We deterministically compute a ∆+1 coloring and a maximal independent set(MIS) in time , where ∆ j is defined as the maximal number of nodes within distance j for a node and ∆ := ∆ 1 . Our greedy coloring and MIS algorithms improve the state-of-the-art algorithms running in O(∆+log * n) for a large class of graphs, i.e., graphs of (moderate) neighborhood growth with h ≥ 36. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. Our algorithm runs in time O(log χ + log * n) for ∆ ∈ Ω(log 1+1/ log * n n) and 1+1/ log * n n). For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest ∆ + 1 coloring algorithm running in time O(log ∆ + √ log n). The algorithm works without knowledge of χ and uses less than ∆ colors, i.e., (1 − 1/O(χ ))∆ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account. We also improve on the state of the art deterministic computation of (2, c)-ruling sets.
Schneider et al. REF investigated distributed coloring where the number of colors used depends on the chromatic number χ (G).
Symmetry breaking depending on the chromatic number or the neighborhood growth
Theor. Comput. Sci.
Theor. Comput. Sci.
Mathematics Computer Science