LOCALITY-PRESERVING HASH FUNCTIONS FOR GENERAL-PURPOSE PARALLEL COMPUTATION

Consider the problem of efficiently simulating the shared-memory parallel random access machine (PRAM) model on massively parallel architectures with physically distributed memory. To prevent network congestion and memory bank contention, it may be advantageous to hash the shared memory address space. The decision on whether or not to use hashing depends on (1) the communication latency in the network and (2) the locality of memory accesses in the algorithm. We relate this decision directly to algorithmic issues by studying the complexity of hashing in the Block PRAM model of Aggarwal, Chandra, and Snir, a shared-memory model of parallel computation which accounts for communication locality. For this model, we exhibit a universal family of hash functions having optimal locality. The complexity of applying these hash functions to the shared address space of the Block PRAM (i.e., by permuting data elements) is asymptotically equivalent to the complexity of performing a square matrix transpose, and this result is best possible for all pairwise independent universal bash families. These complexity bounds provide theoretical evidence that hashing and randomized routing need not destroy communication locality, addressing an open question of Valiant.