Estrella Eisenberg

Swap and Mismatch Edit Distance

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.

Cycle detection and correction

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.

Swap and mismatch edit distance

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).

Closest periodic vectors in Lp spaces

The problem of finding the period of a vector V is central to many applications. Let V′ be a periodic vector closest to V under some metric. We seek this V′, or more precisely we seek the smallest period that generates V′. In this paper we consider the problem of finding the closest periodic vector in L p spaces. The measures of “closeness” that we consider are the metrics in the different L p spaces. Specifically, we consider the L 1, L 2 and L ∞ metrics. In particular, for a given n-dimensional vector V, we develop O(n 2) time algorithms (a different algorithm for each metric) that construct the smallest period that defines such a periodic n-dimensional vector V′. We call that vector the closest periodic vector of V under the appropriate metric. We also show (three) O(n logn) time constant approximation algorithms for the (appropriate) period of the closest periodic vector.