Leonid Barenboim

Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic and Faulty Networks

In the distributed message passing model a communication network is represented by an n-vertex graph G = (V,E) of maximum degree Δ. Computation proceeds in discrete synchronous rounds consisting of sending and receiving messages and performing local computations. The running time of an algorithm is the number of rounds it requires. In the static setting the network remains unchanged throughout the entire execution. In the dynamic setting the topology of the network changes, and a new solution has to be computed after each change. In the faulty setting the network is static, but some vertices or edges may lose the computed solution as a result of faults. The goal of an algorithm in this setting is fixing the solution. The problems of (Δ + 1)-vertex-coloring and (2Δ - 1)-edge-coloring are among the most important and intensively studied problems in distributed computing. Despite a very intensive research in the last 30 years, no deterministic algorithms for these problems with sublinear (in Δ) time have been known so far. Moreover, for more restricted scenarios and some related problems there are lower bounds of Ω(Δ) [13, 14, 20, 27]. The question of the possibility to devise algorithms that overcome this challenging barrier is one of the most fundamental questions in distributed symmetry breaking [4, 6, 13, 14, 19, 24]. In this paper we settle this question for (Δ + 1)-vertex-coloring and (2Δ - 1)-edge-coloring by devising deterministic algorithms that require O(Δ3/4 log Δ + log* n) time in the static, dynamic and faulty settings. (The term log* n is unavoidable in view of the lower bound of Linial [21]. Moreover, for (1 + o(1))Δ-vertex-coloring and (2 + o(1))Δ-edge-coloring we devise algorithms with Õ(√Δ + log* n) deterministic time. This is roughly a quadratic improvement comparing to the state-of-the-art that requires O(Δ + log* n) time [4, 19, 24]. Our results are actually more general than that since they apply also to a variant of the list-coloring problem that generalizes ordinary coloring. Our results are obtained using a novel technique for coloring partially-colored graphs (also known as fixing). We partition the uncolored parts into a small number of subgraphs with certain helpful properties. Then we color these subgraphs gradually using a technique that employs constructions of polynomials in a novel way. Our construction is inspired by the algorithm of Linial [21] for ordinary O(Δ2)-coloring. However, it is a more sophisticated construction that differs from [21] in several important respects. These new insights in using systems of polynomials allow us to significantly speed up the O(Δ)-coloring algorithms. Moreover, they allow us to devise algorithms with the same running time also in the more complicated settings of dynamic and faulty networks.

A Fast Network-Decomposition Algorithm and Its Applications to Constant-Time Distributed Computation

A partition C1,C2,...,Cq of G=V,E into clusters of strong respectively, weak diameter d, such that the supergraph obtained by contracting each Ci is l-colorable is called a strong resp., weak d, l-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong

Nearly Optimal Local Broadcasting in the SINR Model with Feedback

exp\{O\sqrt{ \log n \log \log n}\}

Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic, and Faulty Networks

,

A fast network-decomposition algorithm and its applications to constant-time distributed computation

exp\{O\sqrt{ \log n \log \log n}\}

Deterministic Distributed (Delta + o(Delta))-Edge-Coloring, and Vertex-Coloring of Graphs with Bounded Diversity

-network-decompositions can be computed in distributed deterministic time

Fully-Dynamic Graph Algorithms with Sublinear Time Inspired by Distributed Computing

exp\{O\sqrt{ \log n \log \log n}\}

Distributed Symmetry-Breaking Algorithms for Congested Cliques

. Even more importantly, they demonstrated that network-decompositions can be used for a great variety of applications in the message-passing model of distributed computing. Much more recently Barenboim 2012 devised a distributed randomized constant-time algorithm for computing strong network decompositions with d=O1. However, the parameter l in his result is On1/2+e. In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong O1, One-network-decompositions. As a corollary we derive a constant-time randomized One-approximation algorithm for the distributed minimum coloring problem. This improves the best previously-known On1/2+e approximation guarantee. We also derive other improved distributed algorithms for a variety of problems. Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic -time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer 2010.

Load-Balancing Adaptive Clustering Refinement Algorithm for Wireless Sensor Network Clusters

We consider the SINR wireless model with uniform power. In this model the success of a transmission is determined by the ratio between the strength of the transmission signal and the noise produced by other transmitting processors plus ambient noise. The local broadcasting problem is a fundamental problem in this setting. Its goal is producing a schedule in which each processor successfully transmits a message to all its neighbors. This problem has been studied in various variants of the setting, where the best currently-known algorithm has running time

Brief Announcement: Distributed Symmetry-Breaking with Improved Vertex-Averaged Complexity

O\bar{\Delta} + \log^2 n