Optimal routing of parentheses on the hypercube

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 certain subclasses of deterministic context-free languages, such as Dyck languages, and to a number of problems dealing with algebraic expressions.