Palindromic Decompositions with Gaps and Errors

Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes. We first present an algorithm for obtaining a palindromic decomposition of a string of length n with the minimal total gap length in time O(n log n * g) and space O(n g), where g is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal \delta-palindromes (i.e. palindromes with \delta errors under the edit or Hamming distance) and g allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time O(n (g + \delta)) and space O(n g).Comment: accepted to CSR 201

Finding approximate palindromes in strings

We introduce a novel definition of approximate palindromes in strings, and provide an algorithm to find all maximal approximate palindromes in a string with up to $k$ errors. Our definition is based on the usual edit operations of approximate pattern matching, and the algorithm we give, for a string of size $n$ on a fixed alphabet, runs in $O(k^2 n)$ time. We also discuss two implementation-related improvements to the algorithm, and demonstrate their efficacy in practice by means of both experiments and an average-case analysis

Palindrome Recognition In The Streaming Model

In the Palindrome Problem one tries to find all palindromes (palindromic substrings) in a given string. A palindrome is defined as a string which reads forwards the same as backwards, e.g., the string "racecar". A related problem is the Longest Palindromic Substring Problem in which finding an arbitrary one of the longest palindromes in the given string suffices. We regard the streaming version of both problems. In the streaming model the input arrives over time and at every point in time we are only allowed to use sublinear space. The main algorithms in this paper are the following: The first one is a one-pass randomized algorithm that solves the Palindrome Problem. It has an additive error and uses $O(\sqrt n$) space. The second algorithm is a two-pass algorithm which determines the exact locations of all longest palindromes. It uses the first algorithm as the first pass. The third algorithm is again a one-pass randomized algorithm, which solves the Longest Palindromic Substring Problem. It has a multiplicative error using only $O(\log(n))$ space. We also give two variants of the first algorithm which solve other related practical problems

Comparing Degenerate Strings

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string S is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1,... , kn-1. Our main result is an (N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in S in (min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in S. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach

Parallel Detection of all Palindromes in a String

This paper presents two efficient concurrent-read concurrent-write parallel algorithms that find all palindromes in a given string: 1. An O(log n) time, n-processor algorithm over general alphabets. In case of constant size alphabets the algorithm requires only n= log n processors, and thus achieves an optimal-speedup. 2. An O(log log n) time, n log n= log log n-processor algorithm over general alphabets. This is the fastest possible time with the number of processors used. These new results improve on the known parallel palindrome detection algorithms by using smaller auxiliary space and either by making fewer operations or by achieving a faster running time. 1 Introduction Palindromes are symmetric strings that read the same forward and backward. Palindromes have been studied for centuries as word puzzles and more recently have found several important uses in formal languages and computability theory. Formally, a non-empty string w is a palindrome if w = w R , where w R denotes..