Problem Description

The problem provided requires us to find the shortest string that contains two given strings, str1 and str2, as subsequences. A subsequence of a string t is a new string generated from t after removing some (can be none) characters without changing the order of the remaining characters. For example, "ace" is a subsequence of "abcde" while "aec" is not. The task is to construct the shortest common supersequence that has both str1 and str2 as its subsequences. It is important to note that there can be multiple valid supersequences that satisfy the conditions, and any valid one can be returned.

Intuition

The intuition behind the solution starts by recognizing that this is a classic problem of finding the Longest Common Subsequence (LCS) of the two strings.

1. The LCS assists us in identifying the common base around which we can organize the other characters in the supersequence. Since the LCS is the longest sequence found in both strings, we need to include it only once to cover its occurrence in both str1 and str2.

2. Starting with an empty result string, we move through str1 and str2 in parallel, adding characters from both strings to the result. Whenever we find a pair of characters in str1 and str2 that are part of the LCS, we only add this character once to the result.

3. To be more practical, we create a dynamic programming table, denoted as f, that stores the lengths of LCS for different substrings of str1 and str2. This table can be built by iterating over all characters of str1 and str2 where f[i][j] represents the length of the LCS between str1[:i] and str2[:j].

4. Once we have filled this table, we can backtrack from f[m][n] (where m is the length of str1 and n is the length of str2) to construct the shortest common supersequence by choosing characters from str1 or str2 or both when they are the same and they match the LCS character.

By using the dynamic programming approach to find the LCS and careful backtracking to build the actual supersequence, we can construct a supersequence that is the shortest possible combination of str1 and str2, containing both as subsequences.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution utilizes dynamic programming (DP) to compute the shortest common supersequence. Here's how the steps break down in the provided code:

1. A two-dimensional DP array f with dimensions (m+1) x (n+1) is initialized with zeros, where m and n are the lengths of str1 and str2, respectively. This array will eventually contain the lengths of the longest common subsequences for all possible substrings of str1 and str2.

2. We fill in the DP table f using a nested loop. Here, f[i][j] will be filled with the length of the LCS of substrings str1[:i] and str2[:j]. If str1[i - 1] is equal to str2[j - 1], this means the characters match and can be part of the LCS, so we set f[i][j] to be 1 plus the length of the LCS at f[i - 1][j - 1]. Otherwise, we take the maximum of the lengths of the two possible LCS, by either including the last character of str1 (f[i - 1][j]) or str2 (f[i][j - 1]).

3. After constructing the DP table, we backtrack to build the shortest common supersequence. The list ans is used to store the characters of the supersequence in reverse as we'll be starting from the bottom-right corner of the DP table and moving towards the top-left corner.

4. The backtracking logic works as follows:

   • If i or j reaches 0, it means we have reached the end of str1 or str2, so we append the remaining characters of the other string.
   • If f[i][j] is equal to f[i-1][j], it means the current character of str1 is not part of the LCS, so we include str1[i-1] and move left in the table (decrement i).
   • If f[i][j] is equal to f[i][j-1], it's the character from str2 that's not part of the LCS, so we add str2[j-1] to the answer and move up (decrement j).
   • If none of the above conditions meet, it means both characters from str1 and str2 are part of the LCS, so we add either one to the answer and move diagonally up-left in the table (decrement both i and j).

5. Because ans contains the characters in reverse order (from backtracking), we reverse it back to get the correct order of the shortest common supersequence and return it as a string.

This algorithm ensures the generation of the shortest string which is a supersequence of both str1 and str2, using DP for efficiency in finding the LCS, and backtracking to build the supersequence from the LCS.

Example Walkthrough

Let's take two strings str1 = "abac" and str2 = "cab" to demonstrate how to find the shortest common supersequence using the dynamic programming approach.

1. First, we initialize an empty DP table f with dimensions (4+1) x (3+1) since len(str1) is 4 and len(str2) is 3. Each cell in f will represent the longest common subsequence (LCS) length for substrings ending at str1[i-1] and str2[j-1].

2. We then populate the table f as follows:

   • Start with i = 1 and j = 1. If str1[i - 1] == str2[j - 1], set f[i][j] to f[i - 1][j - 1] + 1. Otherwise, set f[i][j] to max(f[i - 1][j], f[i][j - 1]).
   • Repeat this process to fill the entire table. Since "abac" has no common characters with "cab" in the first position, the first row and column will be filled with incremental counts.

   The filled DP table f would look something like this:

     c a b
   0 0 0 0
   a 0 1 1 1
   b 0 1 2 2
   a 0 1 2 2
   c 0 1 2 2

   This table tells us that the length of the LCS for "abac" and "cab" is 2.

3. Starting from f[4][3], we backtrack to construct the supersequence. Initializing i = 4, j = 3, we go backwards and:

   • Note that the characters str1[3] (c) and str2[2] (a) aren't equal. Since f[4][3] == f[3][3], we add "c" from str1 to our answer and decrement i to 3.
   • Now, str1[2] (a) matches str2[2] (a), so we add "a" from either string to our answer and decrement both i and j.
   • At f[2][1], the characters str1[1] (b) and str2[0] (c) don't match. Since f[2][1] == f[1][1], we add "b" from str1 and decrement i.
   • Finally, str1[0] matches str2[0] (c), so we add "c" to our answer and decrement each index.

   The result string (ans) at this point is "cbac" in reverse order.

4. Reversing ans, we get "cabc" which is the shortest common supersequence of str1 and str2.

In this way, by using dynamic programming to determine the LCS of the two strings and carefully backtracking from the end, we have successfully constructed the shortest string containing str1 = "abac" and str2 = "cab" as subsequences.

Solution Implementation

Time and Space Complexity

The given Python code defines a method shortestCommonSupersequence to find the shortest common supersequence of two strings str1 and str2. To analyze the computational complexities, we need to consider both the time taken to execute the code, which depends on the number of operations performed, and the space required to store the data during execution.

Time Complexity

The time complexity of the algorithm is determined by the nested loop structure which iterates over the lengths of the two input strings str1 and str2. There are two main steps in this algorithm:

1. Filling out the dynamic programming table f, which has dimensions (m + 1) * (n + 1), where m is the length of str1 and n is the length of str2. Each cell in the table is filled in constant time, so the time required to fill this table is directly proportional to the number of cells, resulting in a time complexity of O(m * n).

2. Constructing the shortest common supersequence by traversing back from the bottom-right corner of the table to the top-left, appending characters accordingly. The length of this path is m + n in the worst case because each step either moves left or up in the table, and there can be at most m upward moves and n left moves. Therefore, the time complexity for the reconstruction step is O(m + n).

The overall time complexity of the algorithm is the sum of these two parts, predominantly the first since it involves iterating over a two-dimensional array. Thus, the total time complexity is O(m * n) + O(m + n), which simplifies to O(m * n) because m * n dominates m + n for large values of m and n.

Space Complexity

The space complexity is dictated by the space required to store the dynamic programming table f, which has (m + 1) * (n + 1) cells, each storing an integer. Hence, the space complexity is O(m * n) to accommodate this table. Additional space is used for the list ans which is used to construct the shortest common supersequence. In the worst case, ans could grow to a size of m + n, when all characters from both str1 and str2 are used in the supersequence. However, the space complexity remains O(m * n) because m * n dominates m + n.

In summary, the time complexity of the code is O(m * n), and the space complexity of the code is also O(m * n).

Learn more about how to find time and space complexity quickly using problem constraints.