List Ranking on Meshes and Networks

The list-ranking problem is considered for parallel computers which communicate through an inter-connection network. Each PU holds k 1 nodes of a set of linked lists. We give deterministic and ran-domized algorithms, which considerable improve earlier results. For the case k = 1, the algorithms are based on repeatedly removing the complement of a ruling set. On two-dimensional meshes, the deterministic version takes 100 n steps; the randomized version 70n steps. Extensions for larger k, require 17kn and 10 k n, steps respectively. For suuciently large k, list-ranking can be solved much faster. For a large class of networks, a ran-domized algorithm takes just twice the number of steps required by a k-k routing. The only conditions are that: (1) k = !(k), where k is so large that the time consumption of k-k routing is determined by the bisection bound, and (2) the routing time slightly increases with the number of PUs in the network. For special networks we can prove stronger results. Particularly, for n n meshes, the list ranking problem is solved in (1=2+o(1))kn steps, if k = !(1). For hypercubes with N PUs, assuming all-port communication, the algorithm requires only (2 + o(1)) k steps, if k = !(log 2 N). We show that list ranking requires at least the time for k-k routing. So, these latter results are within a factor of two from optimal. For meshes we even match the lower bound up to lower-order terms.