Algorithm of the day -

The Longest Common Prefix (LCP) Array Algorithm of the Suffix Array with Ukkonen's Suffix Tree Construction

Summary

This algorithm constructs a suffix tree and uses it to find the longest common prefix between any two suffixes in a given string, allowing for efficient string matching and comparison

Use Case

This algorithm can be used in a text editor to provide features such as auto-completion and spell-checking, by quickly finding the longest common prefix between the current word and a list of possible completions

Steps

  1. Construct the suffix tree using Ukkonen's algorithm
  2. Perform a depth-first search on the suffix tree to find the longest common prefix between any two suffixes
  3. Store the lengths of the longest common prefixes in an array, known as the LCP array
  4. Use the LCP array to answer queries about the longest common prefix between any two suffixes in O(1) time

Complexity

The time complexity of this algorithm is O(n), where n is the length of the input string, and the space complexity is also O(n), as we need to store the suffix tree and the LCP array

Code Example

import numpy as np
def ukkonen_suffix_treeconstruction(text):
    # Initialize the suffix tree
    tree = {}
    # Iterate over all suffixes of the text
    for i in range(len(text)):
        suffix = text[i:]
        node = tree
        # Iterate over all characters in the suffix
        for char in suffix:
            if char not in node:
                node[char] = {}
            node = node[char]
    return tree
def lcp_array_construction(suffix_tree):
    # Initialize the LCP array
    lcp_array = np.zeros(len(suffix_tree), dtype=int)
    # Perform a depth-first search on the suffix tree
    def dfs(node, depth):
        if not node:
            return
        for child in node:
            dfs(node[child], depth + 1)
            # Update the LCP array
            lcp_array[depth] = max(lcp_array[depth], depth)
    dfs(suffix_tree, 0)
    return lcp_array