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 (i.e., the Supported-CONGEST model), we show that any Laplacian system on an n-node graph with shortcut quality SQ(G) can be solved after n^{o(1)} SQ(G) log(1/?) rounds, where ? is the required accuracy. This almost matches our lower bound that guarantees that any correct algorithm on G requires ??(SQ(G)) rounds, even for a crude solution with ? ? 1/2. Several important implications hold in the unknown-topology (i.e., standard CONGEST) case: for excluded-minor graphs we get an almost universally optimal algorithm that terminates in D ? n^{o(1)} log(1/?) rounds, where D is the hop-diameter of the network; as well as n^{o(1)} log (1/?)-round algorithms for the case of SQ(G) ? n^{o(1)}, which holds for most networks of interest. Conditioned on improvements in state-of-the-art constructions of low-congestion shortcuts, the CONGEST results will match the Supported-CONGEST 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/?). The unifying thread of these results, and our main technical contribution, is the study of a 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 (with the additional overhead due to standard barriers), as well as a simple algorithm for bounded-treewidth graphs with a quadratic dependence on the congestion ?. This primitive can be readily used to accelerate the Laplacian solver of Forster, Goranci, Liu, Peng, Sun, and Ye, and we believe it will find further independent applications in the future