On the cyclic regularities of strings

Detecting regularities on grammar-compressed strings

We solve the problems of detecting and counting various forms of regularities in a string represented as a Straight Line Program (SLP). Given an SLP of size $n$ that represents a string $s$ of length $N$, our algorithm compute all runs and squares in $s$ in $O(n^3h)$ time and $O(n^2)$ space, where $h$ is the height of the derivation tree of the SLP. We also show an algorithm to compute all gapped-palindromes in $O(n^3h + gnh\log N)$ time and $O(n^2)$ space, where $g$ is the length of the gap. The key technique of the above solution also allows us to compute the periods and covers of the string…

Computing Abelian regularities on RLE strings

Two strings x and y are said to be Abelian equivalent if x is a permutation of y, or vice versa. If a string z satisfies z = xy with x and y being Abelian equivalent, then z is said to be an Abelian square. If a string w can be factorized into a sequence v_1,...,v_s of strings such that v_1 ,..., v_{s-1} are all Abelian equivalent and vs is a substring of a permutation of v_1, then w is said to have a regular Abelian period (p,t) where p = |v_1| and t = |v_s|. If a substring w_1[i..i+l-1] of a string w_1 and a substring w_2[j..j+l-1] of another string w_2 are Abelian equivalent, then the subst…

An $O(n^2)$ Algorithm for Computing Longest Common Cyclic Subsequence

The {\em longest common subsequence (LCS)} problem is a classic and well-studied problem in computer science. LCS is a central problem in stringology and finds broad applications in text compression, error-detecting codes and biological sequence comparison. However, in numerous contexts, words represent cyclic sequences of symbols and LCS must be generalized to consider all circular shifts of the strings. This occurs especially in computational biology when genetic material is sequenced form circular DNA or RNA molecules. This initiates the problem of {\em longest common cyclic subsequence (LC…

String Inference from the LCP Array

The suffix array, perhaps the most important data structure in modern string processing, is often augmented with the longest common prefix (LCP) array which stores the lengths of the LCPs for lexicographically adjacent suffixes of a string. Together the two arrays are roughly equivalent to the suffix tree with the LCP array representing the tree shape. In order to better understand the combinatorics of LCP arrays, we consider the problem of inferring a string from an LCP array, i.e., determining whether a given array of integers is a valid LCP array, and if it is, reconstructing some string or…

使用采样高效计算周期和覆盖

Efficient Computation of Periods and Covers Using Sampling

本文提出利用字符距离采样（CDS）技术，仅选首字符为枢轴，高效计算字符串的周期与最短覆盖，分别实现38%-43%和63%-72%的加速，展示了CDS在字符串分析中的潜力。

Solving Cyclic Longest Common Subsequence in Quadratic Time

We present a practical algorithm for the cyclic longest common subsequence (CLCS) problem that runs in O(mn) time, where m and n are the lengths of the two input strings. While this is not necessarily an asymptotic improvement over the existing record, it is far simpler to understand and to implement.