Sorting n/sup 2/ numbers on n*n meshes

Abstract

We show that by folding data from an n x n mesh onto an n x ( n k ) submesh, sorting on the submesh, and finally unfolding back onto the entire n x n mesh it is possible to sort on bidirectional and strict unidirectional meshes using a number of routing steps that is very close to the distance lower bound for these architectures. The technique may also be applied to reconfigurable bus architectures to obtain faster sorting algorithms.

ART: robustness of meshes and tori for parallel and distributed computation

“…Various algorithms have been developed for mesh-connected computers and their variants, such as tori and n-ary d-cubes, based on the assumption that a fault-free mesh (or torus, n-ary d-cube) is available [3,10,11,13,16,15,17]. Since fault tolerance is very important to parallel processing, a variety of techniques for adaptive fault-tolerant routing or reconfiguring faulty arrays have also been proposed [5,6,12,18,19,23].…”

“…The robust sorting algorithm proposed in [22] is actually a special case of ART-II since we emulate the sorting algorithm proposed in [10], which performs Θ(n) operations along a dimension on the average. In [14], we have shown that 1-1 sorting in an n n bypass mesh with o(n 1=4 ) faults can be performed in 2:5n + o(n) communication time using row order or snakelike order by emulating the Schnorr/Shamir sorting algorithm [11,13,17]). The average number of steps along a dimension in the Schnorr/Shamir sorting algorithm is only Θ(n 1=4 ), so the number of faults that can be tolerated without increasing the leading constant of the running time is only o(n 1=4 ) when direct emulation is used.…”

“…The average number of steps along a dimension in the Schnorr/Shamir sorting algorithm is only Θ(n 1=4 ), so the number of faults that can be tolerated without increasing the leading constant of the running time is only o(n 1=4 ) when direct emulation is used. In [20,24], we have also shown that 1-1 sorting in an n n mesh with o(n) faults can be performed in 3n + o(n) communication time using row order or snakelike order by emulating a variant of the Schnorr/Shamir sorting algorithm [11,13,17]). We increase the number of faults tolerated to o(n) by modifying the Schnorr/Shamir sorting algorithm so that the average number of steps along a dimension is increased to o(n).…”

“…These generalizations can be achieved by emulating the sorting algorithm for fault-free meshes and n-ary d-cubes given in [10], which performs Θ(n) steps along a dimension on the average. When h = 1, we can fold the data items to the middle of the virtual submesh and use 2-2 sorting of preceding method as the algorithms given in [10,13,20,22,24] computation steps, which can be emulated using CET with a slowdown factor of 1 + o (1). These results also lead to efficient permutation routing and the details are omitted in this paper.…”

“…The scalability, compact layout, constant/small node-degree, desirable algorithmic properties, and many other advantages have made meshes, tori, and n-ary d-cubes the most popular topologies for the interconnection of parallel processors. A very large variety of algorithms have been proposed for these networks [3,10,11,13,16,15,17]. These algorithms usually assume that a faultfree mesh or torus is available, and most of them cannot be applied to faulty meshes or tori directly, even in the presence of only a small number of faulty elements.…”