Aho-Corasick algorithm - Software Development PDF Download

Introduction

String matching is a fundamental problem in computer science and is widely used in various applications such as text processing, pattern recognition, and data mining. The Aho-Corasick algorithm is an efficient algorithm for solving the string matching problem. It was developed by Alfred V. Aho and Margaret J. Corasick in 1975. The Aho-Corasick algorithm combines the efficiency of a trie data structure with the power of finite automata to perform multiple pattern matching in a given text efficiently.

Construction of the Trie

The Aho-Corasick algorithm begins with the construction of a trie data structure. A trie is a tree-like data structure that stores a collection of strings, where each node represents a character and the edges represent the transitions between characters.
The trie is constructed as follows:

1. Initialize an empty root node of the trie.
2. For each pattern string in the given set of patterns:
   • Start from the root node.
   • For each character in the pattern string:
     • If there is no edge labeled with the current character, create a new node and add an edge labeled with the character from the current node to the new node.
     • Move to the new node.
   • Mark the last node of the pattern string as a pattern-ending node.

Construction of an Automaton

Once the trie is constructed, we enhance it with the ability to efficiently transition between different patterns. This is achieved by adding failure links and output links to the trie nodes.
The construction of the automaton involves the following steps:

1. Initialize the failure links of all nodes in the trie to point back to the root node.
2. Perform a breadth-first search (BFS) traversal of the trie starting from the root node.
3. For each node 'v' visited during the BFS traversal, do the following:
   • For each child node 'u' of 'v':
     • Set the failure link of 'u' to the longest proper suffix node of 'v' that is reachable through the failure links. If such a suffix node does not exist, set the failure link of 'u' to the root node.
     • If the failure link of 'u' points to a pattern-ending node, mark 'u' as an output node.
     • Enqueue 'u' for the next level of BFS traversal.

BFS-based Construction

The construction of the automaton using a breadth-first search (BFS) traversal ensures that the failure links are correctly set. It guarantees the correctness of the algorithm and efficient matching during runtime.
The BFS-based construction follows the following steps:

1. Create a queue data structure to perform the BFS traversal.
2. Enqueue the root node into the queue.
3. While the queue is not empty:
   • Dequeue a node 'v' from the queue.
   • For each child node 'u' of 'v':
     • Enqueue 'u' into the queue.
     • Set the failure link of 'u' based on the steps mentioned earlier.
     • Set the output link of 'u' if 'u' is an output node.

Applications

The Aho-Corasick algorithm has several applications in string matching problems. Some of the common applications include:

1. Find all strings from a given set in a text:
   • Given a set of patterns and a text, the Aho-Corasick algorithm efficiently finds all occurrences of the patterns in the text.
2. Finding the lexicographically smallest string of a given length that doesn't match any given strings:
   • Given a set of patterns and a length L, the algorithm finds the lexicographically smallest string of length L that does not contain any of the patterns as a substring.
3. Finding the shortest string containing all given strings:
   • Given a set of patterns, the algorithm finds the shortest string that contains all the patterns as substrings.
4. Finding the lexicographically smallest string of length L containing k strings:
   • Given a set of patterns and two integers L and k, the algorithm finds the lexicographically smallest string of length L that contains at least k patterns as substrings.

Practice Problems with Solutions

Problem 1: Given a set of strings and a text, count the occurrences of each pattern in the text.

To solve this problem, construct the Aho-Corasick automaton and count the occurrences of each pattern by traversing the automaton.

Problem 2: Given a set of strings, find the lexicographically smallest string that doesn't match any of the given strings.

Construct the Aho-Corasick automaton and perform a depth-first search (DFS) traversal to find the lexicographically smallest string that doesn't match any pattern.

Problem 3: Given a set of strings, find the shortest string that contains all the given strings as substrings.

Construct the Aho-Corasick automaton and use dynamic programming to find the shortest string that contains all patterns.

Conclusion

In conclusion, the Aho-Corasick algorithm is a powerful tool for efficient string matching in competitive programming. By constructing a trie-based automaton, it provides fast pattern matching capabilities with various applications. Understanding and implementing this algorithm will undoubtedly enhance your problem-solving skills in string-related tasks.