kb/data/en.wikipedia.org/wiki/Maximal_pair-0.md

---
title: "Maximal pair"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Maximal_pair"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:36:08.929308+00:00"
instance: "kb-cron"
---

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 ==

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 
  
    
      
        
          p
          
            1
          
        
      
    
    {\displaystyle p_{1}}
  
 and 
  
    
      
        
          p
          
            2
          
        
      
    
    {\displaystyle p_{2}}
  
 respectively, and both of length 
  
    
      
        l
      
    
    {\displaystyle l}
  
, is specified by a triple 
  
    
      
        (
        
          p
          
            1
          
        
        ,
        
          p
          
            2
          
        
        ,
        l
        )
      
    
    {\displaystyle (p_{1},p_{2},l)}
  
, such that, given a string 
  
    
      
        S
      
    
    {\displaystyle S}
  
 of length 
  
    
      
        n
      
    
    {\displaystyle n}
  
, 
  
    
      
        S
        [
        
          p
          
            1
          
        
        .
        .
        
          p
          
            1
          
        
        +
        l
        −
        1
        ]
        =
        S
        [
        
          p
          
            2
          
        
        .
        .
        
          p
          
            2
          
        
        +
        l
        −
        1
        ]
      
    
    {\displaystyle S[p_{1}..p_{1}+l-1]=S[p_{2}..p_{2}+l-1]}
  
 (meaning that the substrings have identical contents), but 
  
    
      
        S
        [
        
          p
          
            1
          
        
        −
        1
        ]
        ≠
        S
        [
        
          p
          
            2
          
        
        −
        1
        ]
      
    
    {\displaystyle S[p_{1}-1]\neq S[p_{2}-1]}
  
 (they have different characters to their left) and 
  
    
      
        S
        [
        
          p
          
            1
          
        
        +
        l
        ]
        ≠
        S
        [
        
          p
          
            2
          
        
        +
        l
        ]
      
    
    {\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 
  
    
      
        (
        2
        ,
        6
        ,
        3
        )
      
    
    {\displaystyle (2,6,3)}
  
 (the red and blue substrings) and 
  
    
      
        (
        6
        ,
        10
        ,
        3
        )
      
    
    {\displaystyle (6,10,3)}
  
 (the green and blue substrings), and 
  
    
      
        (
        2
        ,
        10
        ,
        3
        )
      
    
    {\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 
  
    
      
        Θ
        (
        n
        +
        z
        )
      
    
    {\displaystyle \Theta (n+z)}
  
 time using a suffix tree, if there are 
  
    
      
        z
      
    
    {\displaystyle z}
  
 such structures.


== References ==


== External links ==
Project for the computation of all maximal repeats in one ore more strings in Python, using suffix array.