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Register a new userDissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling
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Eric Fusy
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We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees, with interesting consequences for enumeration, mesh compression and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best known uniform random sampler for labelled planar graphs [{bf Fusy, Analysis of Algorithms 2005}]. It also provides an encoding for the set $mathcal{P}(n)$ of $n$-edge 3-connected planar graphs that matches the entropy bound $frac1nlog_2|mathcal{P}(n)|=2+o(1)$ bits per edge (bpe). This solves a theoretical problem recently raised in mesh compression, as these graphs abstract the combinatorial part of meshes with spherical topology. We also achieve the {optimal parametric rate} $frac1nlog_2|mathcal{P}(n,i,j)|$ bpe for graphs of $mathcal{P}(n)$ with $i$ vertices and $j$ faces, matching in particular the optimal rate for triangulations. Our encoding relies on a linear time algorithm to compute an orientation associated to the minimal Schnyder wood of a 3-connected planar map. This algorithm is of independent interest, and it is for instance a key ingredient in a recent straight line drawing algorithm for 3-connected planar graphs [bf Bonichon et al., Graph Drawing 2005].
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For a collection of papers in memory of Elwyn Berlekamp (1940-2019), John Conway (1937-2020), and Richard Guy (1916-2020). The Sprague-Grundy theory for finite games without cycles was extended to general finite games by Cedric Smith and by Aviezri Fraenkel and coauthors. We observe that the same framework used to classify finite games also covers the case of locally finite games (that is, games where any position has only finitely many options). In particular, any locally finite game is equivalent to some finite game. We then study cases where the directed graph of a game is chosen randomly, and is given by the tree of a Galton-Watson branching process. Natural families of offspring distributions display a surprisingly wide range of behaviour. The setting shows a nice interplay between ideas from combinatorial game theory and ideas from probability.
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We study a natural question about sparse random matrices which arises from an application in distributed computing: what is the distance from a fixed vector to the column span of a sparse random matrix $A in R^{n times m}$? We answer this question for several ensembles of sparse random matrices in which the average number of non-zero entries per column, $d$, is smaller than $log(n)$. Key to our analysis is a new characterization of linear dependencies in sparse random matrices. We show that with high probability, in certain random matrices, including rectangular matrices with i.i.d.~Bernoulli entries and $m geq (1 + epsilon)n$, and symmetric random matrices with Bernoulli entries, any linear dependency must be caused by one of three specific combinatorial structures. We show applications of our result to analyzing and designing em gradient codesem, replication schemes used in distributed machine learning to mitigate the effect of slow machines, called em stragglersem. We give the first known construction for a gradient code that achieves near-optimal error for both random and adversarial choices of stragglers.
Efficient automatic protein classification is of central importance in genomic annotation. As an independent way to check the reliability of the classification, we propose a statistical approach to test if two sets of protein domain sequences coming from two families of the Pfam database are significantly different. We model protein sequences as realizations of Variable Length Markov Chains (VLMC) and we use the context trees as a signature of each protein family. Our approach is based on a Kolmogorov--Smirnov-type goodness-of-fit test proposed by Balding et al. [Limit theorems for sequences of random trees (2008), DOI: 10.1007/s11749-008-0092-z]. The test statistic is a supremum over the space of trees of a function of the two samples; its computation grows, in principle, exponentially fast with the maximal number of nodes of the potential trees. We show how to transform this problem into a max-flow over a related graph which can be solved using a Ford--Fulkerson algorithm in polynomial time on that number. We apply the test to 10 randomly chosen protein domain families from the seed of Pfam-A database (high quality, manually curated families). The test shows that the distributions of context trees coming from different families are significantly different. We emphasize that this is a novel mathematical approach to validate the automatic clustering of sequences in any context. We also study the performance of the test via simulations on Galton--Watson related processes.
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