Low-Congestion Shortcut and Graph Parameters

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 $\tilde{O}(q + f)$-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 $\tilde{O}(n^{1/4})$ in $\tilde{O}(n^{1/4})$ rounds for graphs of $D=3$, and that with quality $\tilde{O}(n^{1/3})$ in $\tilde{O}(n^{1/3})$ rounds for graphs of $D=4$ respectively. These results obviously deduce two MST algorithms running in $\tilde{O}(n^{1/4})$ and $\tilde{O}(n^{1/3})$ 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.