We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a γ>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+√2γ times the shortest-path distance, and yet the total weight of the tree is at most 1+√2/γ times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex. © 1995 Springer-Verlag New York Inc.

We give an efficient deterministic parallel approximation algorithm for the minimum-weight vertex- and set-cover problems and their duals (edge/element packing). The algorithm is simple and suitable for distributed implementation. It fits no existing paradigm for fast, efficient parallel algorithms-it uses only “local„ information at each step, yet is deterministic. (Generally, such algorithms have required randomization.) The result demonstrates that linear-programming primal-dual approximation techniques can lead to fast, efficient parallel algorithms. The presentation does not assume knowledge of such techniques. © 1994 Academic Press, Inc.