The time complexity of solving the Longest Common Subsequence problem using Dynamic Programming is : (m and n are lengths of subsequences) 

The correct answer is O(m.n).

Key Points

• The Longest Common Subsequence (LCS) problem is a classic computer science problem that can be solved using Dynamic Programming (DP).
• The time complexity of solving the LCS problem using DP is O(m.n), where m and n are the lengths of the two sequences being compared.
• This is because the DP approach involves filling up a 2D table of size m x n where each cell represents the length of the LCS of the substrings considered up to that point.
• Here is a step-by-step explanation of the DP approach to solve the LCS problem:

/ Function to find the length of the Longest Common Subsequence
function lcs(X, Y) {
    let m = X.length;
    let n = Y.length;
    let L = Array(m + 1).fill().map(() => Array(n + 1).fill(0));

    / Building the L[m+1][n+1] table in bottom-up fashion
    for (let i = 1; i <= m; i++) {
        for (let j = 1; j <= n; j++) {
            if (X[i - 1] === Y[j - 1]) {
                L[i][j] = L[i - 1][j - 1] + 1;
            } else {
                L[i][j] = Math.max(L[i - 1][j], L[i][j - 1]);
            }
        }
    }

    / L[m][n] contains the length of LCS for X[0..m-1], Y[0..n-1]
    return L[m][n];
}

/ Example usage
let X = "AGGTAB";
let Y = "GXTXAYB";
console.log("Length of LCS is", lcs(X, Y));  / Output: 4
    

Additional Information

• The LCS problem is used in various applications such as diff tools for comparing files, bioinformatics for sequence alignment, and more.
• Even though the time complexity is O(m.n), which can be computationally expensive for large sequences, the DP approach is efficient for moderate-sized inputs.
• There are more advanced techniques like space optimization to reduce the memory usage from O(m.n) to O(min(m,n)), making the algorithm more practical for large inputs.