A fast distributed approximation algorithm for minimum spanning trees

Abstract

We present a distributed algorithm that constructs an O(log n)-approximate minimum spanning tree (MST) in any arbitrary network. This algorithm runs in timeÕ(D(G)+ L(G, w)) where L(G, w) is a parameter called the local shortest path diameter and D(G) is the (unweighted) diameter of the graph. Our algorithm is existentially optimal (up to polylogarithmic factors), i.e., there exist graphs which need Ω(D(G) + L(G, w)) time to compute an H -approximation to the MST for any H ∈ [1, Θ(log n)]. Our result also shows that there can be a significant time gap between exact and approximate MST computation: there exists graphs in which the running time of our approximation algorithm is exponentially faster than the time-optimal distributed algorithm that computes the MST. Finally, we show that our algorithm can be used to find an approximate MST in wireless networks and in random weighted networks in almost optimalÕ(D(G)) time.

Efficient distributed approximation algorithms via probabilistic tree embeddings

“…The parameter S can be shown to capture the hardness of distributed approximation quite precisely. Using the hardness results of Elkin [15], one can show that there exists a family of n-node graphs where (S) time is needed by any distributed approximation algorithm to approximate the MST within an H -factor, for any H ∈ [1, O(log n)] [20]. Since the MST problem is a special case of the GSF problem, the above bound also applies to the GSF problem.…”

“…The rest of the results hold WHP Definition 1 (Shortest Path Diameter (SPD) [20]). The SPD is denoted by S(G, w) (or S for short) and defined as S = max u,v∈V l(u, v).…”

“…The existential optimality of our algorithm is with respect to S instead of n as in the case of Awerbuch's distributed MST algorithm [2], for example. S can be much smaller than √ n; e.g., in networks where edge weights are chosen uniformly and independently from an arbitrary distribution, [20]. Note that˜ ( √ n) is a lower bound on the time needed to compute exact MST or Steiner tree [20,30].…”

“…S can be much smaller than √ n; e.g., in networks where edge weights are chosen uniformly and independently from an arbitrary distribution, [20]. Note that˜ ( √ n) is a lower bound on the time needed to compute exact MST or Steiner tree [20,30]. 2.…”

“…For this reason, in the distributed context, such algorithms are motivated even for network optimization problems that are not NP-hard, e.g., minimum spanning tree, shortest paths etc. There is a large body of work on distributed approximation algorithms for classical graph optimization problems such as minimum spanning tree, shortest path, minimum edgecoloring, minimum dominating set, minimum vertex cover, maximum matching, and positive linear programs [5,12,15,[18][19][20][21][22]25,27,33]. We refer to the surveys by Elkin [13] and Dubhashi et al [11] that summarize many results regarding efficient distributed approximation algorithms and the hardness of distributed approximation for various classical optimization problems.…”