Space efficient streaming algorithms for the distance to monotonicity and asymmetric edit distance

Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem. We study this problem in the data stream model, where the algorithm is allowed to make a single left-to-right pass through the array and the key resource to be minimized is the amount of additional memory used. We present an algorithm which, for any δ > 0, given streaming access to an array of length n provides a (1 + δ)-multiplicative approximation to the distance to monotonicity (n minus the length of the LIS), and uses only O((log n)/δ) space. The previous best known approximation using polylogarithmic space was a multiplicative 2-factor. The improved approximation factor reflects a qualitative difference between our algorithm and previous algorithms: previous polylogarithmic space algorithms could not reliably detect increasing subsequences of length as large as n/2, while ours can detect increasing subsequences of length βn for any β > 0. More precisely, our algorithm can be used to estimate the length of the LIS to within an additive δn for any δ > 0 while previous algorithms could only achieve additive error n(1/2− o(1)). Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time. It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS. We also show how our technique can be applied to other problems solvable by dynamic programs. For example, we give a streaming algorithm for approximating LCS(x, y), the length of the longest common subsequence between strings x and y, each of length n. Our algorithm works in the asymmetric setting (inspired by [AKO10]), in which we have random access to y and streaming access to x, and runs in small space provided that no single symbol appears very often in y. More precisely, it gives an additive-δn approximation to LCS(x, y) (and hence also to E(x, y) = n − LCS(x, y), the edit distance between x and y when insertions and deletions, but not substitutions, are allowed), with space complexity O(k(log n)/δ), where k is the maximum number of times any one symbol appears in y. We also provide a deterministic 1-pass streaming algorithm that outputs a (1 + δ)-multiplicative approximation This work was supported in part by NSF under CCF 0832787. This work was supported by the Early Career LDRD program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. for E(x, y) (which is also an additive δn-approximation), in the asymmetric setting, and uses O( √ (n log n)/δ) space. All these algorithms are obtained by carefully trading space and accuracy within a standard dynamic program.