Algebraic methods in the congested clique

Abstract

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n 1−2/ω ) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -triangle and 4-cycle counting in O(n 0.158 ) rounds, improving upon the O(n 1/3 ) algorithm of Dolev et al. [DISC 2012], -a (1 + o(1))-approximation of all-pairs shortest paths in O(n 0.158 ) rounds, improving upon theÕ(n 1/2 )-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n 0.158 ) rounds, which is the first non-trivial solution in this model.In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

Quantum Distributed Algorithm for the All-Pairs Shortest Path Problem in the CONGEST-CLIQUE Model

“…For each i ∈ [m] define the set A 1 i = {x i ∈ X | g i (x i ) = 1} and assume for convenience that |A 1 i | > 0. 4 Let u be an arbitrary node of the network. Assume now that node u can evaluate the functions g 1 , .…”

“…Background. The CONGEST-CLIQUE model is a model in distributed computing that has recently been the subject of intensive research [4,8,9,18,19,20,21,29,30,31,32,35,25,34,16]. In this model n nodes communicate with each other over a fully connected network (i.e., a clique) by exchanging messages of O(log n) bits in synchronous rounds.…”

“…This was improved by Censor-Hillel at al. [4], who gave aÕ(n 1/3 )-round exact algorithm for the general APSP (i.e., the APSP over directed graphs with polynomial weights). While faster algorithms based on fast matrix multiplication have been designed for the APSP over graphs with small weights or for approximating the shortest paths [4,26], the aboveÕ(n 1/3 )-round is still not only the best known exact algorithm for the general APSP, but also the best known exact algorithm for SSSP in the CONGEST-CLIQUE model.…”

“…As already mentioned, triangle detection and matrix multiplication are closely related to the APSP problem. There are several results considering those problems in the CONGEST or CONGEST-CLIQUE models [8,4,26,24,33,5,6]. In the CONGEST-CLIQUE model, in particular, anÕ(n 1/3 )-round algorithm for listing all triangles is proposed by Dolev et al [8].…”

“…Our explanations focus on computing the lengths of the shortest paths. Using standard techniques (see for instance[4]), the approach can be adapted to return the shortest paths as well, at a cost of increasing the complexity only by a polylogarithmic factor.…”