Cycle detection and correction

Abstract

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.

Adaptive Exact Learning in a Mixed-Up World: Dealing with Periodicity, Errors and Jumbled-Index Queries in String Reconstruction

“…We say that S is a d-corrupted periodic string if there exists a periodic string S ′ of period p, such that |S| = |S ′ | and δ(S ′ , S) ≤ d, where δ is the Hamming distance. We refer to p as an approximate period of S. Notice that, depending on d, there might exist multiple possible strings S ′ that originate S. We are interested in reconstructing S, as opposed to S ′ , since we can use one of the existing algorithms to enumerate all possible strings S ′ (see [5,7,62]), without incurring additional queries.…”

“…Regarding corrupted periodic strings, different applications suffer from different types of corruption. In particular, the following error metrics have been considered in the literature: Pseudo-local metrics such as swap distance [6] or Interchange (Cayley) distance [7]; and the Levenshtein edit distance [68]. It would be interesting to see whether our reconstruction algorithms can be adapted to these more general error distances.…”

Faster Recovery of Approximate Periods over Edit Distance

“…The following fact characterizes T in terms of ED. 1 Also the APR problem under the Hamming distance was considered [2] for which an O(n log n)-time algorithm was presented [1] for the threshold n (2+ )·p with > 0. Proof.…”

“…1 Also the APR problem under the Hamming distance was considered [2] for which an O(n log n)-time algorithm was presented [1] for the threshold Fact 4. We have min{ED(S, U ∞ ) :…”

“…[8,10]). This task can be stated as the approximate period recovery (APR) problem that was defined by Amir et al [2]. In this problem, we are given a word; we suspect that it was initially periodic, but then errors might have been introduced in it.…”