Speedup of Distributed Algorithms for Power Graphs in the CONGEST Model

Abstract

We obtain improved distributed algorithms in the CONGEST message-passing setting for problems on power graphs of an input graph G. This includes Coloring, Maximal Independent Set, and related problems. For R = f(Δ^k,n), we develop a general deterministic technique that transforms R-round LOCAL model algorithms for G^k with certain properties into O(R ⋅ Δ^{k/2-1})-round CONGEST algorithms for G^k. This improves the previously-known running time for such transformation, which was O(R⋅Δ^{k-1}). Consequently, for problems that can be solved by algorithms with the required properties and within polylogarithmic number of rounds, we obtain quadratic improvement for G^k and exponential improvement for G². We also obtain significant improvements for problems with larger number of rounds in G. Notable implications of our technique are the following deterministic distributed algorithms:  
- We devise a distributed algorithm for O(Δ⁴)-coloring of G² whose number of rounds is O(log Δ + log^* n). This improves exponentially (in terms of Δ) the best previously-known deterministic result of Halldorsson, Kuhn and Maus.[M. M. Halldorson et al., 2020] that required O(Δ + log^{*}n) rounds, and the standard simulation of Linial [N. Linial, 1992] algorithm in G^k that required O(Δ ⋅ log^* n) rounds. 
- We devise an algorithm for O(Δ²)-coloring of G² with O(Δ ⋅ log Δ + log^*n) rounds, and (Δ²+1)-coloring with O(Δ^{1.5} ⋅ log Δ + log^*n) rounds. This improves quadratically, and by a power of 4/3, respectively, the best previously-known results of Halldorsson, Khun and Maus. [M. M. Halldorson et al., 2020]. 
- For k > 2, our running time for O(Δ^{2k})-coloring of G^k is O(k⋅Δ^{k/2-1}⋅log Δ⋅log^* n). Our running time for O(Δ^k)-coloring of G^k is Õ(k⋅Δ^{k-1}⋅log^* n). This improves best previously-known results quadratically, and by a power of 3/2, respectively. 
- For constant k > 2, our upper bound for O(Δ^{2k})-coloring of G^k nearly matches the lower bound of Fraigniaud, Halldorsson and Nolin. [P. Fraigniaud et al., 2020] for checking the correctness of a coloring in G^k.