Low-congestion shortcut and graph parameters

Abstract

Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of {tilde{Omega }}(sqrt{n} + D) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class {mathcal {C}}, an f-round algorithm for constructing shortcuts of quality q for any instance in {mathcal {C}} results in {tilde{O}}(q + f)-round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for {mathcal {C}}. The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking {tilde{O}}(sqrt{n}+D)-round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.