Наибольшая общая подстрока: алгоритмы и применение

Computer science problem

In computer science, a longest common substring of two or more strings is a longest string that is a substring of all of them. There may be more than one longest common substring. Applications include data deduplication and plagiarism detection.

Given two strings, of length and of length , find a longest string which is substring of both and
A generalization is the k common substring problem. Given the set of strings , where and Find for each , a longest string which occurs as substring of at least strings.

One can find the lengths and starting positions of the longest common substrings of and in time with the help of a generalized suffix tree. A faster algorithm can be achieved in the word RAM model of computation if the size of the input alphabet is in In particular, this algorithm runs in time using space. Solving the problem by dynamic programming costs The solutions to the generalized problem take space and time with dynamic programming and take time with a generalized suffix tree.

The longest common substrings of a set of strings can be found by building a generalized suffix tree for the strings, and then finding the deepest internal nodes which have leaf nodes from all the strings in the subtree below it. The figure on the right is the suffix tree for the strings "ABAB", "BABA" and "ABBA", padded with unique string terminators, to become "ABAB$0", "BABA$1" and "ABBA$2". The nodes representing "A", "B", "AB" and "BA" all have descendant leaves from all of the strings, numbered 0, 1 and 2. Building the suffix tree takes time (if the size of the alphabet is constant). If the tree is traversed from the bottom up with a bit vector telling which strings are seen below each node, the k common substring problem can be solved in time. If the suffix tree is prepared for constant time lowest common ancestor retrieval, it can be solved in time.