Doubly Logarithmic Communication Algorithms for Optical-Communication Parallel Computers

In this paper, we consider the problem of interprocessor communication on parallel computers that have optical communication networks. We consider the completely connected optical-communication parallel computer (OCPC), which has a completely connected optical network, and also the mesh-of-optical-buses parallel computer (MOB-PC), which has a mesh of optical buses as its communication network. The particular communication problem that we study is that of realizing an h-relation. In this problem, each processor has at most h messages to send and at most h messages to receive. It is clear that any 1-relation can be realized in one communication step on an OCPC. However, the best previously known p-processor OCPC algorithm for realizing an arbitrary h-relation for h > 1 requires $\Theta(h + \log p)$ expected communication steps. (This algorithm is due to Valiant and is based on earlier work of Anderson and Miller.) Valiant's algorithm is optimal only for $h=\Omega(\log p)$, and it is an open question of Geréb-Graus and Tsantilas whether there is a faster algorithm for h=o(log p). In this paper, we answer this question in the affirmative and we extend the range of optimality by considering the case in which $h\leq \log p$. In particular, we present a $\Theta(h + \log\log p)$-communication-step randomized algorithm that realizes an arbitrary h-relation on a p-processor OCPC. We show that if $h\leq \log p$, then the failure probability can be made as small as $p^{-\alpha}$ for any positive constant $\alpha$. We use the OCPC algorithm as a subroutine in a $\Theta(h + \log\log p)$-communication-step randomized algorithm that realizes an arbitrary h-relation on a $p\times p$-processor MOB-PC. Once again, we show that if $h\leq \log p$, then the failure probability can be made as small as $p^{-\alpha}$ for any positive constant $\alpha$.