Aho-Corasick Algorithm
In computer science, the Aho-Corasick Algorithm is a string searching algorithm that efficiently finds multiple patterns in a given text. It constructs a finite-state machine in the form of a trie (prefix tree) with failure links, allowing it to search for all patterns simultaneously in linear time.
Overview
The Aho-Corasick Algorithm is a classical string matching algorithm used for efficiently finding multiple patterns within a given text. It constructs a finite-state machine in the form of a trie (prefix tree) with failure links, which allows it to search for all patterns simultaneously in linear time.
Video Explanation
Time Complexity:
- Construction Time: O(m), where
mis the total number of characters in all patterns. - Search Time: O(n), where
nis the length of the text.
How It Works
- Trie Construction: All patterns are inserted into a trie where each node represents a character in the patterns.
- Failure Links: Failure links are added to each node, pointing to the longest suffix which is also a prefix. This allows the algorithm to efficiently transition between nodes when a mismatch occurs.
- Pattern Search: The text is processed one character at a time, following the trie structure to match the patterns. If a mismatch occurs, failure links are used to skip unnecessary comparisons.
Example
Given patterns ["he", "she", "his", "hers"] and the text "ushers", the Aho-Corasick algorithm constructs a trie with failure links to search for these patterns. The algorithm finds matches for "he", "she", and "hers".
Code Implementation
Python
from collections import deque, defaultdict
class AhoCorasick:
def __init__(self, patterns):
"""Initialize the Aho-Corasick automaton for the given patterns."""
self.trie = {}
self.output = defaultdict(list)
self.fail = {}
# Build the trie
self._build_trie(patterns)
# Add failure links
self._build_failure_links()
def _build_trie(self, patterns):
"""Insert patterns into the trie."""
self.trie = {'root': {}}
new_node = 0
for idx, pattern in enumerate(patterns):
node = 'root'
for char in pattern:
if char not in self.trie[node]:
new_node += 1
self.trie[node][char] = str(new_node)
self.trie[str(new_node)] = {}
node = self.trie[node][char]
self.output[node].append(pattern)
def _build_failure_links(self):
"""Construct the failure links for Aho-Corasick."""
queue = deque()
self.fail['root'] = 'root'
# Initialize failure links for the first level of the trie
for char, node in self.trie['root'].items():
self.fail[node] = 'root'
queue.append(node)
# BFS for setting failure links
while queue:
curr_node = queue.popleft()
for char, next_node in self.trie[curr_node].items():
queue.append(next_node)
# Find failure link for current node
fail_node = self.fail[curr_node]
while fail_node != 'root' and char not in self.trie[fail_node]:
fail_node = self.fail[fail_node]
if char in self.trie[fail_node]:
self.fail[next_node] = self.trie[fail_node][char]
else:
self.fail[next_node] = 'root'
# Combine output of fail state
self.output[next_node].extend(self.output[self.fail[next_node]])
def search(self, text):
"""Search for patterns in the given text."""
node = 'root'
results = []
for i, char in enumerate(text):
while node != 'root' and char not in self.trie[node]:
node = self.fail[node]
if char in self.trie[node]:
node = self.trie[node]
else:
node = 'root'
# Check if any pattern ends at this character
if self.output[node]:
for pattern in self.output[node]:
results.append((i - len(pattern) + 1, pattern))
return results
# Example usage
patterns = ["he", "she", "his", "hers"]
text = "ushers"
ac = AhoCorasick(patterns)
matches = ac.search(text)
print("Matches found:", matches)
C++
#include <iostream>
#include <vector>
#include <queue>
#include <map>
using namespace std;
class AhoCorasick {
struct Node {
map<char, int> children;
int fail;
vector<int> output;
};
vector<Node> trie;
vector<string> patterns;
public:
AhoCorasick(const vector<string>& patterns) {
trie.push_back(Node()); // Root node
this->patterns = patterns;
build_trie();
build_failure_links();
}
void build_trie() {
for (int idx = 0; idx < patterns.size(); ++idx) {
string pattern = patterns[idx];
int node = 0;
for (char ch : pattern) {
if (!trie[node].children.count(ch)) {
trie[node].children[ch] = trie.size();
trie.push_back(Node());
}
node = trie[node].children[ch];
}
trie[node].output.push_back(idx);
}
}
void build_failure_links() {
queue<int> q;
for (auto& [ch, next_node] : trie[0].children) {
trie[next_node].fail = 0;
q.push(next_node);
}
while (!q.empty()) {
int curr_node = q.front();
q.pop();
for (auto& [ch, next_node] : trie[curr_node].children) {
int fail = trie[curr_node].fail;
while (fail != 0 && !trie[fail].children.count(ch)) {
fail = trie[fail].fail;
}
if (trie[fail].children.count(ch)) {
trie[next_node].fail = trie[fail].children[ch];
} else {
trie[next_node].fail = 0;
}
trie[next_node].output.insert(trie[next_node].output.end(),
trie[trie[next_node].fail].output.begin(),
trie[trie[next_node].fail].output.end());
q.push(next_node);
}
}
}
vector<pair<int, string>> search(const string& text) {
int node = 0;
vector<pair<int, string>> results;
for (int i = 0; i < text.size(); ++i) {
char ch = text[i];
while (node != 0 && !trie[node].children.count(ch)) {
node = trie[node].fail;
}
if (trie[node].children.count(ch)) {
node = trie[node].children[ch];
} else {
node = 0;
}
for (int pattern_idx : trie[node].output) {
results.emplace_back(i - patterns[pattern_idx].size() + 1, patterns[pattern_idx]);
}
}
return results;
}
};
int main() {
vector<string> patterns = {"he", "she", "his", "hers"};
string text = "ushers";
AhoCorasick ac(patterns);
vector<pair<int, string>> matches = ac.search(text);
cout << "Matches found:\n";
for (const auto& [idx, pattern] : matches) {
cout << "Pattern \"" << pattern << "\" found at index " << idx << endl;
}
return 0;
}
Advantages
- Multiple Pattern Search: Efficiently searches for multiple patterns in a single pass through the text.
- Linear Time Complexity: After preprocessing the patterns, the search runs in linear time relative to the length of the text.
- Failure Links: These allow the algorithm to efficiently backtrack and reuse previously computed results when mismatches occur.
Applications
- Text Search: Useful in applications such as search engines, text editors, and data mining where multiple patterns need to be searched.
- Intrusion Detection Systems: Often used to detect multiple predefined patterns in network traffic.
- DNA Sequence Analysis: Helps in finding multiple patterns within genetic sequences.
Conclusion
The Aho-Corasick Algorithm is an optimal solution for problems involving multiple pattern matching. Its combination of a trie and failure links ensures that it can handle large datasets and multiple queries efficiently.
Related Topics
- KMP Algorithm: Single-pattern foundation that Aho-Corasick generalizes for multi-pattern matching.
- Trie Data Structure: Aho-Corasick is built on a Trie with failure links ( build time).
- Rabin-Karp Algorithm: Alternative multi-pattern matching approach using rolling hashes.
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