Ohad Lipsky

Pattern matching with address errors: rearrangement distances

Historically, approximate pattern matching has mainly focused at coping with errors in the data, while the order of the text/pattern was assumed to be more or less correct. In this paper we consider a class of pattern matching problems where the content is assumed to be correct, while the locations may have shifted/changed. We formally define a broad class of problems of this type, capturing situations in which the pattern is obtained from the text by a sequence of rearrangements. We consider several natural rearrangement schemes, including the analogues of the l 1 and l 2 distances, as well as two distances based on interchanges. For these, we present efficient algorithms to solve the resulting string matching problems.

Approximate matching in the L 1 metric

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.

Improved sketching of hamming distance with error correcting

\sum_{j=0}^{m-1} |{t}_{i+j} - {p}_{j}| \leq k

Pattern matching with address errors: Rearrangement distances

. We provide an algorithm that solves the k-L1-distance problem in time

Closest Pair Problems in Very High Dimensions

O(n\sqrt{k\log k})

Approximate Pattern Matching with the L 1 , L 2 and L ∞ Metrics

. The algorithm applies a bounded divide-and-conquer approach and makes novel uses of non-boolean convolutions.

The approximate swap and mismatch edit distance

We address the problemof sketching the hamming distance of data streams. We develop Fixable Sketches which compare data streams or files and restore the differences between them. Our contribution: For two streams with hamming distance bounded by k we show a sketch of size O(k log n) with O(log n) processing time per new element in the stream and how to restore all locations where the two streams differ in time linear in the sketch size. Probability of error is less than 1/n.

Approximated Pattern Matching with the L 1 , L 2 and L ∞ Metrics

Historically, approximate pattern matching has mainly focused at coping with errors in the data, while the order of the text/pattern was assumed to be more or less correct. In this paper we consider a class of pattern matching problems where the content is assumed to be correct, while the locations may have shifted/changed. We formally define a broad class of problems of this type, capturing situations in which the pattern is obtained from the text by a sequence of rearrangements. We consider several natural rearrangement schemes, including the analogues of the l1 and l2 distances, as well as two distances based on interchanges. For these, we present efficient algorithms to solve the resulting string matching problems.

Approximate string matching with swap and mismatch

The problem of finding the closest pair among a collection of points in \(\Re^d\) is a well-known problem. There are better-than-naive-solutions for constant d and approximate solutions in general. We propose the first better-than-naive-solutions for the problem for large d. In particular, we present algorithms for the metrics L 1 and L ∞ with running times of O(n (ω + 3)/2) and O(n (ω + 3)/2logD) respectively, where O(n ω ) is the running time of matrix multiplication and D is the diameter of the points.

Approximate swap and mismatch edit distance

AbstractGiven an alphabet Σ={1,2,…,|Σ|} text string T∈Σn and a pattern string P∈Σm, for each i=1,2,…,n−m+1 define Lp(i) as the p-norm distance when the pattern is aligned below the text and starts at position i of the text. The problem of pattern matching with Lp distance is to compute Lp(i) for every i=1,2,…,n−m+1. We discuss the problem for d=1,2,∞. First, in the case of L1 matching (pattern matching with an L1 distance) we show a reduction of the string matching with mismatches problem to the L1 matching problem and we present an algorithm that approximates the L1 matching up to a factor of 1+ε, which has an