Finding least-weight subsequences with fewer processors

Abstract

It is known that many problems which can be solved sequentially by dynamic programming, are in the class NC. Indeed most of these problems takes O(log² n) time on parallel models like CREW PRAM, but the number of processors involved is usually a high degree polynomial and the total work (i.e., the processor-time product) is very unfavorable in comparing with the work (i.e., the time) in the sequential case. Recently there has been a lot of progress in speeding up dynamic programming sequentially by restricting the weight functions to satisfy the quadrangle inequalities or the inverse quadrangle inequalities, yet little was heard about any improvement in the parallel complexity. Thus, it is time to see whether such kind of restrictions can lead to any improvement in the processor complexity of these problems. In this paper, we study the least-weight subsequence problem which is a typical problem for one-dimensional dynamic programming. The well known sequential solution to this problem takes O(n log n) time, while the conventional parallel algorithm uses O(log² n) time on a CREW PRAM with n ³ processors. Our new result is that with the inverse quadrangle inequality, the problem can be solved in O(log² n log log n) time on a CREW PRAM with n/log log n processors. Notice that the processor-time complexity is close to the sequential time complexity.