Asymmetric Streaming Algorithms for Edit Distance and LCS

Abstract

The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we consider these problems in the streaming model where one string is available via oracle queries and the other string comes as a stream of characters. Our main contribution is a constant factor approximation algorithm in this setting for ED with memory O(n δ ) for any δ > 0. In addition to this, we present an upper bound of Õ( √ n) on the memory needed to approximate ED or LCS within a factor

Lower Bounds and Improved Algorithms for Asymmetric Streaming Edit Distance and Longest Common Subsequence

“…We then consider deterministic approximation of LCS. Here, the work of [GG10,EJ08] gives a lower bound of Ω 1 R n ε log |Σ| εn for any R pass streaming algorithm achieving a 1 + ε approximation of LIS, which also implies a lower bound of Ω 1 R n ε log 1 ε for asymmetric streaming LCS when |Σ| ≥ n. These bounds match the upper bound in [GJKK07] for LIS and LNS, and in [FHRS20,CJLZ20] for LCS. However, a major drawback of this bound is that it gives nothing when |Σ| is small (e.g., |Σ| ≤ εn).…”

“…Second, because it is a relaxation of the standard streaming model, one can hope to design better algorithms for ED or to beat the strong lower bounds for LCS in this model. The latter point is indeed verified by two recent works [FHRS20,CJLZ20], which give a deterministic one pass algorithm achieving a O(2 1/δ ) approximation of ED, using space Õ(n δ /δ) and time Õδ (n 4 ) for any constant δ > 0, as well as deterministic one pass algorithms achieving 1 ± ε approximation of ED and LCS, using space Õ( √ n ε ) and time Õε (n 2 ). A natural question is how much we can improve these results.…”

“…Edit distance. Our algorithm for edit distance builds on and improves the algorithm in [FHRS20,CJLZ20]. The key idea of that algorithm is to use triangle inequality.…”

“…Let ỹ be the concatenation of y[l i : r i ] from i = 1 to b. Then using triangle inequality, [FHRS20] showed that ED(y, ỹ) + b i=1 d i is a 2α + 1 approximation of ED(x, y). The Õ(n δ ) space is achieved by recursively applying this idea, which results in a O(2 1/δ ) approximation.…”

Optimal Space and Time for Streaming Pattern Matching

“…In this streaming model, we have random access to one string and streaming access to the other. Recently, Farhadi et al [10] presented asymmetric streaming algorithms for edit distance and longest common subsequence. Notably, that work presents an algorithm with a constant factor approximation for edit distance with memory Õ(n δ ) for any δ > 0.…”