We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: -- We present a @math round deterministic algorithm for @math -edge-coloring, where @math denotes the maximum degree. Modulo the @math factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of @math -edge-coloring in @math rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. -- We show that sinkless orientation---i.e., orienting edges such that each node has at least one outgoing edge---on @math -regular graphs can be solved in @math rounds randomized and in @math rounds deterministically. These prove the corresponding lower bounds by [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of for @math -coloring @math -regular trees. -- We present a randomized @math round algorithm for orienting @math -arboricity graphs with maximum out-degree @math . This can be also turned into a decomposition into @math forests when @math and into @math pseduo-forests when @math . Obtaining an efficient distributed decomposition into less than @math forests was stated as the 10th open problem in the book by Barenboim and Elkin.
Throughout, we work with the standard distributed model called @math , due to Linial @cite_12 : The network is abstracted as a graph @math . There is one processor on each vertex, which initially knows only its neighbors. Per round, each processor can send one message to each of its neighbors.
@cite_12
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This paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion. The computational model is such that each node of the graph is occupied by a processor which has its own ID. Processors are restricted to collecting data from others which are at a distance at most t away from them in t time units, but are otherwise computationally unbounded. This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data.Three results are proved within this model: • A 3-coloring of an n-cycle requires time @math . This bound is tight, by previous work of Cole and Vishkin. • Any algorithm for coloring the d-regular tree of radius r which runs for time at most @math requires at least @math colors. • In an n-vertex graph of largest degree @math , an @math -coloring may be found in time @math .