Optimized Algorithm for Finding the Longest Common Substring between Three Strings

When dealing with string manipulation and specifically finding the longest common substring between three strings, the naive approach can be computationally expensive and may not efficiently address all test cases. This article presents an optimized algorithm that significantly reduces the time complexity from cubic to linear.

Introduction

The problem at hand involves finding the longest common substring among three given strings: s1, s2, and s3. Traditional methods, such as the naive algorithm, generate all possible combinations of substrings, which leads to a time complexity of O(s1 * s2 * s3). This approach is infeasible for large inputs, as it can result in exceeding the time limit on certain test cases.

Naive Algorithm

The naive algorithm works by generating all possible combinations of substrings from s1 and s2, and then using dynamic programming to find the longest common substring (LCSubstring) between the generated substrings and s3. This process is not only inefficiency but also impractical for large strings due to its cubic time complexity.

Optimized Approach

The optimized approach merges the search process into a single linear pass. It involves the following steps:

Step 1: Index Creation

Create an array of vectors, each representing the indexes of a particular alphabet in s3. This step allows us to quickly locate the positions of each character in s3. Iterate through s3 and populate the array with the indexes of each character.

Step 2: String Search Process

For each possible substring in s3, perform the following operations:

Search the corresponding character in s1 and s2. Track the length of the discovered substring and find the matching part in the other string. Update the longest common substring found.

Step 3: Repeat for Both Strings

The process is repeated for both s1 and s2 to ensure that all possible matches are considered.

Time Complexity

The overall time complexity of this approach is approximately O(s1 * s2), making it significantly more efficient than the naive method.

Algorithm Implementation

The following provides a high-level pseudocode to implement the optimized algorithm:

# Include necessary libraries
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#define N 1000000007
using namespace std;
# Helper function to get the corresponding alphabet character
char getalphabet(int n) {
    char alpha  'a';
    for (int i  0; i > s1 >> s2 >> s3;
    vector v[26];
    fill(v, v   26, vector ());
    for (int i  0; i 

Conclusion

This article has presented an optimized algorithm to find the longest common substring between three strings, significantly improving the time complexity. For those familiar with suffix trees, using them can further enhance the performance of such algorithms. The implementation provided can be adapted for various scenarios, including competitive programming and real-world applications involving string manipulation.

Further Reading

For a deeper understanding of string manipulation and suffix trees, you can refer to the following resources:

Suffix tree - Wikipedia