Optimal Simulations in Reconfigurable Arrays

There are several reconfiguring models of parallel computation that are considered in the published literature, depending on their switching capabilities. Three of the more popular ones are the Horizontal-Vertical Reconfigurable Mesh (HV-RN model) [LM89a, LM89b, ML89], the Linear Reconfigurable Mesh (LRN model) [MPRS87, BPRS91], and the General Reconfigurable Mesh (RN model) [Wan91, Sch91]. Can these reconfigurable models (in particular two dimensional arrays) be the basis for the design of massively parallel computers? Perhaps the most basic related question is: given an algorithm which is designed for a large reconfigurable mesh, can it be executed efficiently on a smaller reconfigurable mesh? We call it the simulation problem. Despite the fact that machines already exist and a large number of efficient algorithms are known for reconfigurable arrays, this question was left open for all reconfiguring models of computation. In this work we give several positive answers to the simulation problem. We have the following results for the simulation of an n x n reconfiguring mesh by the p X P reconfiguring mesh. (1) Using standard techniques the simulation in the HV-RN model is carried optimally with slowdown 8«n/p)2). (2) Although using the same method fails for the LRN model, a novel technique is developed, achieving asymptotically optimal simulation of LRN meshes, with slowdown 8«n/p)2). (3) Efficient simulations in the RN model is achieved too, paying an extra factor which is polylogarithmicin the size of the simulated mesh. The slowdown achieved is O«n/p)210gnlog(n/p)). Although the simulation algorithms are complex and require extensive bookkeeping operations, the required space is asymptotically optimal. In addition to the above we give a bunch of inter-model simulation results. These results give some additional insight into the gap in the computational power of the various reconfiguring models [Sch91, BPS91]. ·Department of Mathematics and Computer Science, University of Haifa, Haifa. 31905, Isra.el. E-mail: [email protected] 'Department of Mathematics and Computer Science, University of Haifa, Haifa 31905, Israel. E-mail: [email protected] l(CONTACT AUTHOR:) Department of Computer Science, Technion, Technion City, Haifa 32000, Israel. E-mail: [email protected] . aifa.a