Deterministic stateless centralized local algorithms for bounded degree graphs

Abstract

We design centralized local algorithms for: maximal independent set, maximal matching, and graph coloring. The improvement is threefold: the algorithms are deterministic, stateless, and the number of probes is O(log* n), where n is the number of vertices of the input graph. Our algorithms for maximal independent set and maximal matching improves over previous randomized algorithms by Alon et al. (SODA 2012) and Mansour et al. (ICALP 2012). In these previous algorithms, the number of probes and the space required for storing the state between queries are poly(logn). We also design the first centralized local algorithm for graph coloring. Our graph coloring algorithms are deterministic and stateless. Let Δ denote the maximum degree of a graph over n vertices. Our algorithm for coloring the vertices by Δ+1 colors requires O(log* n) probes for constant degree graphs. Surprisingly, for the case where the number of colors is O(Δ2 logΔ), the number of probes of our algorithm is O(Δ· log* n+Δ2), that is, the number of probes is sublinear if Δ =o9√n), i.e., our algorithm applies for graphs with unbounded degrees. © 2014 Springer-Verlag Berlin Heidelberg.