Trade-Offs in Dynamic Coloring for Bipartite and General Graphs

Abstract

The dynamic coloring problem has gained attention in the recent past. The focus has largely been on obtaining efficient update time algorithms using $\Delta +1$ or more colors and the trade-offs between update time and query time. Another important parameter in dynamic coloring is the number of recolorings per update which is addressed by the works of Barba et al. in WADS’17, and Solomon and Wein in ESA’18. In SODA’18, Bhattacharya et al. presented a randomized algorithm that uses $\Delta +1$ colors and achieves amortized $O(\log \Delta )$ update time. In STACS’20, Henzinger and Peng presented a randomized $(\Delta +1)$-coloring algorithm with amortized O(1) update time. Independently on arXiv, Bhattacharya et al. also presented a randomized $(\Delta +1)$-coloring algorithm with amortized O(1) update time. While works of Bhattacharya et al., and Henzinger and Peng are very efficient in terms of update time, they do not address the number of recolorings per update. We bridge this gap by providing efficient update time algorithms with constant number of recolorings. Moreover our algorithm is deterministic as opposed to the works of Bhattacharya et al. in SODA’18, and Henzinger and Peng in STACS’20. Next, we study bipartite graphs which can be optimally colored in the static setting. We show that even in the incremental setting (where edges are added to the graph and no edge can be deleted), there is a bad update sequence which forces the update time to be at least $\Omega (\log {n})$ in the amortized setting and $\Omega (n)$ in the worst case. This possibly explains the lack of any results on dynamic coloring specific to bipartite graphs. We circumvent this lower bound by proposing two approaches. Firstly, we allow the use of more than two colors and obtain significantly better update time. Second, we introduce the idea of implicit coloring. If the color of a vertex is explicitly stored in a data structure and updated at end of every update then we call such an algorithm as explicit coloring algorithm. All prior work on dynamic coloring uses explicit coloring algorithms. We show that using implicit coloring we can obtain near constant update time and query time for incremental coloring for bipartite case. We also bound the number of recolorings to near constant. We also show an efficient implicit fully dynamic algorithm for bipartite graphs. All our algorithms are deterministic and use simple data structures. Hence, we believe that our algorithms are of practical importance.