2004/10/26 by Elchanan Mossel, Ryan O’Donnell, Mossel, Elchanan +9 · 2 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Optimization and Search Problems #Probability (math.PR) #math.CO #math.PR
paper · pdf · doi:10.48550/arxiv.math/0410560
26 pages
arxiv created 2004/10/26 · openalex publication_date 2004/10/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study non-interactive correlation distillation (NICD), a generalization of the study of noise sensitivity of boolean functions. We extend the model to NICD 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: 1. In the case of a k-leaf star graph, we resolve the open question of whether the success probability must go to zero as k goes to infinity. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial). 2. In the case of the k-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function. 3. In the general case we show that all players should use monotone functions. 4. For certain trees it is better if not all players use the same function. Our techniques include the use of the reverse Bonami-Beckner inequality.