Minor Sparsifiers and the Distributed Laplacian Paradigm

Abstract

We study distributed algorithms built around minor-based vertex sparsifiers, and give the first algorithm in the CONGEST model for solving linear systems in graph Laplacian matrices to high accuracy. Our Laplacian solver has a round complexity of ( (1) ( √ + )), and thus almost matches

Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts

“…First, we note that any distributed Laplacian solver that always correctly outputs an answer on a fixed graph G must take at least Ω(SQ(G)) rounds, giving us a lower bound to compare ourselves with. Our refined lower bound uses the hardness result recently shown by [18] for the spanning connected subgraph problem, applicable for any (i.e., non-worst-case) graph G. Specifically, we show that a Laplacian solver can be leveraged to solve the spanning connected subgraph problem, thereby substantially strengthening the lower bound in [11].…”

“…Indeed, this framework has led to some state of the art algorithms for a wide range of fundamental graph-theoretic problems; e.g., see [5][6][7][8][9][10], and references therein. In the distributed setting, a major breakthrough was recently made in [11]. In particular, the authors developed a distributed algorithm that solves any Laplacian system on an n-node graph after n o (1) ( √ n + D) log(1/ε) rounds of the standard CONGEST model, where D represents the hop-diameter of the underlying network and ε > 0 is the error of the solver.…”

“…The only known technique in distributed computing for designing algorithms that go below the √ n-bound is the low-congestion shortcut framework of Ghaffari and Haeupler [15], and the large ecosystem of tools built around it [16][17][18][19][20][21][22]. However, the "ρ-congested minor" primitive introduced and extensively used in the novel distributed Laplacian solver [11] is out of reach from the current set of tools available in the low-congestion shortcut framework. We address this issue by introducing an analogous primitive called ρ-congested part-wise aggregation, which greatly simplifies the interface used by [11].…”

“…However, the "ρ-congested minor" primitive introduced and extensively used in the novel distributed Laplacian solver [11] is out of reach from the current set of tools available in the low-congestion shortcut framework. We address this issue by introducing an analogous primitive called ρ-congested part-wise aggregation, which greatly simplifies the interface used by [11]. We then extend the low-congestion shortcut framework with new techniques that enables it to near-optimally solve this primitive: we provide both an algorithm that utilizes the very recent hop-constrained expander decompositions for shortcut construction [22] to solve the primitive in general graphs with a linear dependence on ρ, as well as a very simple algorithm with a quadratic ρ-dependence for bounded-treewidth graphs.…”

“…Finally, we settle our original question in the positive by establishing that our new primitive can be readily used to accelerate the distributed Laplacian solver for non-worst-case topologies. Specifically, we show our new techniques are sufficient to lift the existentially optimal algorithm [11] to a universally optimal algorithm-modulo n o (1) factor inherent in the prior approach-for distributedly solving a Laplacian system, meaning that, for any topology, our algorithm is essentially as fast as possible. In other words, for any graph, our algorithm almost matches the best possible (correct) algorithm for that graph.…”