Parallel Solutions Of Indexed Recurrence Equations

A new type of recurrence equations called ''indexed recurrences'' (IR) is defined, in which the common notion of X[i] = op(X[i], X[i - 1]) i = 1...n is generalized to X[g(i)] = op(X[f(i)], X[h(i)]) f, g, h : {1...n} bar right arrow {1...m}. This enables us to model sequential loops of the formfor i = 1 to n do begin x[g(i)] := op(X[f(i)], X[h(i)];)as IR equations. Thus, a parallel algorithm that solves a set of IR equations is in fact a way to transform sequential loops into parallel ones. Note that the circuit evaluation problem (CVP) can also be expressed as a set of IR equations. Therefore an efficient parallel solution to the general IR problem is not likely to be found, as such solution would also solve the CVP, showing that P subset of or equal to NC. In this paper we introduce parallel algorithms for two variants of the IR equations problem:An O(log n) greedy algorithm for solving IR equations where g(i) is distinct and h(i) = g(i) using O(n) processors.An O(log(2) n) algorithm with no restriction on f, g or h, using up to O(n(2)) processors. However; we show that for general IR, op must be commutative so that a parallel computation can be used.