Streaming algorithms for embedding and computing edit distance in the low distance regime

Abstract

The edit distance is a way of quantifying how similar two strings are to one another by counting the minimum number of character insertions, deletions, and substitutions required to transform one string into the other.In this paper we study the computational problem of computing the edit distance between a pair of strings where their distance is bounded by a parameter k ≪ n. We present two streaming algorithms for computing edit distance: One runs in time O(n + k 2 ) and the other n + O(k 3 ). By writing n + O(k 3 ) we want to emphasize that the number of operations per an input symbol is a small constant. In particular, the running time does not depend on the alphabet size, and the algorithm should be easy to implement.Previously a streaming algorithm with running time O(n + k 4 ) was given in the paper by the current authors (STOC'16). The best off-line algorithm runs in time O(n + k 2 ) (Landau et al., 1998) which is known to be optimal under the Strong Exponential Time Hypothesis. ogy, pattern recognition, text processing, information retrieval and many more. The edit distance between x and y, denoted by ∆(x, y), is defined as the minimum number of character insertions, deletions, and substitutions needed for converting x into y. Due to its immense applicability, the computational problem of computing the edit distance between two given strings x and y ∈ Σ n is of prime interest to researchers in various domains of computer science. Sometimes one also requires that the algorithm finds an alignment of x and y, i.e., a series of edit operations that transform x into y.In this paper we study the problem of computing edit distance of strings when given an a priori upper bound k ≪ n on their distance. This is akin to fixed parameter tractability. Arguably, the case when the edit distance is small relative to the length of the strings is the most interesting as when comparing two strings with respect to their edit distance we are implicitly making an assumption that the strings are similar. If they are not similar the edit distance is uninformative. There are few exceptions to this rule, most notably the reduction of instances of formula satisfiability (SAT) to instances of edit distance of exponentially large strings [BI15] where the edit distance of resulting strings is close to their length. However, such instance of the edit distance problem are rather artificial. For typical applications the edit distance of the two strings is much smaller then the length of the strings. Consider for example copying DNA during cell division: Human DNA is essentially a string of about 10 9 letters from {A, C, G, T }, and due to imperfections in the copying mechanism one can expect about 50 edit operations to occur during the process. So in many applications we can be looking for a handful of edit operations in large strings.Landau et al.[LMS98] provided an algorithm that runs in time O(n + k 2 ) and uses space O(n) when size of the alphabet Σ is constant. In general the running time of the algorithm given in [LMS98] is O(n · mi...

Optimal Document Exchange and New Codes for Insertions and Deletions

“…Independently developed fuzzy extractors [9] can also be seen as providing a document exchange scheme for some k polynomially small in n. A randomized scheme by Jowhari [10] independently achieved a size of O(k log n log * n). In two recent break-throughs Chakraborty, Goldenberg, and Kouckỳ [11] designed a low distortion embedding from edit distance to hamming distance which can be used to get a summary of size Θ(k 2 log n) and Bellazougi and Zhang [2] further build on this randomized embedding and achieved a scheme with summary size Θ(k log 2 k + k log n) which is order optimal for 4 k = exp( √ log n). All of these schemes are randomized.…”

Sublinear Algorithms for Gap Edit Distance

“…for the Ulam metric (edit distance with no repetition, which obviously requires a large alphabet) distinguish between t vs Θ(t) in O( n t + √ n) time, achieving a bound that is similar to the folklore sampling algorithm for approximating Hamming distance. There is a long line of work on edit distance and related problems, aiming to achieve fast running time [AN10, AIKH13, Sah17, BEG + 18, HSSS19], low distortion embedding [OR07, KR06,CGK16b,BZ16], small space complexity [CGK16b, BZ16, BJKK04] and parallel algorithms [HSS19]. The work of Andoni, Onak and Krauthgamer [AKO10] achieves a sublinear asymmetric query complexity for approximating edit distance; however it does not lead to any sublinear time algorithm since one of the strings must be read in its entirety.…”

“…Their algorithm computes a constant-size sketch but still requires a linear pass over the data. This result was later improved to hold for general strings [CGK16b] via embedding into Hamming distance, but again in linear time.…”

Space-Efficient Feature Maps for String Alignment Kernels

“…Then, random number u returned from F unc F is converted to a random number from the Cauchy distribution in F unc H as tan(π · (u − 0.5))/β at line 8. [2], [33] is another string embedding using a randomized algorithm. Let S i for i = 1,2,...,N be input strings of alphabet Σ and let L be the maximum length of input strings.…”