On efficient algorithms for bottleneck path problems with many sources

For given edge-capacitated connected graph and two its vertices s and t, the bottleneck (or \(\max \min \)) path problem is to find the maximum value of path-minimum edge capacities among all paths, connecting s and t. It can be generalized by finding the bottleneck values between s and all possible t. These problems arise as subproblems in the known maximum flow problem, having applications in many real-life tasks. For any graph with n vertices and m edges, they can be solved in O(m) and O(t(m, n)) times, respectively, where \(t(m,n)=\min (m+n\log (n),m\alpha (m,n))\) and \(\alpha (\cdot ,\cdot )\) is the inverse Ackermann function. In this paper, we generalize of the bottleneck path problems by considering their versions with k sources. For the first of them, where k pairs of sources and targets are (offline or online) given, we present an \(O((m+k)\log (n))\)-time randomized and an \(O(m+(n+k)\log (n))\)-time deterministic algorithms for the offline and online versions, respectively. For the second one, where the bottleneck values are found between k sources and all targets, we present an \(O(t(m,n)+kn)\)-time offline/online algorithm.