Low-Congestion Shortcut and Graph Parameters
share
The concept of low-congestion shortcuts is initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. Specifically, given a specific graph class X, an f-round algorithm of constructing shortcuts of quality q for any instance in X results in Õ(q + f)-round algorithms of solving several fundamental graph problems such as minimum spanning tree and minimum cut, for X. In this paper, we consider the relationship between the quality of low-congestion shortcuts and three major graph parameters, chordality, diameter, and clique-width. The main contribution of the paper is threefold: (1) We show an O(1)-round algorithm which constructs a low-congestion shortcut with quality O(kD) for any k-chordal graph, and prove that the quality and running time of this construction is nearly optimal up to polylogarithmic factors. (2) We present two algorithms, each of which constructs a low-congestion shortcut with quality Õ(n^1/4) in Õ(n^1/4) rounds for graphs of D=3, and that with quality Õ(n^1/3) in Õ(n^1/3) rounds for graphs of D=4 respectively. These results obviously deduce two MST algorithms running in Õ(n^1/4) and Õ(n^1/3) rounds for D=3 and 4 respectively, which almost close the long-standing complexity gap of the MST construction in small-diameter graphs originally posed by Lotker et al. [Distributed Computing 2006]. (3) We show that bounding clique-width does not help the construction of good shortcuts by presenting a network topology of clique-width six where the construction of MST is as expensive as the general case.
READ FULL TEXT