关键词:
Dynamic graph algorithms
Interval coloring
Lower bound
摘要:
We consider the dynamic graph coloring problem restricted to the class of interval graphs in the incremental and fully dynamic setting. The input consists of a sequence of intervals that are to be either colored, or deleted, if previously colored. For the incremental setting, we consider the well studied optimal online algorithm (KT-algorithm) for interval coloring due to Kierstead and Trotter [ I]. We present the following results on the dynamic interval coloring problem. Any direct implementation of the KT-algorithm requires Omega(Delta(2)) time per interval in the worst case. There exists an incremental algorithm which supports insertion of an interval in amortized O (logn + Delta) time per update and maintains a proper coloring using at most 3 omega - 2 colors. There exists a fully dynamic algorithm which supports insertion of an interval in O (logn + Delta log omega) update time and deletion of an interval in O (Delta(2) logn) update time in the worst case and maintains a proper coloring using at most 3 omega - 2 colors. The KT-algorithm crucially uses the maximum clique size in an induced subgraph in the neighborhood of a given vertex. We show that the problem of computing the induced subgraph among the neighbors of a given vertex has the same hardness as the online boolean matrix vector multiplication problem [2]. We show that Any algorithm that computes the induced subgraph among the neighbors of a given vertex requires at least quadratic time unless the OMy conjecture [2] is false. Finally, we obtain the following result on the OMy conjecture. If the matrix and the vectors in the online sequence have the consecutive ones property, then the OMy conjecture [2] is false. (C) 2020 Elsevier B.V. All rights reserved.