What are the time and space complexities of the standard dynamic programming approach for finding the length of the Longest Common Subsequence (LC

Detailed Answer and Dry Run: Let's analyze the time and space complexity of the provided Python code for LCS. Time Complexity: The core of the algorithm involves two nested loops: The outer loop iterates m times (from i = 1 to m). The inner loop iterates n times (from j = 1 to n). Inside the inner loop, a constant number of operations are performed: Character comparison (text1[i - 1] == text2[j - 1]). Arithmetic operations (addition, max). Array assignments. Therefore, the total number of operations is proportional to m * n. The time complexity is O(m * n). Space Complexity: The algorithm uses a 2D list (or array) named dp to store the results of subproblems. The dp table has m + 1 rows and n + 1 columns. The total number of cells in this table is (m + 1) * (n + 1), which simplifies to O(m * n). Each cell stores an integer. Thus, the space required to store this table is proportional to m * n. The space complexity is O(m * n). Dry Run Example (Conceptual): If text1 has length 5 and text2 has length 7: The dp table will be 6 x 8. The outer loop runs 5 times. The inner loop runs 7 times for each iteration of the outer loop. Total operations for filling the table: approximately 5 * 7 = 35 cell computations. Total space for the table: 6 * 8 = 48 integer cells. This confirms the O(m * n) time and space complexities.