GMT

I just uploaded two new papers to the arXiv, and my co-author updated an old one:

Optimally fast incremental Manhattan plane embedding and planar tight span construction, arXiv:0909.1866. The concrete problem solved here is testing whether a finite metric space can be represented as the distances among a set of points in the Manhattan plane. The corresponding Euclidean problem is easy (find a non-collinear triple of points, place them in a triangle, and use them to determine uniquely the Euclidean location of the remaining points) but the best previous problem for the Manhattan-metric variant took O(n²log²n) time; this paper removes the logs, getting the time bound down to the size of the input distance matrix, using the tight span as a key tool. The tight span is an analogue for finite metric spaces of the convex hull for finite Euclidean point sets. In general it can be computed as the set of bounded faces of a high-dimensional polytope, but that's not very efficient. The main ideas of the paper are to provide a different method of representing and constructing tight spans that are homeomorphic to a subset of the plane, using rectangular complexes and Manhattan orbifolds.

Paired approximation problems and incompatible inapproximabilities, arXiv:0909.1870. To appear at SODA. If two different approximation problems are defined from the same input and you'd be satisfied with a solution to either, it turns out that the quality of approximation you get can be significantly better than for either problem alone. For instance, suppose you want either a coloring or a long path in a graph. Both problems are hard to approximate to within a significantly sublinear approximation ratio (although we don't know how to prove it for the path problem) but approximating one or the other to within √n is easy using depth-first search: if the DFS tree has many levels it contains a long path, and otherwise the partition into levels is a good coloring. Along with several similar examples the paper contains some other pairs of problems (such as set cover and hitting set) where it's not possible to get a better approximation than for either problem alone; that part ws less easy.

Straggler identification in round-trip data streams via Newton's identities and invertible Bloom filters, arXiv:0704.3313. Now updated with some experimental results for the Bloom filter data structure as requested by the referees, I think primarily as a proof-of-concept to show that the algorithms are implementable. When we first did these experiments we thought we saw some interesting phenomena in them, showing that some complications that we'd included for theoretical reasons (we couldn't figure out how to prove that it worked without them) were also necessary to achieve good practical performance. Sadly, that turned out to be a bug in our implementation. But at least the debugged experiments still show that the extra space used for the added part of the data structure is not wasted: it gives us a proportionate increase in the size of the sets we can identify.

Equilateral linkage rationality

Sat, 05 Sep 2009 03:00:16 GMT

Connect the corners of six unit equilateral triangles by hinges in the pattern shown below (so that the graph formed by the edges and hinges is the Cartesian product of two triangles).

The resulting linkage system of triangles and hinges has (up to symmetry of the plane) a single degree of freedom: if we add a single bar restricting the distance between two hinges on one of the axes of symmetry of the system, it becomes rigid. Therefore, as it flexes, the hinges will remain in a pattern like the one shown, in which they are arranged to form three more concentric equilateral triangles of varying sizes.

It's possible for at least two of these variable-sized equilateral triangles to have side lengths that are rational numbers. For instance, in the illustration above, the triangles and hinges were placed in such a way that the triangle formed by the three innermost hinges has side length 2/7, and the triangle formed by the three outermost hinges has side length 13/7. What about the middle triangle? Is it rational? In general, can all three of these nested triangles be rational?

Yes and yes. It's possible to show this by manipulating formulas for the length (I had this all worked out, starting from some trigonometric formulas in terms of the angles of the hinges and converting them into complex number arithmetic), but there's a much simpler geometric demonstration. Look at the figure above, which is arranged to have a vertical axis of symmetry. Suppose that the side length of the largest equilateral triangle is a rational number q, and the side length of the smallest equilateral triangle is a rational number r. Place the figure so that the axis of symmetry lies on the y-axis of the plane. Then the leftmost and rightmost vertices have x-coordinates −q/2 and q/2, respectively. Similarly the left and right vertices of the inner triangle have x-coordinates −r/2 and r/2, respectively. The left and right vertices of the middle triangle are connected to the leftmost and rightmost vertices by edges that are translates of edges from the bottom vertex (on the center line) to the left and right inner vertices; therefore the x-coordinates of the middle triangle are −q/2 + r/2 and q/2 − r/2, and the side length of the middle triangle is q − r. If q and r are both rational, so is their difference. In the case of the configuration shown in the drawing, the middle length is 11/7.

In general there are infinitely many angles for which all three nested triangles have rational side length, dense in the set of all such angles. The technique for generating them resembles that for generating Pythagorean triples.

Incidentally, there's a four-dimensional convex polytope (the Cartesian product of two equilateral triangles) that has this same graph as its skeleton. It has six triangular 2-faces, nine square 2-faces, and six triangular-prism facets. Can you see them all in the drawing above?

ETA: Konrad Swanepoel uses a similar arrangement of Reuleaux triangles to show that three nonoverlapping translates of a convex body can overlap three nonoverlapping 180-degree rotations of the same body.

SODA acceptances

Fri, 04 Sep 2009 03:55:20 GMT

The list of SODA acceptances is out. ~~Sadly, again, it doesn't include abstracts, but they're promised to come later.~~ Now with abstracts! I have one paper there; I'll post more about it when I've had a chance to respond to the feedback from the reviews and prepare a preprint version.

It's hard to tell just from the titles, but I have the vague impression that there's rather less computational geometry than I'm used to seeing at past SODAs.

One very important change (perhaps a consequence of going to electronic proceedings?): “your conference paper can be up to twenty (20) pages”!!

ETA: See also 3d pancakes, algorithmic games, kd-PhD, 3d pancakes again, biased coin, polylogblog, geomblog, motocicleta