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}/spl rarr/{1...m}. This enables us to model sequential loops of the form for 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/spl sube/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/sup 2/ n) algorithm with no restriction on f, g or h, using up to O(n/sup 2/) processors. However we show that for general IR, op must be commutative so that a parallel computation can be used.