Generalized connection networks for parallel processor intercommunication

Computer Science Departmen

A Complexity Theory for VLSI

The established methodologies for studying computational complexity can be applied to the new problems posed by very large-scale integrated (VLSI) circuits. This thesis develops a ''VLSI model of computation'' and derives upper and lower bounds on the silicon area and time required to solve the problems of sorting and discrete Fourier transformation. In particular, the area A and time T taken by any VLSI chip using any algorithm to perform an N-point Fourier transform must satisfy AT2 ≥ c N2 log2 N, for some fixed c > 0. A more general result for both sorting and Fourier transformation is that AT2x = Ω(N1 + x log2x N) for any x in the range 0 x N3/2 log N). The tightness of these bounds is demonstrated by the existence of nearly optimal circuits for both sorting and Fourier transformation. The circuits based on the shuffle-exchange interconnection pattern are fast but large: T = O(log2 N) for Fourier transformation, T = O(log3 N) for sorting; both have area A of at most O(N2 / log1/2 N). The circuits based on the mesh interconnection pattern are slow but small: T = O(N1/2 loglog N), A = O(N log2 N).</p

Sorting on a mesh-connected parallel computer

Computer Science Departmen

On the Average Number of Maxima in a set of Vectors and Applications

A maximal vector of a set is one which is not less than any other vector in all components. We derive a recurrence relation for computing the average number of maximal vectors in a set of n vectors in d-space under the assumption that all (n!)d relative orderings are equally probable. Solving the recurrence shows that the average number of maxima is 0((ln n)d-1) We use this result to construct an algorithm for finding all the maxima that has expected running time linear in n (for sets of vectors drawn under our assumptions.) For a given set of random points, the result in also used to derive an upper bound on the expected number of points from the set which are on the boundary of the convex hull of the set.</p