Optimal Segmented Scan and Simulation of Reconfigurable Architectures on Fixed Connection Networks

Given n elements x0, . . . , xn−1, and given n bits b0, . . . , bn−1, with at least one zero, the segmented scan problem consists in finding the prefixes si = xi ⊗ bis(i−1) mod n, i = 0, . . . , n − 1, where ⊗ is an associative binary operation that can be computed in constant time by a processor. This paper presents: (i) an O(logB) time optimal algorithm for the segmented scan problem on a (2n−1)-node toroidal X-tree, where B is the maximum distance of two successive zeroes in b0, . . . , bn−1; (ii) a novel definition of locally normal algorithms for trees and meshes of trees; (iii) a constant slow-down, optimal, and locally normal simulation algorithm for a class of reconfigurable architectures on the mesh of toroidal X-trees, if the log-time delay model is assumed; (iv) a constant slow-down optimal simulation of locally normal algorithms for meshes of toroidal X-trees on the hypercube.