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).

The k-mismatch problem revisited

Property matching and weighted matching

Pattern Matching with Properties (Property Matching, for short), involves a string matching between the pattern and the text, and the requirement that the text part satisfies some property. It is straightforward to do sequential matching in a text with properties. However, indexing in a text with properties becomes difficult if we desire the time to be output dependent. We present an algorithm for indexing a text with properties in O(n log|Σ + n log log n) time for preprocessing and O(|P| log |Σ| + tocc π ) per query, where n is the length of the text, P is the sought pattern, E is the alphabet, and tocc π is the number of occurrences of the pattern that satisfy some property π. As a practical use of Property Matching we show how to solve Weighted Matching problems using techniques from Property Matching. Weighted sequences have been introduced as a tool to handle a set of sequences that are not identical but have many local similarities. The weighted sequence is a statistical image of this set, where we are given the probability of every symbol's occurrence at every text location. Weighted matching problems are pattern matching problems where the given text is weighted. We present a reduction from Weighted Matching to Property Matching that allows off-the-shelf solutions to numerous weighted matching problems including indexing, swapped matching, parameterized matching, approximate matching, and many more. Assuming that one seeks the occurrence of pattern P with probability ∈ in weighted text T of length n, we reduce the problem to a property matching problem of pattern P in text T' of length O(n(1 6) 2 log 1 6).

Approximate matching is one of the fundamental problems in pattern matching, and a ubiquitous problem in real applications. The Hamming distance is a simple and well studied example of approximate matching, motivated by typing, or noisy channels. Biological and image processing applications assign a different value to mismatches of different symbols.

We consider the problem of approximate matching in the L1 metric – the k-L1-distance problem. Given text T=t0,...,tn−1 and pattern P=p0,...,pm−1 strings of natural number, and a natural number k, we seek all text locations i where the L1 distance of the pattern from the length m substring of text starting at i is not greater than k, i.e. $\sum_{j=0}^{m-1} |{t}_{i+j} - {p}_{j}| \leq k$.

We provide an algorithm that solves the k-L1-distance problem in time $O(n\sqrt{k\log k})$. The algorithm applies a bounded divide-and-conquer approach and makes novel uses of non-boolean convolutions.

Approximate matching in the L 1 metric

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:• Our first result is a deterministic O(nk2 log k/m + n 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 [9, 10].• We then give a randomised and online algorithm which runs in the same time complexity but requires only O(k2 polylog m) space in total.• Next we give a randomised (1 + e)-approximation algorithm for the streaming k-mismatch problem which uses O(k2 polylog m/e2) space and runs in O(polylog m/e2) worst-case time per arriving symbol.• Finally we combine our new results to derive a randomised O(k2 polylog m) space algorithm for the streaming k-mismatch problem which runs in O([EQUATION] log k + polylog m) worst-case time per arriving symbol. This improves the best previous space complexity for streaming k-mismatch from FOCS 2009 [26] 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).

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There is no known algorithm that solves the general case of theapproximate string matching problem with the extended edit distance, where the edit operations are: insertion, deletion, mismatch and swap, in timeo(nm), wheren is the length of the text andm is the length of the pattern. In an effort to study this problem, the edit operations were analysed independently. It turns out that the approximate matching problem with only the mismatch operation can be solved in timeO(n √m logm). If the only edit operation allowed is swap, then the problem can be solved in timeO(n logm logσ), whereσ=min(m, |Σ|). In this paper we show that theapproximate string matching problem withswap andmismatch as the edit operations, can be computed in timeO(n √m logm).

Swap and mismatch edit distance

Assume that a natural cyclic phenomenon has been measured, but the data is corrupted by errors. The type of corruption is application-dependent and may be caused by measurements errors, or natural features of the phenomenon. We assume that an appropriate metric exists, which measures the amount of corruption experienced. This article studies the problem of recovering the correct cycle from data corrupted by various error models, formally defined as the period recovery problem. Specifically, we define a metric property which we call pseudolocality and study the period recovery problem under pseudolocal metrics. Examples of pseudolocal metrics are the Hamming distance, the swap distance, and the interchange (or Cayley) distance. We show that for pseudolocal metrics, periodicity is a powerful property allowing detecting the original cycle and correcting the data, under suitable conditions. Some surprising features of our algorithm are that we can efficiently identify the period in the corrupted data, up to a number of possibilities logarithmic in the length of the data string, even for metrics whose calculation is NP-hard. For the Hamming metric, we can reconstruct the corrupted data in near-linear time even for unbounded alphabets. This result is achieved using the property of separation in the self-convolution vector and Reed-Solomon codes. Finally, we employ our techniques beyond the scope of pseudo-local metrics and give a recovery algorithm for the non-pseudolocal Levenshtein edit metric.

Cycle detection and correction

Finding an approximate period in a given string S of length n is defined as follows. Let S ' be a periodic string closest to S under some distance metric, find the smallest period of S ' . This period is called an approximate period of S under the given metric. Let the distance between the input string S and a closest periodic string under the Hamming distance S ' be k. We develop algorithms that construct an approximate period of S under the Hamming distance in time O ( n k log ? log ? n ) and under the swap distance in time O ( n 2 ) . Finally, we show an O ( n log ? n ) algorithm for finite alphabets, and an O ( n log 3 ? n ) algorithm for infinite alphabets, that approximate the minimum number of mismatches between the input string and a closest periodic string under the Hamming distance.

Approximate periodicity

There is no known algorithm that solves the general case of approximate string matching problem with the extended edit distance, where the edit operations are: insertion, deletion, mismatch, and swap, in time o(nm), where n is the length of the text and m is the length of the pattern.

Assume that a natural cyclic phenomenon has been measured, but the data is corrupted by errors. The type of corruption is application-dependent and may be caused by measurements errors, or natural features of the phenomenon. This paper studies the problem of recovering the correct cycle from data corrupted by various error models, formally defined as the period recovery problem. Specifically, we define a metric property which we call pseudo-locality and study the period recovery problem under pseudo-local metrics. Examples of pseudo-local metrics are the Hamming distance, the swap distance, and the interchange (or Cayley) distance. We show that for pseudo-local metrics, periodicity is a powerful property allowing detecting the original cycle and correcting the data, under suitable conditions. Some surprising features of our algorithm are that we can efficiently identify the period in the corrupted data, up to a number of possibilities logarithmic in the length of the data string, even for metrics whose calculation is NP-hard. For the Hamming metric we can reconstruct the corrupted data in near linear time even for unbounded alphabets. This result is achieved using the property of separation in the self-convolution vector and Reed-Solomon codes. Finally, we employ our techniques beyond the scope of pseudo-local metrics and give a recovery algorithm for the non pseudo-local Levenshtein edit metric.

Estrella Eisenberg

Papers

Swap and mismatch edit distance

Cycle detection and correction

Approximate periodicity

Swap and mismatch edit distance

Cycle detection and correction