The k-mismatch problem revisited

We revisit the complexity of one of the most basic problems in pattern matching. In the k-mismatch problem we must compute the Hamming distance between a pattern of length m and every m-length substring of a text of length n, as long as that Hamming distance is at most k. Where the Hamming distance is greater than k at some alignment of the pattern and text, we simply output "No". We study this problem in both the standard offline setting and also as a streaming problem. In the streaming k-mismatch problem the text arrives one symbol at a time and we must give an output before processing any future symbols. Our main results are as follows: 1) Our first result is a deterministic $O(n k^2\log{k} / m+n \text{polylog} m)$ time offline algorithm for k-mismatch on a text of length n. This is a factor of k improvement over the fastest previous result of this form from SODA 2000 by Amihood Amir et al. 2) We then give a randomised and online algorithm which runs in the same time complexity but requires only $O(k^2\text{polylog} {m})$ space in total. 3) Next we give a randomised $(1+\epsilon)$-approximation algorithm for the streaming k-mismatch problem which uses $O(k^2\text{polylog} m / \epsilon^2)$ space and runs in $O(\text{polylog} m / \epsilon^2)$ worst-case time per arriving symbol. 4) Finally we combine our new results to derive a randomised $O(k^2\text{polylog} {m})$ space algorithm for the streaming k-mismatch problem which runs in $O(\sqrt{k}\log{k} + \text{polylog} {m})$ worst-case time per arriving symbol. This improves the best previous space complexity for streaming k-mismatch from FOCS 2009 by Benny Porat and Ely Porat by a factor of k. We also improve the time complexity of this previous result by an even greater factor to match the fastest known offline algorithm (up to logarithmic factors)

String Matching with Variable Length Gaps

We consider string matching with variable length gaps. Given a string $T$ and a pattern $P$ consisting of strings separated by variable length gaps (arbitrary strings of length in a specified range), the problem is to find all ending positions of substrings in $T$ that match $P$. This problem is a basic primitive in computational biology applications. Let $m$ and $n$ be the lengths of $P$ and $T$, respectively, and let $k$ be the number of strings in $P$. We present a new algorithm achieving time $O(n\log k + m +\alpha)$ and space $O(m + A)$, where $A$ is the sum of the lower bounds of the lengths of the gaps in $P$ and $\alpha$ is the total number of occurrences of the strings in $P$ within $T$. Compared to the previous results this bound essentially achieves the best known time and space complexities simultaneously. Consequently, our algorithm obtains the best known bounds for almost all combinations of $m$, $n$, $k$, $A$, and $\alpha$. Our algorithm is surprisingly simple and straightforward to implement. We also present algorithms for finding and encoding the positions of all strings in $P$ for every match of the pattern.Comment: draft of full version, extended abstract at SPIRE 201

Pattern Matching in Multiple Streams

We investigate the problem of deterministic pattern matching in multiple streams. In this model, one symbol arrives at a time and is associated with one of s streaming texts. The task at each time step is to report if there is a new match between a fixed pattern of length m and a newly updated stream. As is usual in the streaming context, the goal is to use as little space as possible while still reporting matches quickly. We give almost matching upper and lower space bounds for three distinct pattern matching problems. For exact matching we show that the problem can be solved in constant time per arriving symbol and O(m+s) words of space. For the k-mismatch and k-difference problems we give O(k) time solutions that require O(m+ks) words of space. In all three cases we also give space lower bounds which show our methods are optimal up to a single logarithmic factor. Finally we set out a number of open problems related to this new model for pattern matching.Comment: 13 pages, 1 figur

Online Pattern Matching for String Edit Distance with Moves

Edit distance with moves (EDM) is a string-to-string distance measure that includes substring moves in addition to ordinal editing operations to turn one string to the other. Although optimizing EDM is intractable, it has many applications especially in error detections. Edit sensitive parsing (ESP) is an efficient parsing algorithm that guarantees an upper bound of parsing discrepancies between different appearances of the same substrings in a string. ESP can be used for computing an approximate EDM as the L1 distance between characteristic vectors built by node labels in parsing trees. However, ESP is not applicable to a streaming text data where a whole text is unknown in advance. We present an online ESP (OESP) that enables an online pattern matching for EDM. OESP builds a parse tree for a streaming text and computes the L1 distance between characteristic vectors in an online manner. For the space-efficient computation of EDM, OESP directly encodes the parse tree into a succinct representation by leveraging the idea behind recent results of a dynamic succinct tree. We experimentally test OESP on the ability to compute EDM in an online manner on benchmark datasets, and we show OESP's efficiency.Comment: This paper has been accepted to the 21st edition of the International Symposium on String Processing and Information Retrieval (SPIRE2014