Finding a set of repeated, non-overlapping substrings of two input strings using suffix arrays

Question

Input: two strings A and B.

Output: a set of repeated, non overlapping substrings

I have to find all the repeated strings, each of which has to occur in both(!) strings at least once. So for instance, let

A = "xyabcxeeeyabczeee" and B = "yxabcxabee".

Then a valid output would be {"abcx","ab","ee"} but not "eee", since it occurs only in string A.

I think this problem is very related to the "supermaximal repeat" problem. Here is a definition:

Maximal repeated pair : A pair of identical substrings alpha and beta in S such that extending alpha and beta in either direction would destroy the equality of the two strings It is represented as a triplet (position1,position2, length)

Maximal repeat : “A substring of S that occurs in a maximal pair in S”. Example: abc in S = xabcyiiizabcqabcyrxar. Note: There can be numerous maximal repeated pairs, but there can be only a limited number of maximal repeats.

Supermaximal repeat “A maximal repeat that never occurs as a substring of any other maximal repeat” Example: abcy in S = xabcyiiizabcqabcyrxar.

An algorithm for finding all supermaximal repeats is described in "Algorithms on strings, trees and sequences", but only for suffix trees.

It works by: 1.) finding all left-diverse nodes using DFS

For each position i in S, S(i-1) is called the left character i. Left character of a leaf in T(S) is the left character of the suffix position represented by that leaf. An internal node v in T(S) is called left-diverse if at least two leaves in v’s subtree have different left characters.

2.) applying theorem 7.12.4 on those nodes:

A left diverse internal node v represents a supermaximal repeat a if and only if all of v's children are leaves, and each has a distinct left character

Both strings A and B probably have to be concatenated and when we check v's leaves in step two we also have to impose an additional constraint, that there has to be at least one distinct left character from strings A and B. This can be done by comparing their position against the length of A. If position(left character) > length(A), then left character is in A, else in B.

Can you help me solve this problem with suffix + lcp arrays?

score 0 · Answer 1 · answered Jul 25 '13 at 18:21

It sounds like you are looking for the set intersection of all substrings of your two input strings. In that case, single letter substrings should also be returned. Let s1 and s2 be your strings, s1 the shorter of the two. After doing some thinking for a while about this, I don't think you can get much better than the intuitive O(n^3m) or O(n^3) algorithm, where n is the length of s1 and m is the length of s2. I don't think suffix trees can help you here.

for(int i=0 to n-1){
    for(int j=1 to n-i){
        if(contains(s2,substring(s1,i,j))) emit the substring
    }
}

The runtime comes from the (n^2)/2 loop iterations, each doing a worst-case O(nm) contains operation (possibly O(n) depending on implementation). But its not really quite this bad since there will be a constant much smaller than one out front, since the length of the substring will actually range between 1 and n.
If you don't want single character matches, you could just initialize j as 2 or something higher.

BTW: Don't actually create new strings with substring, find/create a contains function that will take indicies and the original string and just look at the characters between, inclusive of i, exclusive of j.

Finding a set of repeated, non-overlapping substrings of two input strings using suffix arrays

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