Going against the Grain

We review a general, space and time eecient technique for delivering a sequence of values computable by a recurrence relation, in the order opposing the data-dependencies of the recurrence. The technique provides a series of time/space tradeoos we characterize by parameter K > 0. Namely, N values can be delivered against the grain in O(KN) time and O(KN 1=K) space. This basic idea is not new, but here we present it in a framework exposing its essential nature and we give a concise yet easily understood explanation of it in terms of counting in a radix-N 1=K number system. We then show how to apply this paradigm to a couple of problems in sequence comparison, a domain where it has here to fore not been used. We show that in the limiting case of K = log 2 N , the method coincides with the well-known divide-and-conquer algorithm of Hirschberg. Thus, our observation provides a continuum of time/space tradeoos for all comparison problems that have appealed to the Hirschberg paradigm for greater space eeciency, e.g., sequence comparison with concave weights, approximate matching of regular expressions, etc. In particular, an O(M N +(M +N)W log(M +N)) algorithm for computing a representation of all suboptimal alignments is a corollary of our technique with the important property that only O(M + N + W log(M + N)) working storage is required versus the O(M + N + F) space of an earlier result by Chao. Here F is the size of a graph modeling all of the suboptimal alignments, and W is its width. Since F may be O(M N) this savings can be important in contexts where only summaries of the suboptimal region is required, e.g., the number of suboptimal paths with score greater than some threshold. Another application to computing locally optimal alignments as deened by Sellers illustrates the broad utility of the technique. 1 The Paradigm 1.1 Introduction Consider a one-variable recurrence: f n = h(f n?1 ; n) if n > 0 d 0 otherwise where the values f n are from domain D, d 0 is a value in D, and h : D@ 7 ! D is a computable function. Further assume that O(S) memory is required to store a value from the domain D and that h(d; n) is computable in O(T) time for any