On Sparsity Awareness in Distributed Computations (2105.06068v4)

Abstract: We extract a core principle underlying seemingly different fundamental distributed settings, showing sparsity awareness may induce faster algorithms for problems in these settings. To leverage this, we establish a new framework by developing an intermediate auxiliary model weak enough to be simulated in the CONGEST model given low mixing time, as well as in the recently introduced HYBRID model. We prove that despite imposing harsh restrictions, this artificial model allows balancing massive data transfers with high bandwidth utilization. We exemplify the power of our methods, by deriving shortest-paths algorithms improving upon the state-of-the-art. Specifically, we show the following for graphs of $n$ nodes: A $(3+\epsilon)$ approximation for weighted APSP in $(n^{1/2}+n/\delta)\tau_{mix}\cdot 2^{O(\sqrt\log n)}$ rounds in the CONGEST model, where $\delta$ is the minimum degree of the graph and $\tau_{mix}$ is its mixing time. For graphs with $\delta=\tau_{mix}\cdot 2^{\omega(\sqrt\log n)}$, this takes $o(n)$ rounds, despite the $\Omega(n)$ lower bound for general graphs [Nanongkai, STOC'14]. An $(n^{7/6}/m^{1/2}+n^2/m)\cdot\tau_{mix}\cdot 2^{O(\sqrt\log n)}$-round exact SSSP algorithm in the CONGNEST model, for graphs with $m$ edges and a mixing time of $\tau_{mix}$. This improves upon the algorithm of [Chechik and Mukhtar, PODC'20] for significant ranges of values of $m$ and $\tau_{mix}$. A CONGESTED CLIQUE simulation in the CONGEST model improving upon the state-of-the-art simulation of [Ghaffari, Kuhn, and SU, PODC'17] by a factor proportional to the average degree in the graph. An $\tilde O(n^{5/17}/\epsilon^9)$-round algorithm for a $(1+\epsilon)$ approximation for SSSP in the HYBRID model. The only previous $o(n^{1/3})$ round algorithm for distance approximations in this model is for a much larger factor [Augustine, Hinnenthal, Kuhn, Scheideler, Schneider, SODA'20].