[Paper Review] A Fast Network-Decomposition Algorithm and its Applications to Constant-Time Distributed Computation

Leonid Barenboim, Michael Elkin+1 more|arXiv (Cornell University)|May 21, 2015

Complexity and Algorithms in Graphs41 references4 citations

TL;DR

This paper presents a randomized distributed algorithm that computes strong $(O(1), O(n^{ heta}))$-network decompositions in constant time for any constant $ heta > 0$, significantly improving prior constant-time results with a much better color count ($O(n^{ heta})$ vs. $O(n^{1/2+ heta})$). The method leverages iterative clustering and random coloring to achieve constant-time computation while maintaining low cluster diameter and color count, enabling improved approximation algorithms for fundamental problems like minimum coloring and dominating set.

ABSTRACT

A partition $(C_1,C_2,...,C_q)$ of $G = (V,E)$ into clusters of strong (respectively, weak) diameter $d$, such that the supergraph obtained by contracting each $C_i$ is $\ell$-colorable is called a strong (resp., weak) $(d, \ell)$-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong $(exp\{O(\sqrt{ \log n \log \log n})\}, exp\{O(\sqrt{ \log n \log \log n})\})$-network-decompositions can be computed in distributed deterministic time $exp\{O(\sqrt{ \log n \log \log n})\}$. The result of Awerbuch et al. was improved by Panconesi and Srinivasan in 1992: in the latter result $d = \ell = exp\{O(\sqrt{\log n})\}$, and the running time is $exp\{O(\sqrt{\log n})\}$ as well. Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with $d = O(1)$. However, the parameter $\ell$ in his result is $O(n^{1/2 + ε})$. In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong $(O(1), O(n^ε))$-network-decompositions. As a corollary we derive a constant-time randomized $O(n^ε)$-approximation algorithm for the distributed minimum coloring problem, improving the previously best-known $O(n^{1/2 + ε})$ 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).

Motivation & Objective

• To develop a constant-time distributed algorithm for strong network decompositions with significantly improved color count compared to prior constant-time approaches.
• To address the open problem of designing a deterministic polylogarithmic-time approximation algorithm for the minimum dominating set problem in distributed networks.
• To improve the approximation ratio for the distributed minimum coloring problem beyond the previously best-known $O(n^{1/2+ heta})$ guarantee.
• To enable efficient constant-time solutions for a range of fundamental distributed problems using the new network decomposition framework.

Proposed method

• The algorithm constructs clusters iteratively using a randomized coloring scheme that ensures low-diameter clusters and bounded number of colors.
• It employs a hierarchical clustering process where vertices are grouped into clusters of constant strong diameter through multiple rounds of local computation and message passing.
• The method uses a novel random partitioning strategy to assign vertices to clusters such that the supergraph of contracted clusters is $O(n^{ heta})$-colorable.
• It applies a recursive decomposition technique with a parameter $ heta$ to control the trade-off between cluster diameter and number of colors, achieving constant running time.
• The algorithm leverages the fact that each cluster's neighborhood can be processed independently, allowing parallel execution and constant-time convergence.
• It integrates centralized polynomial-time approximation algorithms (e.g., for set cover) to compute local dominating sets within clusters, ensuring bounded approximation ratios.

Experimental results

Research questions

• RQ1 Can a strong network decomposition with constant cluster diameter and $O(n^{ heta})$ colors be computed in constant distributed time for any constant $\theta > 0$?
• RQ2 Can this improved network decomposition lead to better approximation algorithms for the distributed minimum coloring problem?
• RQ3 Can the new decomposition technique be used to design a deterministic polylogarithmic-time approximation algorithm for the minimum dominating set problem?
• RQ4 Does the framework enable constant-time approximation for the $t$-spanner problem with improved approximation ratios?

Key findings

• The paper presents a randomized distributed algorithm that computes strong $(O(1), O(n^{ heta}))$-network decompositions in $\exp\{O(1/\theta)\}$ time, achieving constant running time for any constant $\theta > 0$.
• This improves upon prior constant-time algorithms that required $O(n^{1/2+\theta})$ colors, reducing the color count to $O(n^{\theta})$ for any $\theta > 0$.
• As a corollary, the paper obtains a constant-time randomized $O(n^{\theta})$-approximation algorithm for the distributed minimum coloring problem, improving the previous $O(n^{1/2+\theta})$ guarantee.
• The framework enables a deterministic polylogarithmic-time approximation algorithm for the minimum dominating set problem, resolving an open problem posed by Lenzen and Wattenhofer (2010).
• The method yields a $O(n^{1/k})$-approximation for the $t$-spanner problem in $\exp\{O(k)\} + O(t)$ randomized time, with polynomial local computation.
• The algorithm uses centralized $O(\log \Delta)$-approximation techniques for local subproblems (e.g., dominating sets in clusters), ensuring bounded approximation ratios without increasing the overall running time.