Quickest paths: parallelization and dynamization

Let N = (V, E, c, /) be a network, where G = (V, E), (VI = n and (El = m, is a directed graph, c(e) > 0 is the capacity and I(e) 2 0 is the lead time for each edge e E E. The transmission time to send u units of data from a given source s to a destination t using path p is T(a,p) = I(p) + &, where 1(p) is the sum of the lead times of the edges in p, and c(p) is the minimum capacity of the edges in p. The quickest path problem is to find a path of minimum transmission time to transmit the u units of data from s to t. The problem has applications to fast data transmissions in communication networks. In this paper we present the first parallel algorithms for solving the quickest path problem in the case where the network is sparse (i.e. m = O(n)). We also give the first algorithms for solving the dynamic quickest path problem. In this problem, the network, the lead times and the capacities on its edges, as well as the amount of data to be transmitted, change over time. The goal is to build a data structure so that one can compute very fast the quickest path to transmit a given amount of data from any node s to any node t and also, after a dynamic change (edge lead time or edge capacity modification, edge deletion) on the input network, to be able to update the data structure in appropriately short time. Furthermore, we improve upon the best, sequential result for the single pair quickest path problem which needs O(rm + rn logn) time [l, 61, where r is the number of distinct edge capacities.