An algorithmic Friedman--Pippenger theorem on tree embeddings and applications to routing

• Authors: D. Dellamonica, Jr.;Y. Kohayakawa
• Venue: SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
• Year: 2006

An (n, d)-expander is a graph G = (V, E) such that for every X ⊆ V with |X| ≤ 2n − 2 we have |ΓG(X)| ≥ (d + 1)|X|. A tree T is small if it has at most n vertices and has maximum degree at most d. Friedman and Pippenger (1987) proved that any (n, d)-expander contains every small tree. The elegant proof discovered by those authors does not yield an efficient algorithm for obtaining the tree. In this extended abstract, we give an alternative, polynomial formulation for a key concept in their proof, and thus obtain an efficient algorithm for the Friedman---Pippenger theorem.As an application, we offer a polynomial time algorithm for routing connection requests in wide-sense non-blocking networks of constant depth, following an approach put forward in a seminal paper of Feldman, Friedman, and Pippenger (1988). We thus provide a simple, efficient companion routing scheme for the explicitly described networks of essentially optimal size given by Wigderson and Zuckerman (1999). The applicability of our methods to deterministically constructible networks is in contrast with some previous work of Aggarwal et al. (1996).