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Graph Drawing acceptances

2009-07-14T21:59:11Z

The list of papers accepted to Graph Drawing is now online here, and a booklet of abstracts is also available.

If you want your work represented at Graph Drawing, there's still time to submit it as a poster for the poster session. The poster submission deadline is July 31.

More corporate academic-publishing shenanigans

2009-07-02T22:21:36Z

Not Elsevier, this time. The rumor is that SAGE Publications, the corporate publisher of the journal Political Theory, have bypassed the journal's editorial board and unilaterally imposed a new editor. As one commenter (6/17 6:44 on the first thread below) states, "The idea that the editorship of the journal is to be determined directly by them, apparently with no formal consultation with members of the existing editorial community, is like the idea of a faculty search being run by a couple of corporate honchos from a University's Board of Trustees, without consultation with current members of the faculty of the relevant department."

See here, here, and here for discussion, but so far (despite a signed statement by one of the editorial board members) there's a lot more heat than light.

FOCS accepts

2009-07-02T17:21:15Z

The FOCS accepted papers (with abstracts) have been posted. Since not every conference is doing this, I think it bears repeating that the addition of abstracts to these lists is a very welcome innovation of recent years.

And while I'm making a short link-only post, an amusing photo gallery that I was pointed to recently: the Russian solution to the art-gallery problem.

Three sets of photos

2009-07-01T04:55:51Z

With all my traveling, I'd become delinquent at processing my photos, but I've caught up again (it made it easier that I edited the sets down to half a dozen photos each):

A school concert in which my son played viola, held at an auditorium at the local Lutheran college.

My recent trip to the Netherlands consisting of photos from the sculpture garden at the Kröller-Müller museum, a hike through Zuid-Kennemerland National Park, and the view from Marc van Kreveld and Bettina Speckmann's apartment.

Montpellier, France including a dinner with some other WG participants.

Report from WG

2009-06-29T23:40:56Z

I just returned from Montpellier, France, where I was invited to speak at WG 2009, the 35th International Workshop on Graph-Theoretic Concepts in Computer Science.

This was my first trip to France, and Montpellier was a very charming place to visit. It is in the south, and though not quite on the Mediterranean itself it has a very Mediterranean feeling. The conference organizers put together an interesting and educational excursion for us in which we learned some of the local history by visiting three local landmarks (the opera house, an old pharmacy, and the top of the gateway arch) and drinking a different wine in each. We learned that Montpellier is a new city for France, being only a little over 1000 years old, but it is home to one of the oldest medical schools in Europe. It's on the old pilgrim trail from Rome to Compostela, markers for which run through the town, and although the language is no longer spoken in the area there are several inscriptions for tourists written in Occitan. The nearby mountains were a refuge for the Cathars when they were persecuted by the Catholics, and there are many open squares in the city due to the monasteries and other buildings that were torn down in the French revolution. My hotel was on one side of the old quarter, now mostly a shopping district, and the conference was in an old movie theater on the other side of the quarter, so I had a very pleasant walk every day to the conference (only getting lost twice), and my bad high-school French was enough to get by with the people who didn't or wouldn't speak English (rather more of them than I've encountered in other parts of Europe).

The conference itself is about graph algorithms, both for problems on arbitrary graphs and (a larger fraction of the papers) for important special classes of graphs. There were too many interesting talks to describe them all in detail, so let me just mention a few.

Daniel Král started the conference with the other invited talk, on graphs of bounded expansion. This is a notion of sparsity in graphs, based on shallow minors, graph minors (subgraphs of contractions) in which two vertices may only be contracted together if they started out within some constant distance of each other in the original graph. There is a trichotomy theorem for the density of shallow minors: among the densest shallow minors of a given family of graphs, the numbers of edges must grow roughly as an integer power of the number of vertices, the exponent of which can only be 0 (for graphs with bounded numbers of edges), 1 (sparse graphs, with comparable numbers of edges and vertices), or 2 (dense graphs). Additionally, in the case where the exponent is 2, all graphs can be formed as shallow minors. Graphs of bounded expansion form the case of exponent 1, in which the shallow minors are all sparse; they generalize both minor-closed graph families (in which all minors are sparse) and bounded-degree graphs (since a shallow minor of a bounded-degree graph also has bounded degree). As Král described, graph properties describable as first-order logic formulae may be tested efficiently on these graphs, but there seems to be plenty of opportunity for more algorithmic work on them. The two main papers on this subject are in the European Journal of Combinatorics, by Nešetřil and Ossona de Mendez: one on structure theory and a second on algorithms. There also seems to be some connection with Král's work with Dvořák and Thomas at SODA 2009 on efficient 3-coloring of triangle-free surface-embedded graphs (via a data structure for finding paths of bounded length) but I'm not sure of the details of this part.

My favorite contributed talk of the first day was by Pim van 't Hof, a Ph.D. student of Broersma at Durham, presenting his work with Marcin Kamiński
and Daniël Paulusma on finding induced paths of given parity in claw-free graphs. Since line graphs are claw-free, this generalizes the problem of finding arbitrary paths of given parity in arbitrary graphs. Non-induced paths of given parity are easy enough to find: the shortest path has one parity, and a path of the other parity exists if and only if at least one of the biconnected components on a path between a given pair of terminals is non-bipartite (one direction is easy and the other can be proved by induction using open ear decomposition). However, the induced claw-free generalization is not as easy. I don't know that it's a very important problem, but it was a well-presented method that used some deep results of Chudnovksy and Seymour on the three-in-a-tree problem to reduce the problem to one in which all vertices belong to induced paths between the given terminals, and then split the problem into a case analysis using the known structure of perfect claw-free graphs (in the case that the graph is perfect) or the existence of an odd hole (in the case that it is not).

The talks on the morning of the second day were all about parametrized complexity and exponential time algorithmics. The most interesting to me (and the most closely related to my own recent research) was the last of the morning's talks, by Fedor Fomin, Daniel Lokshtanov, and Saket Saurabh, on algorithms for finding an embedding of a metric space into a line minimizing the total distortion. The metric spaces they consider are the ones generated by distances in an unweighted graph (hence the relevance to WG); the difficult part of the problem is determining the ordering of the vertices on the line, so the problem could be solved in factorial time, but they reduce this time bound to 5^n + o(n). The method involves partitioning the line into blocks whose length is the eventual distortion bound, so that each vertex only has a constant number of choices for the block it belongs to, and then performing some more complicated placement algorithms within the blocks. It's one of only a very few algorithms that I know of in which the optimal distortion of an embedding can be calculated exactly (another being my upcoming WADS paper); it would be of interest to extend the result from unweighted graphs to arbitrary metrics.

My own talk was in the afternoon; I surveyed problems (such as the partition of an orthogonal polygon into a minimum number of rectangles) the solution to which involves constructing a graph from the input (the bipartite intersection graph of line segments bounded by two reflex vertices of the polygon) and uses the special properties of this graph to find an efficient algorithmic solution (maximum independent sets in bipartite graphs via König's theorem). See here for talk slides.

I almost missed the first talk of the third day, due to my mistaken belief that I could find an alternative route from my hotel to the conference site without referring to a map, but I was very glad that I recovered from my disorientation in time for its start. Torben Hagerup spoke, on the problem of determining whether a given tree is the minimum spanning tree of a weighted graph, or (almost equivalently) of finding the maximum-weight edge on each of a collection of paths in a given tree, a key step in the randomized linear time minimum spanning tree algorithm of Karger, Klein, and Tarjan. This spanning tree verification problem had been previously solved in linear time in two papers, by Tarjan, Rauch, and Dixon and again by King, but Hagerup's solution was even simpler (though still not without some complication). By using least common ancestors and a technique of hierarchical clustering into subtrees from the previous papers based on Borůvka's algorithm, the problem becomes one of finding maximum-weight edges on ancestor-descendant paths in a tree in which all paths have logarithmic length, so sets of edges in these paths (for instance, the set of suffix maxima according to the weight function) can be represented in single machine words. Hagerup then derives some clever bit manipulation tricks for solving the problem in a linear number of word operations, based around the algebraic properties of an interesting operation down(x,y) that returns the nonzero bits of y for which the next larger bit of x is smaller than the next larger bit of y. Unlike our new SIGACT chair, I'm a big fan of this sort of bit-parallel computation — I think the effort of algorithm design in this area is well-spent because these techniques really do speed up actual programs and yet can be quite nontrivial to discover. Or, if you prefer an argument by authority, Knuth likes them too: on my way to France, I had coincentally watched Knuth discuss similar manipulations in his "Platologic Computation" talk (available here for Windows or here for iTunes).

All in all, a good conference, and one I would attend much more regularly if only it weren't so far away.

Copyright, permanence, jossage

2009-06-22T14:22:55Z

Via Michael Nielsen: Don't Ask, Don't Tell: Rights Retention for Scholarly Articles. My mother, a poet, thinks it very strange when she hears about the system of scientific publishing in which we give away the copyrights for all our papers. In poetry, the authors retain their copyrights, and give permission to publishers to publish their poems; the same is true in fiction writing. The system works without problems: it doesn't prevent publishers from going after people who illicitly copy their works, and it doesn't prevent them from getting exclusive publication rights to the works in question. So what, exactly, do we gain by giving away our copyrights? What we lose is the right to distribute our own works online for free; but as this Harvard Law blog post observes, many of us do that anyway, hoping that the publishers won't demand that we take them down again or sue us for noncompliance with their contracts. And mostly it works, but there's always that risk...

Fortunately, there is a solution: free online journals. The Journal of Graph Algorithms and Applications and the new Journal of Computational Geometry both are free as in free beer (no cost to access the papers, no publication fees) but also free in the sense that authors retain copyright and grant the publisher a license to print the paper. Therefore, I am happy to echo Suresh and ~~Ernie~~ Jeff and announce that JoCG is now open for business and accepting submissions.

One question I had with the new journal was, if it's online-only, how permanent are its archives? If whoever's running the journal gets hit by a bus, what happens to all the old papers? In today's business climate one should wonder about that for commercial journals too, I suppose. JGAA has been handling the issue by collecting its old issues into printed volumes, but as I understand it that arrangement has run into difficulties, so I was curious to hear what JoCG intended. Anyway, the answer is that they're using the Open Journal Systems software and LOCKSS data security model, in which university libraries maintain local copies of open content in order to assure its permanence. So I am greatly reassured on that front.

Therefore, as Samir exhorts us, get those SoCG papers into a journal. And now that we finally have a noncommercial alternative to DCG, CGTA, and IJCGA, let's support it by sending our papers there.

Metroville: the confluent drawing puzzle

2009-06-19T17:22:18Z

I just returned from a pleasant post-SoCG visit to Bettina Speckmann in the Netherlands. While there, she presented me with a copy of Metroville, a puzzle based on confluent drawing. (Image below stolen from the manufacturer's web site.)

Metroville consists of a game board with nine rotating pieces on it, each containing a section of train track with turns and junctions. The pieces can be permuted to form eight different "cities", and for each city there are eight puzzles, in which one must rotate the pieces so that the resulting configuration of track can allow a train to pass through a given sequence of cities in order without reversals.

I haven't worked out the details, but I'm pretty sure that larger versions of the game would be NP-complete (that is, it should be NP-complete to test, for a fixed city, whether there exists a set of rotations that allows a given train route to work) by a reduction from 3-SAT very similar to the one in my Phutball paper in which the train zigzags horizontally across the board through tracks representing variables and then zigzags again vertically through tracks representing clauses.

Regardless, it's a fun puzzle. Thanks, Bettina!

Also, slides from Elena Mumford's talk on area-universal cartograms (my SoCG paper with Bettina, Elena, and Bettina's student Kevin Verbeek) are now online.

Report from SoCG

2009-06-11T22:40:30Z

The 25th ACM Symposium on Computational Geometry just finished yesterday. Suresh has already reported on the pre-conference historical review of computational geometry, but I thought I'd mention a few highlights of the conference itself.

One strong theme this year concerned epsilon-nets and geometric set cover problems. An epsilon-net for a weighted set S of n points and a family F of shapes is a set N such that every shape in F that contains at least an epsilon fraction of the total weight of S has a nonempty intersection with N. When F consists of shapes with constant description complexity, one can find epsilon-nets with cardinality 1/epsilon times some more slowly growing function of epsilon, and the growth rate of this function controls the quality of certain approximation algorithms for geometric set cover problems (or their dual hitting set problems) in which one finds a small point set that hits all shapes in F by choosing an epsilon-net, doubling the weights of the points that would hit some unhit set, and repeating until finding an epsilon-net that hits everything. Varadarajan described the relation between this more slowly growing function and the combinatorial complexity of unions of shapes in F; similar results were obtained independently (at STOC 2009) by Aronov, Ezra, and Sharir, who had a paper at SoCG about speeding up the hitting set algorithms. Bukh, Matoušek and Nivasch had a paper on lower bounds on the size of weak epsilon-nets; normally a net must be a subset of the given points but in a weak epsilon-net this constraint is lifted. Mustafa and Ray showed that the epsilon-net approach to hitting sets is not always best: it can be replaced in some circumstances by local search algorithms that provide much more accurate approximations. Miller and Sheehy had a nice derandomization of some old work of mine on centerpoints, which are essentially single-point weak epsilon-nets. And Ken Clarkson spoke about generalized set cover problems in which some points need to be covered by multiple sets; he and his co-authors showed that many of the epsilon-net-based set cover approximations work in this more general setting.

Along with the invited talk on the computational geometry of origami by Robert Lang, there was an "origami session" that featured a proof by Katoh and Tanigawa of the "molecular conjecture" that frameworks of flat rigid panels with hinges (not unlike a piece of origami in the process of being folded) behave generically like more general rigid bodies with hinges, and a proof by Panina and Streinu that, if you cut a wedge from a piece of paper and fold it so that all the fold lines pass through the vertex of the wedge, then any configuration of the folded paper can be reached from any other configuration.

This year I supervised an undergraduate doing a project on watershed boundaries, so I was interested to see the work of van Kreveld and Silveira on river networks. Their problem is one of integrating two types of data, high resolution maps of river networks (showing only their x and y coordinates) and lower-resolution elevation maps. They describe methods for modifying the elevation map so that it makes sense with respect to the river that is supposed to flow over it: the water should flow where it actually does flow (that is, the river's elevation should consistently decrease) and it shouldn't flow where it doesn't (the banks of the river should maintain a higher elevation than the river itself).

Metric embedding is a subject that has been surprisingly absent from SoCG in the past — it's more popular at FOCS/STOC/SODA despite its geometric flavor — so it was good to see some of it this time. Borradaile, Lee, and Sidiropoulos described a randomized embedding from genus-g graphs to planar graphs in which the expected dilation of any distance is O(g²), a big improvement over the previous exponential bound. On the other hand, the best lower bound for this problem is logarithmic in g, so there may be another exponential improvement to be made. Chin, Guo, and Sun also described a hardness result for finding a smallest subset of the Manhattan plane containing the same shortest paths connecting a given set of points as the whole plane; this isn't the tight span (or orthogonal convex hull) because it doesn't require points outside the given set to be connected by paths.

There were lots of other interesting papers, so if I didn't mention yours you can safely assume that it's because I ran out of energy for writing this post rather than because I didn't like it.

Lars Arge did a great job of organizing, and especially in organizing the food. The reception buffet never ran out, the banquet featured a whole roast boar among an impressive spread of other foods, and the daily lunch spread was almost as impressive. I can't claim a very strong performance in the go-kart race excursion after the conference, but it was a lot of fun too.

All in all, a good conference.

Faster ladders in Life without Death

2009-06-07T10:40:58Z

Life without Death is a cellular automaton with the same rule for creating new live cells as Conway's Game of Life, but in which no live cell ever dies. Despite the inability to create oscillators or spaceships (both of which require some cells to die sometimes) it has interesting dynamics due to the existence of ladders, patterns that grow at speed c/3 (that is, one unit of growth for every three time steps of the cellular automaton). Typical random seeds end up with from one to four chaotic regions spreading diagonally from the starting region, spewing ladders on both sides of them so that one ends up with a pattern in which the four quarters of the pattern (as split on roughly diagonal lines) are striped by axis-parallel ladders, separated by the chaotic regions. The chaotic regions separating the striped quarters tend to grow wider, but slowly as the pattern grows, and the boundary of the chaotic regions keeps pace with the ladders (probably in a self-limiting way: there is a pattern that grows at speed 2c/3 along the side of a ladder and then erupts in chaos at the tip, so if the chaotic growth regions ever fell too far behind they would catch up using this mechanism).

Patterns of this type can be simulated to hundreds of thousands of time steps using the Hashlife algorithm embedded into Golly.

But as it turns out there is another ladder, one that can grow more quickly (the text below is a pattern format that can be copied and pasted into Golly):

x = 26, y = 15, rule = B3/S012345678
obobobobobobobobobobo$23o$3ob3ob3ob3ob3ob3o$26o$26o$ob3ob3ob3ob3ob3ob
3o$3ob3ob3ob3ob3ob3obo$25o$3ob3ob3ob3ob3ob3obo$ob3ob3ob3ob3ob3ob3o$26o
$26o$3ob3ob3ob3ob3ob3o$23o$obobobobobobobobobobo!

Rather than growing at speed c/3, it grows at speed 4c/9. I've only seen this once out of many chaotic starting seeds, but if it happens once it should happen more than once. My suspicion is that most chaotic patterns should have a shape that eventually comes to be dominated by these faster ladders, but that even hundreds of thousands of steps isn't enough to see this domination.

ETA: animated gif by Simpsons contributor, who set this all off by starting a Wikipedia article on LwoD.

ETA2, 6/13: Already discovered in October 2000 by Dean Hickerson, who also found ladders of speeds 4c/10 and 4c/13. Dean's patterns:

#C 4c/9
x = 258, y = 20, rule = B3/S012345678
246bo6bo$245b4o2b4o$245b3ob2ob3o$230bo2b3o7b14o$228b4ob3o7b4ob4ob4o$
221b2ob14o4b16o$220b7ob5ob4o2b2obob3ob2ob3obo$207b2ob2o9b3ob4ob10ob17o
$60bo145b6o7b8ob4ob3obob3ob4ob4ob4o$59b3ob3o90b3o3b3o32bo2b3ob4ob3o7b
4ob16ob17o$4bo5bo10bo6bo28b10o2b2ob2o82b3obob3o31b3ob15o2b10ob5ob4ob3o
bob3ob2ob3obo$3b3obob3o8b3ob2ob3o9bo9bo7b4ob5ob6o8b2ob2o16b2ob2o16b2ob
2o16b2ob2o4b13o16bo11b8ob5ob4ob3obob3ob4ob10ob17o$b13o4b14o6b3ob3o2b4o
5b9ob4ob3o8b6o15b6o15b6o15b6o3b4ob3ob4o15b3ob3o6bob3ob4ob10ob10ob4ob3o
bob3ob4ob4ob4o$b4ob3ob4o4b4ob4ob4o4b8ob2ob3o6bob3ob4ob8o6b3ob4ob3o2bo
6b3ob4ob3o2bo6b3ob4ob3o2bo6b3ob4ob14o5b3ob2o3b7o5b9ob4ob3obob3ob4ob16o
b17o$15o2b16o3b4ob3ob8o3b15ob4o4b14ob3o3b14ob3o3b14ob3o3b10obob3obob3o
bo4b8ob4ob3obo5b4ob16ob10ob5ob4ob3obob3ob2ob3obo$bob3obob3obo4bob3ob2o
b3obo3b9ob3ob4o4b4ob5ob9o3b4ob5ob8o2b4ob5ob8o2b4ob5ob8o2b4ob4ob15o3b4o
b16ob10ob5ob4ob3obob3ob4ob10ob17o$15o2b16o3bob3ob12o2b9ob4ob3obo3b9ob
4ob3ob11ob4ob3ob11ob4ob3ob11obob4ob3ob4o3b9ob5ob4ob3obob3ob4ob10ob10ob
4ob3obob3ob4ob4ob4o$b4ob3ob4o4b4ob4ob4o3b9ob2ob3obo4bob3ob4ob9o3bob3ob
4ob8o2bob3ob4ob8o2bob3ob4ob8o2bob3ob4ob14o3bob3ob4ob10ob10ob4ob3obob3o
b4ob16ob17o$b13o4b14o3b18o3b21o3b87o4b85o$3bo2bobo2bo8bo2bo2bo2bo7bo2b
o2bo2bo2b2o5bo2bo2bo2bo2bobo2b2o7bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo6bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bobo!

#C 4c/10
x = 274, y = 13, rule = B3/S012345678
224bo11b2o8bob3o4bo4b3ob3o$140bo3bo11b2o42bo10b3o8b4o7b6o5b6ob2ob3obob
3ob3o$129bo3bo5b3ob3obob3ob2ob5o26b3o9b3o8b4o7b6o2bo2b8ob9ob15obob3o$
128b3ob4o2bob15ob4ob2o24b3o4bo3b4o3b2o2b5o2b3ob6ob4ob6ob4ob6ob3obob3ob
2ob8o$9bo11bo3bo19bo39bo33bob3o3b4ob3ob6ob2ob3obob3ob9o17b3obob4ob4obo
b4ob4obob4ob4obob4ob4obob4ob4obob15ob4ob2obo$6bob4o2b3o2b4ob3o15bob4o
2b3o4b4o4b4o3b3o3b3o2b4obo24bo3b6ob2obob8ob4ob15obob4o11bo5b13ob11ob
11ob11ob11ob6ob2ob3obob3ob9o$3b7ob2ob3obob3ob4o2bo8b7ob2ob3ob2ob4ob2ob
4obob3obob3ob2ob7o13bo6b3ob7ob7ob2ob7ob3obob3ob2ob10o8b4o3bob2ob6ob11o
b11ob11ob11ob9ob15obob4o$3b2ob4ob15obob3o7b2ob4ob37ob4ob2o11b4ob2obob
4ob5ob7ob5obob15ob4ob2obo6b2ob5o2b9ob4ob6ob4ob6ob4ob6ob4ob6ob4ob6ob3ob
ob3ob2ob10o$b9ob3obob3ob2ob9o4b9ob3obob4ob2ob4ob2ob3obob3obob3ob9o6b2o
b4ob7ob2obob4ob5ob3ob6ob2ob3obob3ob10o3b9o3b4obob4ob4obob4ob4obob4ob4o
bob4ob4obob4ob4obob15ob4ob2obo$b4obob15ob4ob2obo4b4obob43obob4o4b9ob7o
b7ob2obob8ob4ob15obob4o4b4obob4ob14ob11ob11ob11ob11ob6ob2ob3obob3ob10o
$10ob2ob3obob3ob10o2b10ob2ob3ob2ob4ob2ob4obob3obob3ob2ob10o3b4obob4ob
5ob7ob7ob2ob7ob3obob3ob2ob10o2b12obob2ob6ob11ob11ob11ob11ob9ob15obob4o
$2b30o6b55o4b67o6b103o$4bo2bo2bo2bo2bo2bo2bo2bo2b2o10bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bobo8bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo2bo2bo2bo2bo2bo2bo2bo2bobo10bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bobo!

#C 4c/12
x = 213, y = 14, rule = B3/S012345678
163bo$162b4o9b3o$162b6o5bob3o11b2o$160b5ob4ob4ob5o6b4ob2o$22bo56bo25bo
54b4ob4obob3ob4ob2o4b4ob4o$3b2o16b3o54b3o23b4o51b7ob17o2b5ob4o$b5o5b3o
6b6o50b6o20bob3obo5bo45bob6ob3obob3obob5obob9o$b4o6b4o5bob4o3b2ob2ob2o
4b2ob2ob2o4b2ob2ob2o4b2ob2ob2o3b4obo7bo10b6ob10o2b3o37b6ob9ob6obob6ob
2obob2o8b3o$7o2b6o4b9ob8ob2ob8ob2ob8ob2ob8ob9o4b4o7b3ob4ob3obob3obob3o
bob3obo5bo15b2o7b4ob4obob3ob4ob5ob4ob6obo7b5o$bob4o2b4obo5b4obob4ob2ob
8ob2ob8ob2ob8ob2ob4obob4o5b4o6b35ob5o3bo8b5o4b7ob17ob6ob4ob5ob4ob4o$
16o3b9ob8ob2ob8ob2ob8ob2ob8ob9o3b7o3b5obob3obob3obob3obob3obob3obob3ob
ob3obob2o3b4o5bob6ob3obob3obob5obob11ob4ob7o$b4obobo2b4o5bob4ob4ob2ob
8ob2ob8ob2ob8ob2ob4ob4obo5bob4o2b2obob6ob41ob7o3b16ob6obob6ob2obob6ob
6obo$b14o5b62o5b67o5b53o$3bo2bo2bo2bo9bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo9bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bobo9bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo!

#C 4c/13
x = 507, y = 52, rule = B3/S012345678
39bo$37b4o$36b2ob4o117b2o$34b4ob4o115b4ob2o$34b4ob6o113b4ob4o$33b8obob
2o111b5ob4o$34bob2ob6obo109bobob8o$33b5ob4ob5o105bob6ob2obo$34b4ob8ob
3o101b5ob4ob5o$33b8obob7o99b3ob8ob4o$34bob2ob6obob4o98b7obob8o$33b5ob
4ob5obob2o95b4obob6ob2obo$34b4ob8ob5obo95bob5ob4ob5o$33b8obob7ob5o91b
5ob8ob4o$34bob2ob6obob7ob3o86b4ob7obob8o$33b5ob4ob5obob7o84b2ob7obob6o
b2obo$34b4ob8ob5obob4o82b7obob5ob4ob5o$33b8obob7ob5obobo81b3obob5ob8ob
4o$34bob2ob6obob7ob5obo79bob5ob7obob8o$33b5ob4ob5obob7ob5o75b5ob7obob
6ob2obo$34b4ob8ob5obob7ob3o71b3ob7obob5ob4ob5o$33b8obob7ob5obob7o69bob
7obob5ob8ob4o$34bob2ob6obob7ob5obob4o67b6obob5ob7obob8o$33b5ob4ob5obob
7ob5obo67b3obob5ob7obob6ob2obo$34b4ob8ob5obob7ob5o65bob5ob7obob5ob4ob
5o$33b8obob7ob5obob7ob4o60b5ob7obob5ob8ob4o$34bob2ob6obob7ob5obob7ob2o
56b3ob7obob5ob7obob8o12b3o$33b5ob4ob5obob7ob5obob7o55b7obob5ob7obob6ob
2obo12b4o5bo$34b4ob8ob5obob7ob5obob3o52b6obob5ob7obob5ob4ob5o11b6ob5o$
33b8obob7ob5obob7ob5obo52b2obob5ob7obob5ob8ob4o12bob3obob3obob3obo$34b
ob2ob6obob7ob5obob7ob5o49bob5ob7obob5ob7obob8o10b5ob16o2b3o$33b5ob4ob
5obob7ob5obob7ob3o45b5ob7obob5ob7obob6ob2obo12b4ob3obob3obob3obob3obo
5bo$34b4ob8ob5obob7ob5obob7obo41b3ob7obob5ob7obob5ob4ob5o10b37o3bo$33b
8obob7ob5obob7ob5obob6o40b7obob5ob7obob5ob8ob4o12bob3obob3obob3obob3ob
ob3obob3obob3obo5bo$34bob2ob6obob7ob5obob7ob5obob3o37b5obob5ob7obob5ob
7obob8o10b5ob40ob5o$33b5ob4ob5obob7ob5obob7ob5obo36b2obob5ob7obob5ob7o
bob6ob2obo12b4ob3obob3obob3obob3obob3obob3obob3obob3obob3obo$34b4ob8ob
5obob7ob5obob7ob5o33bob5ob7obob5ob7obob5ob4ob5o10b61o2b3o$33b8obob7ob
5obob7ob5obob7ob3o29b5ob7obob5ob7obob5ob8ob4o12bob3obob3obob3obob3obob
3obob3obob3obob3obob3obob3obob3obo5bo92b2o$34bob2ob6obob7ob5obob7ob5ob
ob7o27b3ob7obob5ob7obob5ob7obob8o10b5ob70o3bo83b5o$33b5ob4ob5obob7ob5o
bob7ob5obob6o24b7obob5ob7obob5ob7obob6ob2obo12b4ob3obob3obob3obob3obob
3obob3obob3obob3obob3obob3obob3obob3obob3obo5bo74b2ob4o13b3o2b3o3b2o
105bo3bo$34b4ob8ob5obob7ob5obob7ob5obob2o23b4obob5ob7obob5ob7obob5ob4o
b5o10b85ob5o39b2ob3o25b4ob4o11b4ob2ob4ob4ob3o2b2o3bo90b3ob3o$6bo26b8ob
ob7ob5obob7ob5obob7ob5obo20b2obob5ob7obob5ob7obob5ob8ob4o12bob3obob3ob
ob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3ob
o30b4ob4o24b4ob7o8b4ob5ob4ob4ob4ob4o3b2o4bo78b7obo13b2ob2o$4b4o26bob2o
b6obob7ob5obob7ob5obob7ob5o17bob5ob7obob5ob7obob5ob7obob8o10b5ob94o2b
3o23b4ob4o12b3ob3o4b8obob2o7b5ob2ob4ob4ob4ob4ob4ob4ob4ob2o3bo69bob2ob
6o9bob6o$3b2ob4o13b2o8b5ob4ob5obob7ob5obob7ob5obob7ob3o13b5ob7obob5ob
7obob5ob7obob6ob2obo12b4ob3obob3obob3obob3obob3obob3obob3obob3obob3obo
b3obob3obob3obob3obob3obob3obob3obob3obo5bo14b8obo12b3ob4o4bob2ob6obo
4b2obob8ob4ob4ob4ob4ob4ob4ob4ob4o2b3o4bo47b3obo5b5ob4ob3o5b6ob2o$b4ob
4o11b4ob2o6b4ob8ob5obob7ob5obob7ob5obob7o11b3ob7obob5ob7obob5ob7obob5o
b4ob5o10b115o3bo9bob2ob6o9bob7o3b5ob4ob5obob6ob2ob4ob4ob4ob4ob4ob4ob4o
b4ob4ob4ob4ob3o2b2o35b4ob4o4b4ob8o3b3ob4ob4o$b4ob6o9b4ob4o3b8obob7ob5o
bob7ob5obob7ob5obob5o9b7obob5ob7obob5ob7obob5ob8ob4o12bob3obob3obob3ob
ob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3ob
ob3obob3obob3obo4b6ob4ob4o2bob6ob2obo4b4ob8ob5ob4ob5ob4ob4ob4ob4ob4ob
4ob4ob4ob4ob4ob4ob4o2b3o3b2o25b4ob3obob9obob5ob8ob4o$8obob2o7b5ob4o4bo
b2ob6obob7ob5obob7ob5obob7ob5obob2o7b4obob5ob7obob5ob7obob5ob7obob8o
10b5ob121ob4ob8ob5ob4ob5o2b8obob5ob8ob2ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4o
b4ob4ob4ob4ob4ob3o2b2o3bo10b13obob2ob6obob5obob8o$bob2ob6obo5bobob8o2b
5ob4ob5obob7ob5obob7ob5obob7ob5obo5bobob5ob7obob5ob7obob5ob7obob6ob2ob
o12b4ob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3ob
ob3obob3obob3obob3obob3obob3obob3obob9obob5ob8ob4o4bob2ob6obob5obob8ob
4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4o3b2o2b2obob3obob2ob
6ob4ob5obob6ob2obo$5ob4ob5obob6ob2obo4b4ob8ob5obob7ob5obob7ob5obob7ob
5obob5ob7obob5ob7obob5ob7obob5ob4ob5o10b127obob2ob6obob5obob8o2b5ob4ob
5obob6ob2ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob6o
b7ob4ob8ob5ob4ob5o$b4ob8ob5ob4ob5o3b7obob7ob5obob7ob5obob7ob5obob7ob5o
b7obob5ob7obob5ob7obob5ob8ob4o12bob3obob3obob3obob3obob3obob3obob3obob
3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob3obob2ob6o
b4ob5obob6ob2obo4b4ob8ob5ob4ob5ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob4ob
4ob4ob4ob4ob4ob4ob3ob4ob3obob9obob5ob8ob4o$b28o7b131o12b156o4b167o$3bo
2bo2bo2bo2bo2bo2bo2bobo11bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo16b2o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo8bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo!

Dean also discovered some slow parasitic vines:

#C 4c/13 parasite
x = 57, y = 35, rule = B3/S012345678
3bo3bo$2b3ob3o$b7obo$bob2ob6o$5ob4ob3o$b4ob8o$8obob4o$bob2ob6obob2o$5o
b4ob5obo$b4ob8ob5o$8obob7ob3o$bob2ob6obob7o$5ob4ob5obob4o$b4ob8ob5obob
o$8obob7ob5obo$bob2ob6obob7ob5o$5ob4ob5obob7ob3o$b4ob8ob5obob7o$8obob
7ob5obob4o$bob2ob6obob7ob5obo$5ob4ob5obob7ob5o$b4ob8ob5obob7ob4o$8obob
7ob5obob7ob2o15b2o$bob2ob6obob7ob5obob7o13b5o$5ob4ob5obob7ob5obob3o14b
4o$b4ob8ob5obob7ob5obo12b7o$8obob7ob5obob7ob5o10b4obo$bob2ob6obob7ob5o
bob7ob3o7b8o$5ob4ob5obob7ob5obob7obo6bob4o$b4ob8ob5obob7ob5obob6o4b8o$
8obob7ob5obob7ob5obob3o4b4obo$bob2ob6obob7ob5obob7ob5obob10o$b4ob4ob5o
bob7ob5obob7ob6obob4o$3b53o$5bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo!

#C 4c/11 parasite
x = 323, y = 44, rule = B3/S012345678
315bo$310b11o$309b4ob3ob4o$309b13o$309bob3obob3obo$308b15o$309b4ob3ob
4o$308b15o$309bob3obob3obo$297b2o9b15o$296b5o6bob4ob3ob4o$283bo13b4o6b
16o$282b3o10b7o3b3obob3obob3obo$116bo152bo11b6o8b4obob21o$111b11o145b
4o10bob4obo4b10obob2ob4ob3ob4o$110b4ob3ob4o144b4obo7b10ob3obob4ob4ob
16o$110b13o132b3o8b9o6b4obob16ob3obob3obob3obo$110bob3obob3obo130b5o9b
ob4ob3o2b10obob2ob4obob21o$109b15o117b3o9b4obo7b10ob3obob4ob4ob10obob
2ob4ob3ob4o$110b4ob3ob4o118b4o7b9obob2ob4obob16ob3obob4ob4ob16o$109b
15o103b2o10b6o6bobob4ob4ob10obob2ob4obob16ob3obob3obob3obo$110bob3obob
3obo104b4o8b4obo5b12ob3obob4ob4ob10obob2ob4obob21o$98b2o9b15o103b4o6b
10obob2ob4obob16ob3obob4ob4ob10obob2ob4ob3ob4o$97b5o6bob4ob3ob4o89b3o
10b7o3b3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob16o$85b2o9bob4o6b16o88b
5o8b4obobo2b13ob3obob4ob4ob10obob2ob4obob16ob3obob3obob3obo$84b5o5b9o
3b3obob3obob3obo75b3o10bob4o6b10obob2ob4obob16ob3obob4ob4ob10obob2ob4o
bob21o$21bo50b2o9bob4o3b3ob4obob21o73b4o10b9ob3obob4ob4ob10obob2ob4obo
b16ob3obob4ob4ob10obob2ob4ob3ob4o$16b11o44b5o5b10ob12obob2ob4ob3ob4o
62b2o10b7o7b4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob16o$15b4o
b3ob4o31b2o9bob4o3b3ob4obob5obob4ob4ob16o59b5o9bob4ob2o3b10obob2ob4obo
b16ob3obob4ob4ob10obob2ob4obob16ob3obob3obob3obo$15b13o30b5o5b10ob12ob
ob11ob3obob3obob3obo49bo10b4obo7b10ob3obob4ob4ob10obob2ob4obob16ob3obo
b4ob4ob10obob2ob4obob21o$15bob3obob3obo18b2o9bob4o3b3ob4obob5obob4ob5o
b4obob21o47b3o8b9o4bob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4o
b10obob2ob4ob3ob4o$14b15o16b5o5b10ob12obob10ob12obob2ob4ob3ob4o35bo10b
6o8bob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob
16o$15b4ob3ob4o16bob4o3b3ob4obob5obob4ob5ob4obob5obob4ob4ob16o33b4o8b
4obo6b11ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3o
bob3obob3obo$3b2o9b15o13b10ob12obob10ob12obob11ob3obob3obob3obo34b4o7b
9obob2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob4obo
b21o$2b5o6bobob3obob3obo5b3o4b3ob4obob5obob4ob5ob4obob5obob4ob5ob4obob
21o19b2o10b7o4b2obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob4obob
16ob3obob4ob4ob10obob2ob4ob3ob4o$3b4o6b16o3b4o3b13obob10ob12obob10ob
12obob2ob4ob3ob4o19b5o8b4obo4b13ob3obob4ob4ob10obob2ob4obob16ob3obob4o
b4ob10obob2ob4obob16ob3obob4ob4ob16o$b7o3b3ob4ob3ob4o4b11obob4ob5ob4ob
ob5obob4ob5ob4obob5obob4ob4ob16o5bo13b4o6b10obob2ob4obob16ob3obob4ob4o
b10obob2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob3obob3obo$b4obob
21o3bob3obobob10ob12obob10ob12obob11ob3obob3obob3obo5b3o10b7o3b3obob4o
b4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2o
b4obob21o$9obob2obob3obob3obo3b12ob4obob5obob4ob5ob4obob5obob4ob5ob4ob
ob21o3b6o8b4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob
4obob16ob3obob4ob4ob10obob2ob4ob3ob4o$bob4ob4ob16o3b4obob13obob10ob12o
bob10ob12obob2ob4ob3ob4o4bob4obo4b10obob2ob4obob16ob3obob4ob4ob10obob
2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob16o$10ob3ob4ob3ob
4o3b12obob4ob5ob4obob5obob4ob5ob4obob5obob4ob4ob16o2b10ob3obob4ob4ob
10obob2ob4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob4ob
ob16ob3obob3obob3obo$b4obob20o4bob3obobob10ob12obob10ob12obob11ob3obob
3obob3obo4b4obob16ob3obob4ob4ob10obob2ob4obob16ob3obob4ob4ob10obob2ob
4obob16ob3obob4ob4ob10obob2ob4obob20o$b25o6b91o4b193o$3bo2bo2bo2bo2bo
2bo2bobo10bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bobo8bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b
o2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo
2bo2bo2bobo!

Dean has also recently solved the long-open problem of finding quadratic growth patterns: the following pattern starts growing at period 25196 some time around generation 64000. It fills a single quadrant, so four copies of it can be used to fill the whole plane to constant density.

x = 47, y = 47, rule = B3/S012345678
3b2o26b2o$3b4o23b5o$3b4o24b4o$b7o21b7o$b4obo22b4obo$8o20b8o$bob4o22bob
4o$8o20b8o3bo$b4obo22b4obo2b2o$8o20b8o2bo$bob4o22bob4o$8o20b7o2bobobob
o$b4obo22b17o$8o20b10ob3ob3o$bob4o22b18o$8o20b19o$b4obo21b8ob3ob3obo$
8o22b14o$bob4o23b2obobobobobo$8o$b4obo$8o$bob4o$8o$b4obo$8o$bob4o$8o$b
4obo$8o$bob4o$8o$b4obo$8o$bob4o$8o$b4obo$8o$bob4o$7o2bobobobobobobobob
obobobobobobobobobo$b45o$10ob3ob3ob3ob3ob3ob3ob3ob3ob3o$b46o$47o$8ob3o
b3ob3ob3ob3ob3ob3ob3ob3o$2b42o$2b2obobobobobobobobobobobobobobobobobob
obo!

Another very simple seed eventually settles down in two of its four quadrants exhibiting glide symmetry in those quadrants:

x = 3, y = 3, rule = B3/S012345678
2o2$2bo!

Two great tastes that... no wait

2009-06-01T20:46:54Z

Boing Boing is showing a great information visualization concerning a case in which a college president has been accused of plagiarising his Ph.D. thesis, here. It consists of nothing more sophisticated than a yellow highlighter and a grid of thumbnail images of pages, but I think it conveys the message very effectively.

Fixed radius near neighbors

2009-05-31T04:14:40Z

Someone recently added this new Wikipedia article on the fixed-radius near-neighbor search problem. I cleaned it up a little (it obviously needs a lot more work); in the process of the cleanup I changed the definition of the problem to better match the one in the Bentley reference. But before I got to it, the definition was that one should report all pairs with distance at most Δ in a given point set.

It occurred to me that the all-pairs fixed radius problem of the original version of the article has a simple solution with time linear in the input and output size (for any fixed dimension, assuming constant time integer truncation and hash table operations): use a hash table to group the points into cubical buckets with side length Δ, and for each point look at all potential neighbors in adjacent buckets. If some bucket contains many points, you might spend a lot of time examining pairs of points that are too far apart, but in that case there are many nearby close pairs against which the time can be charged: both the time and the number of reported pairs are proportional to the sum of the square of the bucket sizes.

This must have been known, probably in the 1960s or 1970s; it's a close relative of several linear-time randomized bucketing closest-pair algorithms, but even simpler. Bentley's 1975 survey mentions this sort of approach to nearest neighbor problems under the heading of "cell techniques", but without the hashing, without any analysis, and with a claim that these techniques require uniformly distributed points. Anyone know a reference for the linear time analysis?

In contrast, by the way, there can be no linear-time bucketing algorithm for finding the nearest neighbor of every point unless one is using a model of computation that can sort in linear time. The reason is that, if one can find all nearest neighbors, one can use something like Borůvka's algorithm to sort in one dimension: the nearest neighbor graph forms a set of non-overlapping paths, and one may replace each path by a single representative point and continue recursively.

New Journal of Computational Geometry

2009-05-30T01:07:57Z

This looks very promising, as does this.

*starts thinking about which of my computational geometry papers are in need of journal versions*

Congratulations, Dr. Wortman!

2009-05-29T22:19:51Z

My Ph.D. student Kevin Wortman passed his thesis defense this morning, and is now (or will be when they hold the graduation ceremony in a week) Dr. Wortman.

Kevin (not to be confused with my other recent co-authors Kevin and Kevin, nor with the Utah mathematician Kevin Wortman) has been working with me since 2005, when we had a paper in SoCG on minimum dilation stars, or, less formally, the problem of selecting an airline hub that minimizes the maximum ratio between the route length between two cities through the hub and the straight-line distance. Since then, we've written more papers about fast approximations to the minimum dilation star problem, and minimum dilation stars for metric spaces (to appear at WADS), both of which became parts of Kevin's thesis. The minimum dilation star has led to the definition of a new triangle center, and an interpretation of dilation as a smoothed distance function is a key component of another paper with Kevin on generalized Voronoi diagrams. Kevin's graph drawing algorithms also formed the basis of the somewhat monstrous graph drawing below, a more legible version of the drawings in this post that I needed for my recent work on squaregraphs.

Kevin has a job offer from California State University, Fullerton (also in Orange County, but with heavier teaching loads and less of the heavy emphasis on research that the University of California has) and will start there in August. Landing a job like that in the current economic climate is not easy (we just received word of a hard hiring freeze on our own campus) but well deserved.

Congratulations, Kevin!

Squaregraphs

2009-05-29T06:44:00Z

I have another preprint on the arXiv: “Combinatorics and geometry of finite and infinite squaregraphs” (arXiv:0905.4537), with Hans-Jürgen Bandelt and Victor Chepoi. It's a long one, 46 pages, and as the title says is about the properties of squaregraphs, planar graphs in which all the interior faces are quadrilaterals and all the interior vertices have degree at least four.

About a year ago I made several postings here about chord diagrams, the patterns formed by multiple chords in a circle (or, if you like, simple line arrangements in the hyperbolic plane). Now I can explain why: it's because of this paper. As I described here, if one has a hyperbolic line arrangement with no three mutually intersecting lines (a chord diagram with no ...a...b...c...a...b...c... pattern), its planar dual is a squaregraph, and all finite squaregraphs can be constructed in this way.

Coloring the chords of a chord diagram (so that no two chords of the same color cross each other) corresponds to embedding a squaregraph into a Cartesian product of trees. It's possible to give the vertices of a tree constant-sized labels in such a way that one can in constant time determine the distance of two vertices by looking at their labels, and this coloring idea allows the same distance labeling techniques to be applied to quadtrees, but with a label size that depends on the number of trees in the Cartesian product, or equivalently on the number of colors in the coloring. The squaregraph dual to Ageev's 5-chromatic chord diagram shows that this sort of labeling scheme sometimes needs as many as five trees, but never more than that.

Here's another curious fact about squaregraphs that we needed as a lemma, and is vaguely related to my recent post about visualizing breadth-first search: if one performs a BFS on a squaregraph, starting at any vertex, it's always possible to connect each level of the BFS tree into a cycle, and draw these cycles concentrically around the starting vertex, in such a way that the cycle edges don't cross the squaregraph edges. That is, in graph drawing terminology, the cycles and the squaregraph have a simultaneous embedding.

Exercise: find a planar bipartite graph G, and a starting vertex v, such that the levels of the BFS tree rooted at v cannot be connected into cycles and simultaneously embedded with G.

There's also lots more in the paper about forbidden subgraph characterizations of squaregraphs, algorithms for finding small sets of vertices that generate the whole graph by median operations, infinite squaregraphs and their duality relation with infinite hyperbolic line arrangements (a little messier than we had hoped: duals of hyperbolic line arrangements can be disconnected, and connected duals of hyperbolic line arrangements aren't general enough; but a general class of infinite squaregraphs is formed by the connected components of duals of infinite triangle-free hyperbolic line arrangements), etc.

Topological Bentley–Ottman?

2009-05-27T04:13:39Z

I just wrote a major expansion of the Wikipedia article on the Bentley–Ottmann algorithm, the plane sweep algorithm for listing all crossings among a set of line segments. As usual, feedback would be very welcome; I already know that it could still use illustrations, and the article on the closely related Shamos–Hoey algorithm for detecting a single crossing is completely missing.

One surprise for me as I was researching this: Chazelle and Edelsbrunner, in their 1992 JACM paper on deterministic O(n log n + k) algorithms for constructing arrangements of line segments, leave as an open problem whether it is possible to list all crossings deterministically in the same O(n log n + k) time bound as their algorithm but with the O(n) space bound of Bentley–Ottmann, rather than the larger O(n + k) required for storing the whole arrangement in memory. I had thought that this was one of the problems solved by topological sweeping, but if so I don't know what the right reference is: the original Edelsbrunner–Guibas topological sweeping paper appears to be only for line arrangements, and the Asano–Guibas–Tokuyama SoCG '91 paper extends the topological sweeping method to the intersection of a line arrangement with a convex region of the plane, but not to arbitrary line segment arrangements. Can this really still be open?

ETA: Seems not to be open; see comments.

Graph Drawing deadline approaching

2009-05-26T18:55:23Z

A reminder: the Graph Drawing Symposium submission deadline is approaching. Get your graph drawing submissions ready!

We recently changed the submission deadline, from May 31 to June 3, to articulate better with some other conference deadlines. Unlike previous years' extensions we've tried to make this change sufficiently early that it will also be useful for providing extra time to write new submissions.

Acute-dihedral triangulation of the cube

2009-05-25T04:44:29Z

VanderZee et al have solved an open problem in mesh generation from one of my earlier papers (with Sullivan and Üngör): is there a partition of a cube into tetrahedra, meeting triangle-to-triangle and edge-to-edge, with the property that ell the dihedral angles of every tetrahedron are acute? In our earlier work, we were only able to show the existence of acute triangulations of this type for infinite slabs, but the new paper shows that they do exist for cubes. The triangulation they find is well-behaved in other ways as well: the angles are bounded well away from right angles, and every tetrahedron contains its circumcenter.

This work raises the possibility that every polyhedron has an acute triangulation, but that remains an open problem still.

Wolfram Alpha: no competition for Google

2009-05-19T23:32:26Z

This afternoon there was a minor aftershock of Sunday's LA earthquake, and I noticed that the time on the USGS earthquake report was pretty much the same as when I felt it several tens of miles away. So I became curious: how fast do earthquakes travel, anyway?

This seemed like a good test case for Wolfram Alpha, a fancy new search engine from the makers of Mathematica, so I thought I'd try it. But "earthquake speed" as a query led it to think I was asking about two action movies with those titles, "how fast does an earthquake travel?" just confused it, and substituting "seismic wave" for earthquake at least avoided the movie misinterpretation but didn't get me any better results. "Seismic wave" alone as a query gave the results "Seismology: Additional functionality for this topic is under development... Leave your email address to be notified when it is ready." And I still didn't find out what I had set out to.

Finally, I gave up, went to the Google search box in my browser's toolbar, and typed the same query I'd started with, "earthquake speed". The first hit was exactly on-topic, and the snippet Google displayed with the hit showed that it was on-topic. The answer: it varies, but for the roughly 56 km between the epicenter and my location it should have taken between 4 seconds (for the fastest P waves) and 16 seconds (for the slowest S waves).

Combined with Wolfram Alpha's Babylonian arithmetic fiasco, this does not fill me with a lot of confidence in their service...

Short conclusion: I still feel lucky.

I am not a number!

2009-05-17T21:11:56Z

Apparently any email containing a URL to this journal (http://11011110.livejournal.com), even as plain text, is being interpreted as "URL Obfuscation" by the AT&T Research email servers, and permanently blocked. I haven't done enough experiments to tell whether they don't like any URLs, or just not mine, but my guess is that it has something to do with the purely-numeric domain name component.

Visualizing BFS as a spiral

2009-05-17T01:04:30Z

It's a pretty obvious observation that the graphic conventions we use when illustrating mathematical objects can have a lot to do with how we think of them. The standard way the computer science textbooks draw trees is like a biological tree, but upside-down, with the root at the top and the leaves at the bottom:

When we view a tree in this way, the traversal order given by breadth-first search forms a geometric pattern that scans left-to-right across the vertices in a single level of the tree (conventionally, these vertices lie along a horizontal line), then the next level, and so on, much as one reads English text left-to-right and line-by-line. And this level-by-level ordering can be useful for conveying the idea that breadth-first search sorts vertices by their distances from the root.

But the breadth-first search algorithm itself, as it is usually described, does not progress level-by-level. Rather, it maintains a queue of open vertices that, in general, spans more than one level, and removes vertices from the tail of the queue and adds their unqueued neighbors to the head of the queue without regard for the boundaries between levels of the tree. The same properties, of performing the same steps at every vertex of a tree regardless of whether the vertex starts or ends a new level, and of not actually having any data that represents the boundaries between levels, is also true of an alternative BFS algorithm based on self-recursive iterators that I described a few years ago (this alternative algorithm typically uses much less space, but a constant factor more time, than a conventional BFS). The transition that these algorithms make from one level to the next is smooth, without the big discontinuities that the standard level-by-level visualization might make one think of.

Last week, while working on the Wikipedia article on the Calkin–Wilf tree, I wanted to find a visual way of representing this smoothness. The Calkin–Wilf tree is a binary tree with 1/1 at the root, and where the children of a/b are a/(a + b) and (a + b)/b; it contains every positive rational number exactly once, so one can list the positive rationals without duplication by performing a breadth-first search of this tree. And in fact this breadth-first search can be done by a simple formula, without any need for queues or recursion: the next number after q is 1/(2 floor(q) + 1 − q). Notice, again, the lack of jumps in this sequence: the formula works the same no matter whether the next number belongs to the same level or is the start of another level in the tree. Eventually, I realized that if the tree were drawn radially, the BFS sequence could be visualized as a nice smooth spiral:

The same spiral layout can be used for any tree, and provides a nice graphical counterpart to the lack of jumps in the BFS algorithm. One can extend the metaphor: the head and tail of the BFS queue lie close together on two parallel tracks, and the BFS algorithm moves both of them in tandem while adding neighbors of the tail (on the inner track) to the head (on the outer track).

Some technical detail about the drawing: if one wants to preserve vertex spacing along the spiral, as the number of nodes per level grows exponentially, one should use a logarithmic spiral, but this will cause the lengths of the tree edges to grow quickly. Instead, I used an approximation to an Archimedean spiral formed by two sets of concentric semicircles with even and odd integer radii; the odd-radius semicircles lie in the bottom half of the drawing and the even radii lie in the top half. For the tree drawing, I used a style described (for concentric circles rather than spirals) in a paper I wrote with chouyu_31 in which one fixes the angles of the edges and then extends them to whatever length will make the vertices lie on the correct arm of the spiral. The angles were (as in that paper) chosen to maximize the sharpest angles of the drawing, while not allowing any two subtrees to turn back towards each other, but if I were to redraw this I might choose different angles that lead to a more uniform vertex placement.

A confluent drawing

2009-05-13T22:34:45Z

In the revisions of one of my papers, I needed an illustration of the graph shown below, which comes from an NP-completeness reduction. This is a confluent drawing: two vertices are connected by an edge whenever there is a smooth path between them. (The paths are allowed to self-intersect, so for instance all 30 outer vertices form a clique.)

I think it's a great illustration of the power of confluent graph drawing: the graph has 37 vertices, 536 edges, and an average degree of nearly 29, but despite being so dense the graph has a lot of structure and the drawing makes the structure apparent.

Carnival

2009-05-08T06:50:35Z

The 52nd Carnival of Mathematics is up and it's a good one.

Elsevier

2009-05-07T23:12:39Z

Given the revelation that Elsevier published not just one, but six fake journals as advertising for drug companies, the El Naschie affair, the history of gun-running, and the general issue of commercial journal publishers profiting from academic research and locking away the results from the public to nobody's benefit but their own, I think it's time to take a public position that I will neither submit my own papers to Elsevier journals nor referee for them.

The two journals for which this decision would most affect me are (as Jeff already noted) CGTA, and the European Journal on Combinatorics. CGTA has been a very friendly place to publish computational geometry papers, and for now there are no non-commercial alternatives devoted to computational geometry. And EuJC is the main outlet for research in the cubical graph theory I've been working on lately. However, I'd rather have a clear conscience about how I publish my work than continue being part of the problem.

I do currently have one paper in submission to a different Elsevier journal (as well as one still to appear in CGTA); I've considered withdrawing the submitted paper over this issue, but it's already gone through a round of review and I don't want to feel like I've wasted some other academic's time over this issue. So those will stay (although I welcome feedback telling me to do otherwise) but no more.

ETA: More on Elsevier's fake-journal unit and on not publishing with Elsevier.

ETA2: John Baez does some price comparisons.

The metric space of star metric spaces

2009-05-05T01:20:15Z

Suppose that ƒ is a function that maps a set X to the positive real numbers. Then we can define a metric space from ƒ by defining the distance from x to y to be ƒ(x)+ƒ(y) (with d(x,x)=0 in exception to this formula); this automatically satisfies the required symmetry, positivity, and triangle inequality properties. The resulting metric space can be represented as the shortest-path distances in a star (tree with one non-leaf node h, its "hub"), by placing X at the leaves of the star and setting the length of the edge from leaf x to hub h to be ƒ(x). We can include h itself in the metric space, setting ƒ(h)=0. If we do include the hub, the spaces coming from this construction can then be characterized by the existence of a point h that belongs to a shortest path between any other two distinct points: for all x and y with x ≠ y, d(x,y)=d(x,h)+d(h,y). The function ƒ can be recovered as ƒ(x) = d(x,h).

My latest arXiv preprint, Optimal Embedding Into Star Metrics (with Kevin Wortman, arXiv:0905.0283, to appear at WADS) looks at the problem of embedding an arbitrary metric into a star metric of this type in order to minimize the amount by which distances are distorted in the embedding. We may as well scale the target star metric so that all of its distances are greater than the distances in the input metric, so the problem becomes one of, given a metric space (X,d), finding a function ƒ on X that defines a star metric with distances at least equal to d, minimizing the dilation max_x,y ƒ(x)+ƒ(y)/d(x,y). As we show, the problem can be solved polynomially by transforming it into a parametric shortest path problem in an auxiliary graph defined from X.

But what is this space of star metrics into which (X,d) can be embedded? The requirement that distances be non-decreasing translates into a property that, for every x and y, ƒ(x)+ƒ(y) ≥ d(x,y), but we don't want the distances in ƒ to be unnecessarily large, so we should restrict our attention to minimal functions, with the property that each x has a y for which the inequality above turns into an equality. If no such y existed, we could reduce ƒ(x) leaving all the other values unchanged and get a better overall star. That is, for each x there should exist y such that ƒ(x)+ƒ(y) = d(x,y). (This formula works directly only when X is finite; a similar formula involving suprema is needed when X may be infinite.)

As it turns out, if T_X is the family of minimal star metrics into which (X,d) can be embedded, then T_X itself can be given the structure of a metric space, and (X,d) can be embedded into T_X with zero distortion. Given two functions ƒ and g in T_X, define the distance between them to be the sup-norm or L_∞ norm d(ƒ,g) = sup_x|ƒ(x) − g(x)|. Each point x of X defines a star with x as the hub, defined by the function ƒ_x(y) = d(x,y), and the sup-norm distance d(ƒ_x,ƒ_y) equals the original distance d(x,y). This space T_X of minimal star metrics itself has other nice properties (it contains a line segment connecting any two points, and any pairwise-intersecting collection of metric balls has a common intersection) that can be summarized by saying that it is injective or hyperconvex. Every metric space (X,d) can be embedded without distortion into an injective space, and the space of minimal star metrics is the smallest injective space into which (X,d) can be embedded, its tight span.

Therefore, mapping (X,d) in a distance-nondecreasing way into a star metric can be viewed as picking a single point that represents the point cloud X within a larger metric space, the tight span T_X. Kevin and I previously studied the corresponding problem of picking a single star center to minimize dilation in bounded-dimension Euclidean spaces in an earlier paper; the new paper solves the same problem for injective spaces of unbounded dimension. But dilation is not the only way to measure the quality of a point representing a point cloud. Finding the star metric for X that minimizes the maximum distance between any two points (the 1-center or circumradius problem in a tight span) turns out not to be very interesting: the solution is half the diameter of X (the largest number in its distance matrix). But finding a star metric for X that minimizes the average distance (the 1-median or Fermat–Weber problem in a tight span) is less trivial. It appears to be the case that: (1) the optimal solution is determined by the maximum-weight 2-factor for X (that is, the maximum weight collection of disjoint cycles through the points in X, allowing 2-cycles whose weight is twice the distance between the endpoints); (2) the maximum 2-factor has the form of a collection of odd cycles and 2-cycles; (3) within each odd cycle, the distance from a vertex x to the hub is half of an alternating sum of the edge lengths in the cycle, in which the two edges incident to x are both added and otherwise edges of the cycle alternate between being added and subtracted; and (4) there may be some freedom to choose the distances between endpoints of a 2-factor and the hub, as long as the sum of the two distances equals the distance between the endpoints. If all this is true, the star embedding for X that minimizes average distance can also be found in polynomial time. (If it's not true, the optimal embedding can still be found in polynomial time by linear programming, but it's good to have a more combinatorial algorithm.)

One may also be wondering, at this point, are star metrics injective, and if not, what is the tight span of a star metric? The first answer is no: nontrivial injective spaces must have infinitely many points and star metrics may be finite. But even an infinite star metric can never be injective, because in a star metric the only point that lies on a shortest path between any two other points is the hub whereas in an injective metric space any two points have a whole shortest path of points between them. The tight span of a star metric is a hedgehog space, the space formed by connecting each point of the star to the hub by a line segment of the appropriate length.

PS: due to continued crankiness from martinmusatov I've set all comments from non-LJ-friends to be screened. This means you can all still comment (and, except for MM, you are all encouraged to do so) but your comment will not be visible to others until I have checked it.

Between the Folds

2009-05-01T04:19:56Z

I just came back from seeing Between the Folds, a beautiful feature-length documentary film on the art and science of origami, at the Newport Beach Film Festival, featuring star turns by, among others, tomster0 and Erik and Marty Demaine. My kids were laughing at loud at Paul Jackson's description of three-legged elephants (part of a rant about realism in art) and my wife was nodding in recognition at Erik's description of how excited one gets when one first solves a hard problem and how likely one's initial solution is to be wrong. I guess it's still on the film festival tour, so you'll have to figure out where it's playing next or wait to see it on DVD, but I highly recommend it.