Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts

Abstract

In this work we refine the analysis of the distributed Laplacian solver recently established by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS '21), via the Ghaffari-Haeupler framework (SODA '16) of low-congestion shortcuts. Specifically, if > 0 represents the error of the solver, we derive two main results:

The Laplacian Paradigm in the Broadcast Congested Clique

“…Using such methods is very natural in distributed models because a matrix-vector multiplication can be carried out in a single round if each processor stores one coordinate of the vector. In recent years, this methodology has been successfully employed in the CONGEST model [GKKL + 18, BFKL21] and in particular, solvers for Laplacian systems with near-optimal round complexity have been developed for the CONGEST model -in networks with arbitrary topology [FGLP + 21] and in bounded-treewidth graphs [AGL21] -and for the HYBRID model [AGL21]. In this paper, we switch the focus to the BCC model and show that it allows a faster implementation of the basic Laplacian primitive.…”