Maximal pair

In computer science, a maximal pair within a string is a pair of matching substrings that are maximal, where "maximal" means that it is not possible to make a longer matching pair by extending the range of both substrings to the left or right.

Example

Index	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Character	x	a	b	c	y	a	b	c	w	a	b	c	y	z

For example, in this table, the substrings at indices 2 to 4 (in red) and indices 6 to 8 (in blue) are a maximal pair, because they contain identical characters (abc), and they have different characters to the left (x at index 1 and y at index 5) and different characters to the right (y at index 5 and w at index 9). Similarly, the substrings at indices 6 to 8 (in blue) and indices 10 to 12 (in green) are a maximal pair.

However, the substrings at indices 2 to 4 (in red) and indices 10 to 12 (in green) are not a maximal pair, as the character y follows both substrings, and so they can be extended to the right to make a longer pair.

Formal definition

Formally, a maximal pair of substrings with starting positions ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ respectively, and both of length ${\displaystyle l}$, is specified by a triple ${\displaystyle (p_{1},p_{2},l)}$, such that, given a string ${\displaystyle S}$ of length ${\displaystyle n}$, ${\displaystyle S[p_{1}..p_{1}+l-1]=S[p_{2}..p_{2}+l-1]}$ (meaning that the substrings have identical contents), but ${\displaystyle S[p_{1}-1]\neq S[p_{2}-1]}$ (they have different characters to their left) and ${\displaystyle S[p_{1}+l]\neq S[p_{2}+l]}$ (they also have different characters to their right; together, these two inequalities are the condition for being maximal). Thus, in the example above, the maximal pairs are ${\displaystyle (2,6,3)}$ (the red and blue substrings) and ${\displaystyle (6,10,3)}$ (the green and blue substrings), and ${\displaystyle (2,10,3)}$ is not a maximal pair.

Related concepts and time complexity

A maximal repeat is the string represented by a maximal pair. A supermaximal repeat is a maximal repeat never occurring as a proper substring of another maximal repeat. In the above example, abc and abcy are both maximal repeats, but only abcy is a supermaximal repeat.

Maximal pairs, maximal repeats and supermaximal repeats can each be found in ${\displaystyle \Theta (n+z)}$ time using a suffix tree,^{[1][dead link]} if there are ${\displaystyle z}$ such structures.