Steps to Find the LCIS

To implement the LCIS, we can use a dynamic programming approach. Here's a step-by-step guide:

Initialization

Dynamic Programming Table

Iterate through each character of A and B. If A[i-1] B[j-1], update the dp value to dp[i-1][j-1] 1 if the previous character of A is less than the current character of B. If they are not equal, carry forward the maximum value from the previous row or the same column.

Track the Length

Keep track of the maximum length of the LCIS found during the iterations.

Reconstruction

After filling the dp table, you can backtrack to reconstruct the LCIS. This step involves tracing back through the dp table to get the actual subsequence.

Example Code Implementation

Here's a Python implementation of the above logic:

def longest_common_increasing_subsequence(A, B):
    m, n  len(A), len(B)
    dp  [[0] * (n   1) for _ in range(m   1)]
    max_length  0
    end_index  0
    # Fill the dp table
    for i in range(1, m   1):
        for j in range(1, n   1):
            if A[i - 1]  B[j - 1]:
                if i - 2 > 0 and j - 2 > 0 and A[i - 2]  max_length:
                max_length  dp[i][j]
                end_index  j
    # Reconstruct the LCIS
    lcis  []
    for i in range(m, 0, -1):
        for j in range(end_index, 0, -1):
            if dp[i][j]  max_length and not lcis or A[i - 1] > lcis[-1]:
                (A[i - 1])
                max_length - 1
                end_index  j - 1
                break
    return ''.join(lcis)

Example Usage

Example: A 'abc' B 'abceac' print(longest_common_increasing_subsequence(A, B)) Output: abc

Explanation of the Code

The dp table stores the lengths of the longest common increasing subsequences found up to each character. The nested loops compare characters of both strings and update the dp table based on matching characters and the increasing condition. Finally, the LCIS is reconstructed by backtracking through the dp table.

Complexity

Time Complexity: O(m * n) where m and n are the lengths of the two strings.

Space Complexity: O(m * n) for the dp table.

This approach efficiently finds the longest common increasing subsequence for two strings, making it a valuable tool in various applications such as bioinformatics, text analysis, and more.