Constant-Time Dynamic \((\Delta+1)\)-Coloring and Weight Approximation for Minimum Spanning Forest: Dynamic Algorithms Meet Property Testing

Abstract

With few exceptions (namely, algorithms for maximal matching, \(2\)-approximate vertex cover, and certain constant-stretch spanners), all known fully dynamic algorithms in general graphs require (amortized) \(\Omega(\log n)\) update/query time. Showing for the first time that techniques from property testing can lead to constant-time fully dynamic graph algorithms we prove the following results: (1) We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper \((\Delta+1)\)-vertex coloring of a graph with maximum degree at most \(\Delta\). This improves upon the previous \(O(\log \Delta)\)-time algorithm by Bhattacharya et al. (SODA 2018). We show that our result does not only have optimal running time, but is also optimal in the sense that already deciding whether a \(\Delta\)-coloring exists in a dynamically changing graph with maximum degree at most \(\Delta\) takes \(\Omega(\log n)\) time per operation. (2) We give two fully dynamic algorithms that maintain a \((1+\varepsilon)\)-approximation of the weight \(M\) of the minimum spanning forest of a graph \(G\) with edges weights in \([1,W]\). Our deterministic algorithm takes \(O({W^2 \log W}/{\varepsilon^3})\) worst-case time, which is constant if both \(W\) and \(\varepsilon\) are constant. This is somewhat surprising as a lower bound by Patrascu and Demaine (SIAM J. Comput. 2006) shows that it takes \(\Omega(\log n)\) time per operation to maintain the exact weight of the MSF that holds even for \(W=1\). Our randomized (Monte-Carlo style) algorithm works with high probability and runs in worst-case \(O(\frac{1}{\varepsilon^4}\log^2(\frac{1}{\varepsilon}))\) time if \(W= O({(m^*)^{1/3}}/{\log^3 n})\), where \(m^*\) is the minimum number of edges in the graph throughout all the updates. It works even against an adaptive adversary.