Efficient parallel divide-and-conquer for a class of interconnection topologies

Abstract

In this paper, we propose an efficient scheduling algorithm for expanding any divide-and-conquer (D&C) computation tree on k-dimensional mesh, hypercube, and perfect shuffle networks with p processors. Assume that it takes t _n time steps to expand one node of the tree and t _c time steps to transmit one datum or convey one node. For any D&C computation tree with N nodes, height h, and degree d (maximal number of children of any node), our algorithm requires at most (N/p+h)t _n+ϕdht _c time steps, where ϕ is O(log₂ p) on a hypercube or perfect shuffle network and is $O\left( {\sqrt[k]{p}} \right)$ on a n _k−1x···xn ₀ mesh network, where $n_{k - 1} = \cdot \cdot \cdot = n_0 = \sqrt[k]{p}$. This algorithm is general in the sense that it does not know the values of N, h, and d, and the shape of the computation tree as well, a priori. Most importantly, we can easily obtain a linear speedup by nearly a factor of p, especially when N ≫ ph(1+ϕdt _c /t _n ).

This research was supported in part by the Defense Advanced Research Projects Agency, Information Science and Technology Office, under the title Research on Parallel Computing issued by DARPA/CMO under Contract MDA972-90-C-0035, ARPA Order No. 7330, in part by the National Science Foundation and the Defense Advanced Research Projects Agency under Cooperative Agreement NCR-8919038 with the Corporation for National Research Initiatives, and in part by the Office of Naval Research under Contract N00014-90-J-1939.