The problem of (vertex) (∆+1)-coloring a graph of maximum degree ∆ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA'18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a (∆ + 1)-coloring with O(log ∆) expected amortized update time. In this paper, we present a (∆ + 1)-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high probability. 1
Pebble games are single-player games on DAGs involving placing and moving pebbles on nodes of the graph according to a certain set of rules. The goal is to pebble a set of target nodes using a minimum number of pebbles. In this paper, we present a possibly simpler proof of the result in [CLNV15] and strengthen the result to show that it is PSPACE-hard to determine the minimum number of pebbles to an additive n 1/3−ε term for all ε > 0, which improves upon the currently known additive constant hardness of approximation [CLNV15] in the standard pebble game. We also introduce a family of explicit, constant indegree graphs with n nodes where there exists a graph in the family such that using constant k pebbles requires Ω(n k ) moves to pebble in both the standard and black-white pebble games. This independently answers an open question summarized in [Nor15] of whether a family of DAGs exists that meets the upper bound of O(n k ) moves using constant k pebbles with a different construction than that presented in [AdRNV17].
In this paper, we study new batch-dynamic algorithms for the k-clique counting problem, which are dynamic algorithms where the updates are batches of edge insertions and deletions. We study this problem in the parallel setting, where the goal is to obtain algorithms with low (polylogarithmic) depth. Our first result is a new parallel batch-dynamic triangle counting algorithm with O(∆ √ ∆ + m) amortized work and O(log * (∆ + m)) depth with high probability, and O(∆ + m) space for a batch of ∆ edge insertions or deletions. Our second result is an algebraic algorithm based on parallel fast matrix multiplication. Assuming that a parallel fast matrix multiplication algorithm exists with parallel matrix multiplication constant ω p , the same algorithm solves dynamic k-clique counting with O min ∆m
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