c-Perfect Hashing Schemes for Binary Trees, with Applications to Parallel Memories

We study the problem of mapping tree-structured data to an ensemble of parallel memory modules. We are given a “conflict tolerance” c, and we seek the smallest ensemble that will allow us to store any nvertex rooted binary tree with no more than c tree-vertices stored on the same module. Our attack on this problem abstracts it to a search for the smallest c-perfect universal graph for complete binary trees. We construct such a graph which witnesses that only O ¡ c (1¡1=c) ¢ 2 (n+1)=(c+1) ¢ memory modules are needed to obtain the required bound on conflicts, and we prove that ­ ¡ 2 (n+1)=(c+1) ¢ memory modules are necessary. These bounds are tight to within constant factors when c is fixed—as it is with the motivating application