Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts
In this paper, we refine the (almost) existentially optimal distributed Laplacian solver recently developed by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS '21) into an (almost) universally optimal distributed Laplacian solver. Specifically, when the topology is known, we show that any Laplacian system on an n-node graph with shortcut quality SQ(G) can be solved within n^o(1)SQ(G) log(1/ε) rounds, where ε is the required accuracy. This almost matches our lower bound which guarantees that any
... orrect algorithm on G requires Ω(SQ(G)) rounds, even for a crude solution with ε≤ 1/2. Even in the unknown-topology case (i.e., standard CONGEST), the same bounds also hold in most networks of interest. Furthermore, conditional on conjectured improvements in state-of-the-art constructions of low-congestion shortcuts, the CONGEST results will match the known-topology ones. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique (NCC) model. In this model, we show the existence of a Laplacian solver with round complexity n^o(1)log(1/ε). The unifying thread of these results, and our main technical contribution, is the study of novel congested generalization of the standard part-wise aggregation problem. We develop near-optimal algorithms for this primitive in the Supported-CONGEST model, almost-optimal algorithms in (standard) CONGEST, as well as a very simple algorithm for bounded-treewidth graphs with slightly worse bounds. This primitive can be readily used to accelerate the FOCS'21 Laplacian solver. We believe this primitive will find further independent applications.