Bottleneck paths and trees and deterministic Graphical games

Gabow and Tarjan showed that the Bottleneck Path (BP) problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree (BST) problem, i.e., finding a directed spanning tree rooted at a given vertex whose largest edge weight is minimized, can both be solved deterministically in O(m log^∗ n) time, where m is the number of edges and n is the number of vertices in the graph. We present a slightly improved randomized algorithm for these problems with an expected running time of O(mβ(m, n)), where β(m,n) = min{k ≥ 1 | log^(k) n ≤ m/n} < log^∗ n - log^∗ (m/n) + 1. This is the first improvement for these problems in over 25 years. In particular, if m ≥ n log^(k) n, for some constant k, the expected running time of the new algorithm is O(m). Our algorithm, as that of Gabow and Tarjan, work in the comparison model. We also observe that in the word-RAM model, both problems can be solved deterministically in O(m) time. Finally, we solve an open problem of Andersson et al., giving a deterministic O(m)-time comparison-based algorithm for solving deterministic 2-player turn-based zero-sum terminal payoff games, also known as Deterministic Graphical Games (DGG).