Problem Description

Given three strings s1, s2, and s3, we need to determine if s3 can be constructed by interleaving characters from s1 and s2. The interleaving of s1 and s2 should create a sequence where characters from s1 and s2 are alternated, but maintain their relative order from the original strings. There can be more than one character from either string coming in sequence, but overall, the characters from s1 and s2 should come in the exact order they appear in their respective strings.

For example, if s1 = "abc" and s2 = "def", then s3 = "adbcef" would be a valid interleaving. However, s3 = "abcdef" or s3 = "badcfe" would not be valid interleavings, because the relative ordering of the characters from s1 and s2 isn't respected in s3.

Intuition

The intuition for solving this problem lies in dynamic programming. We can imagine forming s3 character by character, making choices at each step of taking the next character from either s1 or s2. If at any point, the characters from s1 or s2 do not match the current character in s3, we cannot proceed further by that route.

The key insight for the solution is to construct a table (or array) that represents whether it is possible to form the prefix s3[0...k] from prefixes of s1 and s2. We initialize an array f of length n+1 (where n is the length of s2), which will help us track if s3 can be formed up to the j-th character of s2. We start by assuming that an empty s3 can be formed without any characters from either string (which is always true).

We then iterate over all characters in both s1 and s2, updating the array at each step. The condition f[j] &= s1[i - 1] == s3[k] checks if we can form the string by taking the next character from s1, while the condition f[j] |= f[j - 1] and s2[j - 1] == s3[k] checks if we can form the string by taking the next character from s2.

This way, we fill up the table iteratively, making sure at every step that we satisfy the condition of interleaving without violating the original order in the given strings. If, by the end of the iteration, we find that f[n] is True, it means that we can form s3 by interleaving s1 and s2. Otherwise, we cannot.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution uses dynamic programming, a method that solves problems by breaking them down into simpler subproblems and stores the results of those subproblems to avoid redundant computations.

Here's a step-by-step breakdown of the algorithm:

1. We begin by defining the lengths of s1 and s2 as m and n, respectively. Immediately, we check if the length of s3 is equal to the sum of the lengths of s1 and s2 (m + n). If not, s3 cannot be an interleaving of s1 and s2, and we return False.

2. We then create an array f of length n+1. f[j] will hold a boolean value indicating whether s3[0...i+j-1] is a valid interleaving of s1[0...i-1] and s2[0...j-1]. Here is a critical observation: f[0] is initialized to True because an empty string is considered an interleaving of two other empty strings by default.

3. We iterate through both strings s1 and s2 using two nested loops, one for index i ranging from 0 to m, and another for index j ranging from 0 to n. Each iteration represents an attempt to match the character in s3 at the current combined index k = i + j - 1.

4. For each pair (i, j), if i is not zero, we update f[j] to keep track of whether s3[k] can still match with the character in s1[i - 1]. This is indicated by the operation f[j] &= s1[i - 1] == s3[k].

5. Similarly, if j is not zero, we update f[j] to see if s3[k] can match with the character s2[j - 1], while also taking into account the value of f[j - 1], which indicates whether the interleaving was possible without considering the current character of s2. This happens with the operation f[j] |= f[j - 1] and s2[j - 1] == s3[k].

6. After both loops terminate, we return the value of f[n]. If f[n] is True, it means that s3 can be formed by interleaving s1 and s2. Otherwise, it means that it is not possible.

The above steps summarize the dynamic programming approach to solving the interleaving string problem. It's worth noting that the space complexity is optimized to O(n) since we are only using a single one-dimensional array to store the results.

Example Walkthrough

Let's take a simple example to illustrate the solution approach. Consider the following strings:

• s1 = "ax"
• s2 = "by"
• s3 = "abxy"

Here's how we'd walk through the algorithm:

1. First, we determine the lengths of s1 (length m = 2), and s2 (length n = 2). The length of s3 is 4, which is equal to m + n (2 + 2), so it's possible for s3 to be an interleaving of s1 and s2.

2. We create an array f with n+1 elements, which includes the initial condition (f[0] = True). Initially f = [True, False, False].

3. We begin iterating over the strings:

• For i = 0 (considering an empty string for s1):

  • For j = 0, f[0] remains True because an empty string can interleave with another empty string to form an empty string.
  • For j = 1, since s2[0] ('b') matches s3[0] ('a'), f[1] remains False.
  • For j = 2, since s2[1] ('y') does not match s3[1] ('b'), f[2] remains False. After checking, f = [True, False, False].

• For i = 1 (considering the first character of s1):

  • For j = 0, s1[0] ('a') matches s3[0] ('a'), so f[0] remains True.
  • For j = 1, since s2[0] ('b') matches s3[1] ('b'), f[1] is updated to True.
  • For j = 2, since s2[1] ('y') does not match s3[2] ('x'), f[2] remains False. After checking, f = [True, True, False].

• For i = 2 (considering the second character of s1):

  • For j = 0, s1[1] ('x') matches s3[2] ('x'), so f[0] remains True.
  • For j = 1, since s2[0] ('b') does not match s3[3] ('y'), f[1] remains True.
  • For j = 2, s2[1] ('y') matches s3[3] ('y'), and since f[1] is True, f[2] is updated to True. After checking, f = [True, True, True].

4. Once we have completed our iterations, the final f array is [True, True, True].

5. At the end, we examine the value of f[n], which in this case is True, indicating that s3 can indeed be formed by interleaving s1 and s2.

This completes the example walkthrough, showing that given the strings s1 = "ax", s2 = "by", and s3 = "abxy", s3 is a valid interleaving of s1 and s2 using the described dynamic programming approach.

Solution Implementation

Time and Space Complexity

The given Python code provides a solution to determine if a string s3 is formed by the interleaving of two other strings s1 and s2. Let's analyze both the time complexity and space complexity of the code.

Time Complexity

The outer loop in the code runs m + 1 times, where m is the length of s1. Within this loop, there's an inner loop that runs n + 1 times, where n is the length of s2. However, observe that for each outer iteration, the inner loop starts at 1 (since j ranges from 0 to n), so the combined iterations for the inner loop are actually m * (n + 1).

Each iteration of the inner loop consists of constant time checks and assignment operations, so its time complexity is O(1). Thus, the total time complexity for the double loop structure is O(m * (n + 1)). Simplifying, this is equivalent to O(m * n) since the addition of a constant 1 does not change the order of growth.

Therefore, the overall time complexity of the code is O(m * n).

Space Complexity

Space complexity considers the additional space used by the algorithm excluding the input sizes. In the code, a one-dimensional Boolean array f is initialized with n + 1 elements. The space usage of this array dominates the space complexity. There is no other data structure that grows with the input size. Thus, the space complexity is based on the size of f, which is O(n).

To summarize:

• Time complexity: O(m * n)
• Space complexity: O(n)

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