Fast approximate shortest paths in the congested clique

Abstract

We design fast deterministic algorithms for distance computation in the Congested Clique model. Our key contributions include:A $$(2+\epsilon )$$(2+ϵ)-approximation for all-pairs shortest paths in $$O(\log ^2{n} / \epsilon )$$O(log2n/ϵ) rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model.A $$(1+\epsilon )$$(1+ϵ)-approximation for multi-source shortest paths from $$O(\sqrt{n})$$O(n) sources in $$O(\log ^2{n} / \epsilon )$$O(log2n/ϵ) rounds on weighted undirected graphs. This is the first sub-polynomial algorithm obtaining this approximation for a set of sources of polynomial size. Our main techniques are new distance tools that are obtained via improved algorithms for sparse matrix multiplication, which we leverage to construct efficient hopsets and shortest paths. Furthermore, our techniques extend to additional distance problems for which we improve upon the state-of-the-art, including diameter approximation, and an exact single-source shortest paths algorithm for weighted undirected graphs in $$\tilde{O}(n^{1/6})$$O~(n1/6) rounds.

Massively Parallel Approximate Distance Sketches

“…Similarly, hopsets have been used extensively in various models of computation for solving approximate SSSP ( [14,9]). Our result on hopset construction in low memory MPC also gives the first (approximate) SSSP algorithm in this model for weighted graphs (in Congested Clique there are more results known [9,14,4,6], but these do not translate obviously to MPC when there is sublinear memory per machine). In a recent result, [6] gave an efficient Congested Clique algorithm that constructs hopsets of size Õ(n 3/2 ) with hopbound O(log 2 (n)/ǫ).…”

“…This is what hopsets do: we discuss them in more detail in Section 2.2, but informally they allow us to reduce the diameter of the graph while preserving distances by adding in a carefully chosen set of weighted "shortcut" edges. Hopset constructions for the Congested Clique were given by Elkin and Neiman [9] (and more recently by [6]) so for Congested Clique we can essentially just combine result of [9] (or [6]) with [24] to get our result (modulo a small number of technicalities).…”

Constant-Round Spanners and Shortest Paths in Congested Clique and MPC

“…In recent years, this model has received a lot of attention, partially due to its connections to modern settings for parallel computation such as the massively parallel computation (MPC) model [10,34]. In particular, distance computation has been extensively studied in the Congested Cliqe model [9,11,14,15,17,24,35,37,41].…”

“…A natural question is whether faster algorithms can be obtained if we settle for approximate solutions. This is indeed possible as shown by a recent line of work [9,11,14,24]. The first algorithm that obtains a faster running time is a poly(log )-round algorithm for (1+ )-approximate Single-Source Shortest Paths (SSSP), based on continuous optimization techniques [9].…”

“…The first algorithm that obtains a faster running time is a poly(log )-round algorithm for (1+ )-approximate Single-Source Shortest Paths (SSSP), based on continuous optimization techniques [9]. A later work [14] used sparse matrix multiplication to obtain polylogarithmictime algorithms for (3 + )-approximation of APSP, and (1 + )approximation of Multi-Source Shortest Paths (MSSP), with ( √ ) sources. All these results hold for weighted undirected graphs.…”