Optimal discovery of repetitions in 2D

Authors:
Alberto Apostolico;Valentin E. Brimkov
Affiliations:
Dipartimento di Ingegneria dell' Informazione, Universití di Padova, Via Gradenigo 6A, I-35131 Padova, Italy and Department of Computer Sciences, Purdue University, Computer Sciences Building ...;Mathematics Department, SUNY Buffalo State College, Buffalo, NY 14222, USA
Venue:
Discrete Applied Mathematics - Special issue: IWCIA 2003 - Ninth international workshop on combinatorial image analysis
Year:
2005

Citing 13
Cited 0

An O(n log n) algorithm for finding all repetitions in a string

Journal of Algorithms
Structural properties of the string statistics problem

Journal of Computer and System Sciences
Optimal parallel pattern matching in strings

Information and Control
Transducers and repetitions

Theoretical Computer Science
Two-dimensional periodicity and its applications

SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
An Alphabet Independent Approach to Two-Dimensional Pattern Matching

SIAM Journal on Computing
An Optimal O(log log n)-Time Parallel Algorithm for Detecting all Squares in a String

SIAM Journal on Computing
Double sequences with complexity mn+11

Automatica (Journal of IFAC)
Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences

Discrete Mathematics
Fibonacci arrays and their two-dimensional repetitions

Theoretical Computer Science
On a conjecture on bidimensional words

Theoretical Computer Science
Truly alphabet-independent two-dimensional pattern matching

SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Optimally fast parallel algorithms for preprocessing and pattern matching in one and two dimensions

SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science

Quantified Score

Hi-index	0.00

Visualization

Abstract

Repetitive substructures in two-dimensional arrays emerge in speeding up searches and have been recently studied also independently in an attempt to parallel some of the classical derivations concerning repetitions in strings. The present paper focuses on repetitions in two dimensions that manifest themselves in form of two ''tandem'' occurrences of a same primitive rectangular pattern W where the two replicas touch each other with either one side or corner. Being primitive here means that W cannot be expressed in turn by repeated tiling of another array. The main result of the paper is an O(n^3logn) algorithm for detecting all ''side-sharing'' repetitions in an nxn array. This is optimal, based on bounds on the number of such repetitions established in previous work. With easy adaptations, these constructions lead to an equally optimal, O(n^4) algorithm for repetitions of the second type.