Distributed Approximation Algorithms for Steiner Tree in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$

Abstract

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization.In this paper we study this problem in the CONGEST ED CLIQUE model of distributed computing and present two deterministic distributed approximation algorithms for the same. The first algorithm computes a Steiner tree in Õ(n 1/3 ) rounds and Õ(n 7/3 ) messages for a given connected undirected weighted graph of n nodes. Note here that Õ(•) notation hides polylogarithmic factors in n. The second one computes a Steiner tree in O(S + log log n) rounds and O(S (n − t) 2 + n 2 ) messages, where S and t are the shortest path diameter and the number of terminal nodes respectively in the given input graph. Both the algorithms admit an approximation factor of 2(1 − 1/ ), where is the number of terminal leaf nodes in the optimal Steiner tree. For graphs with S = ω(n 1/3 log n), the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with S = õ(n 1/3 ), the second algorithm outperforms the first one in terms of the round complexity. In fact when S = O(log log n) then the second algorithm admits a round complexity of O(log log n) and message complexity of Õ(n 2 ). To the best of our knowledge, this is the first work to study the Steiner tree problem in the CONGEST ED CLIQUE model.

Mind the O˜: Asymptotically Better, but Still Impractical, Quantum Distributed Algorithms

“…The main idea is that we can reduce distance products to a binary search in which each step in the search finds negative triangles. This procedure corresponds to ( [18] Proposition 2), which we describe here, restricting to finding the distance product square needed for Equation (2).…”

“…Although in definition 1, we make no assumption on the memory capacities of each node, the trivial n-round strategy uses at least 2 log(n)|E| 2 • log(W) memory at the leader node that solves the problem. For the APSP problem in question, using the Floyd-Warshall algorithm results in memory requirements of 2n 2 log(n) • log(nW) at the leader node.…”

“…Then, g(n) > n, up until about n ≈ 2.6 • 10 11 . As with the quantum APSP, though this algorithm provides the best-known asymptotic complexity of Õ(n 1/3 ) in the classical CONGEST-CLIQUE, it also fails to give any real improvement over the trivial strategy across a very large regime of values of n. Consequently, algorithms making use of this APSP algorithm, such as [2] or [3], suffer from the same problem of impracticality. However, the algorithm only requires 4n 4/3 log(n) log(nW) + n log(n) log(nW) memory per node, which is less than what is required for the trivial strategy, even for n ≥ 4.…”

“…The classical CONGEST-CLIQUE model (henceforth referred to as cCCM) in distributed computing has been carefully studied as a model central to the field, e.g., [1][2][3][4][5][6]. In this model, processors in a network solve a problem whose input is distributed across the nodes under significant communication limitations, described in detail in Section 2.…”

“…This is done by augmenting the APSP algorithm of [7] with an efficient routing table scheme, which is necessary to make use of the shortest path information instead of only the APSP distances, and using the resulting subroutine with existing classical algorithmic frameworks. Beyond asymptotics, we also characterize the complexity of our algorithms as well as those of [2,3,7,12] to include the logarithmic and constant factors involved to estimate the scales at which they would be practical, which was not included in the previous work. It should be noted that, similar to APSP, these problems cannot see quantum speedups in the CONGEST (non-clique) setting, as shown in [10].…”