Benjamin Sudakov

Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality

AbstractIn this paper we studynon-interactive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating.Our new contributions include the following:• In the case of ak-leaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero ask » ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial).• In the case of thek-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function.• In the general case we show that all players should use monotone functions. We also show, somewhat surprisingly, that for certain trees it is better if not all players use the same function. Our techniques include the use of thereverse Bonami-Beckner inequality. Although the usual Bonami-Beckner has been frequently used before, its reverse counterpart seems not to be well known. To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds [2, 3, 6]. On the probabilistic side, we use the “reflection principle” and the FKG and related inequalities in order to study the problem on general trees.

Genome reassembly with high-throughput sequencing data

MotivationRecent studies in genomics have highlighted the significance of structural variation in determining individual variation. Current methods for identifying structural variation, however, are predominantly focused on either assembling whole genomes from scratch, or identifying the relatively small changes between a genome and a reference sequence. While significant progress has been made in recent years on both de novo assembly and resequencing (read mapping) methods, few attempts have been made to bridge the gap between them.ResultsIn this paper, we present a computational method for incorporating a reference sequence into an assembly algorithm. We propose a novel graph construction that builds upon the well-known de Bruijn graph to incorporate the reference, and describe a simple algorithm, based on iterative message passing, which uses this information to significantly improve assembly results. We validate our method by applying it to a series of 5 Mb simulation genomes derived from both mammalian and bacterial references. The results of applying our method to this simulation data are presented along with a discussion of the benefits and drawbacks of this technique.

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, and let Ck denote the set of cycles Cℓ, where 3≤ℓ≤k and ℓ and k have the same parity. Erdős and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ Ck) ∼ z(n,F) — here we write f(n) ∼ g(n) for functions f,g: ℕ → ℝ if limn→∞f(n)/g(n)=1. They proved this when F ={C4} by showing that ex(n,{C4;C5})∼z(n,C4). In this paper, we extend this result by showing that if ℓ∈{2,3,5} and k>2ℓ is odd, then ex(n,C2ℓ∪{Ck}) ∼ z(n,C2ℓ). Furthermore, if k>2ℓ+2 is odd, then for infinitely many n we show that the extremal C2ℓ∪{Ck}-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2ℓ, and furthermore the asymptotic result does not hold when (ℓ,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.