We consider a new class of routing requests or partial permutations for which we give optimal on-line routing algorithms on the hypercube and shuffle-exchange network. For well-formed words of parentheses our algorithm establishes communication between all matching pairs in logarithmic time. It can be applied to the membership problem for Dyck languages and a number of problems for algebraic expressions.

We present a parallel randomized algorithm running on a CRCW PRAM , to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms in O(log 2 n) time with n 1+ffl processors, for any ffl ? 0). If n is the number of vertices, our algorithm takes O(log(n)) time with P = O i n 1:5 \Delta p log(n) j processors and with probability of failure at most 1 n . The algorithm needs 2 \Delta log(m) \Delta log(n) +O(log(n)) random bits. The number of random bits can be decreased to O(log(n)) by increasing the number of processors to n 3 2 +ffl ; for any ffl ? 0: Our parallel algorithm has significantly improved processor efficiency, compared to the previous logorithmic time parallel algorithm of Miller and Reif [MR 91], which requires n 4 randomized processors or n 5 deterministic processors. Keywords: planar graphs, graph isomorph...