Question

Let us consider two sequences of characters \(S_{1}\) and \(S_{2}\). For example, we could have \(S_{1}-\) \(\mathrm{A} \$ \mathrm{CMA}^{*} \mathrm{MN}\) and \(S_{2}=\mathrm{AXMC} 4 \mathrm{ANB}\). Assuming that a subsequence of a sequence can be constructed by deleting any number of characters from any positions, use the dynamic programming approach to create an algorithm that finds the Iongest common subsequence of \(S_{1}\) and \(S_{2}\). This algorithm returns the maximum-length common subsequence of each sequence.

Short answer

The solution creates and fills the dynamic programming table such that each cell dp[i][j] represents the length of the longest common subsequence in substrings S1[0..i] and S2[0..j]. The length of LCS is obtained from cell dp[m][n], where m and n are sizes of S1 and S2 respectively. The LCS string is obtained by moving from cell dp[m][n] towards dp[0][0] while including appropriate characters.

Step-by-step solution

Initialize Table

For two strings S1 of size m and S2 of size n, create a matrix dp of size (m+1)x(n+1). Initialize all values in dp to 0.

Populate the Table

For each character pair being analyzed from S1 and S2, perform the following operations. If S1[i] == S2[j], where i is the index of S1 and j is the index of S2, dp[i][j] = dp[i-1][j-1] + 1. If S1[i] != S2[j], then dp[i][j] = max(dp[i-1][j], dp[i][j-1]). These operations start from the second row and second column, filling up the dp table.

Find the LCS length

The bottom-right cell of the dp table will contain the length of the longest common subsequence.

Construct the LCS

Starting from the bottom-right cell, move up or left in the dp table based on the values. If current cell has value equal to cell left to it, move left. If current cell has value equal to cell above it, move up. If current cell has value one plus its up-left cell, move diagonally up-left and include current character in LCS. Continue this until the first row or first column of dp table is reached. LCS is the sequence of characters included in reverse order.

Key concepts

Dynamic Programming

Dynamic programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It is particularly useful when a problem can be split into overlapping subproblems, which can then be solved independently. The beauty of dynamic programming lies in its ability to systematically record the results of these subproblems in a table (known as a DP table) to avoid redundant calculations.

In the context of finding the longest common subsequence (LCS), dynamic programming shines by allowing us to approach the problem in a manageable and structured way. Through DP, we construct a table that represents all possible comparisons between the two sequences, and we fill this table based on certain rules that compare characters and lengths of subsequences. The final solution to the LCS problem is then deduced from this table, making the approach both systematic and efficient.

Subsequence Algorithm

A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements. The longest common subsequence (LCS) algorithm is designed to find the longest subsequence present in two sequences. To achieve this, the LCS algorithm compares characters from both sequences and identifies the maximum length of matching subsequences.

The dynamic programming approach to the LCS algorithm involves constructing a matrix that incrementally builds the solution by consolidating the knowledge of previous comparisons. The algorithm starts with comparing the first character of each sequence and builds upon those comparisons to determine if the subsequent characters contribute to a longer subsequence. By analyzing the problem in this step-by-step fashion, the algorithm efficiently unravels the most extended subsequence shared between the two sequences.

Programming with Pseudocode

Pseudocode is a high-level description of a computer algorithm that uses the structural conventions of programming languages, but is intended for human reading rather than machine reading. Pseudocode omits specific syntax details and focuses on outlining the algorithm conceptually, which is essential for understanding and implementing algorithms without getting bogged down by particular programming languages.

For the LCS problem, pseudocode provides a language-agnostic blueprint of the solution, highlighting the initialization of the DP table, the conditions for filling in the table, and the method for constructing the LCS itself. Pseudocode allows us to communicate the algorithm's logic clearly and ensures that the intended structure of the program is easily grasped.

Algorithm Complexity Analysis

The complexity analysis of an algorithm evaluates the efficiency of an algorithm in terms of time and space required. When assessing the LCS algorithm implemented with dynamic programming, we consider both the number of operations performed (time complexity) and the amount of memory used (space complexity).

The time complexity of the LCS algorithm is generally O(mn), where m and n are the lengths of the two sequences. This is because each cell in the m x n DP table represents a subproblem that must be solved. The space complexity also follows the O(mn) pattern, given that the DP table stores the lengths of the longest subsequence at each step.

Understanding the algorithm's complexity helps us predict its performance, especially with larger inputs, and is critical for optimizing the algorithm to reduce both running time and memory usage where possible.