Fibonacci arrays and their two-dimensional repetitions

Abstract

Notions related to repetitive substructures in two-dimensional arrays are introduced and studied in an attempt to parallel some of the analogous developments already known for strings. In particular, sequences of “Fibonacci arrays” are defined, capable of exhibiting extremal properties in terms of certain repetitive subpatterns called “tandems”. Two types of tandems are considered. For one type, it is shown that the number of occurrences in an m×n Fibonacci array attains the general upper bound of O(m^2nlogn)

Optimal discovery of repetitions in 2D

“…In a recent work [5], a two-dimensional repetition called a tandem is defined as a configuration consisting of two occurrences of the same (primitive) block that touch each other with one side or corner. Being primitive for a block means that it cannot be expressed itself by repetitive placement of another block.…”

Two-dimensional maximal repetitions

“…In one-dimension, a square is a string which consists of precisely two consecutive occurrences of a substring. Apostolico and Brimkov [3] extend the notion of a square to two dimensions, to form a 2D tandem. They define a 2D tandem as a configuration consisting of two occurrences of the same primitive block that share a side or a corner.…”

“…non-overlapping replicas of some block W [3]. Apostolico and Brimkov prove combinatorially that an n × n matrix can contain Θ(n 4 ) corner-sharing tandems and Θ(n 3 log n) side-sharing tandems [3].…”

“…non-overlapping replicas of some block W [3]. Apostolico and Brimkov prove combinatorially that an n × n matrix can contain Θ(n 4 ) corner-sharing tandems and Θ(n 3 log n) side-sharing tandems [3]. They develop an O(n 3 log n) algorithm for finding side-sharing tandems in an n × n matrix, which can be used to derive an O(n 4 ) algorithm for locating all corner-sharing tandems [4].…”

Optimally Fast CRCW-Pram Testing 2d-Arrays for Existence of Repetitive Patterns

“…An n × n array can contain Θ(n 3 log Φ n) tandems of type (a) and Θ(n 4 ) tandems of Type (b), where Φ is the golden ratio. 5 For more details about tandems the reader is referred to Refs. 5 and 6.…”