Problem Description

The goal of this problem is to count the number of ways in which we can select a non-empty substring from a string s and replace exactly one character in it such that the modified substring matches some substring in another string t. This is equivalent to finding substrings in s that differ by precisely one character from any substring in t.

For example, take s = "computer" and t = "computation". If we look at the substring s[0:7] = "compute" from s, we can change the last character e to a, obtaining s[0:6] + 'a' = "computa". This new substring, computa, is also a substring of t, thus constituting one valid way.

The problem requires us to find the total number of such valid ways for all possible substrings in s.

Intuition

To solve this problem efficiently, we can apply dynamic programming (DP). The intuition for using DP hinges on two key observations:

1. For every pair of indices (i, j) where s[i] is equal to t[j], the substrings ending at these indices can form a part of larger matching substrings, barring the one character swap.

2. If s[i] is not equal to t[j], then we have found the place where a single character difference occurs. All substring pairs ending at (i, j) and having this as their only difference should be counted.

Given these observations, we can define two DP tables:

• f[i][j] that stores the length of the matching substring of s[0..i-1] and t[0..j-1] ending at s[i-1] and t[j-1]. Essentially, it tells us how far back we can extend the matching part to the left, before encountering a mismatch or the start of the strings.

• g[i][j] serves a similar purpose but looks to the right of i and j, telling us how far we can extend the matching part of the substrings starting at s[i] and t[j].

The variable ans accumulates the number of valid ways. For each mismatch found (where s[i] != t[j]), we calculate how many valid substrings can be formed and add this to ans. The number of valid substrings is computed as (f[i][j] + 1) * (g[i + 1][j + 1] + 1). This accounts for the one-character swap by combining the lengths of matching substrings directly to the left and right of (i, j).

We iterate through each pair of indices (i, j) comparing characters from s and t, and whenever we find a mismatch, we perform the above calculation. Finally, we return the value of ans as our answer.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution implements a dynamic programming approach to efficiently solve the problem by considering all possible substrings of s and t and determining if they differ by exactly one character.

Here's a step-by-step breakdown of the solution code:

1. Initialize Variables: The solution begins by initializing an accumulator ans to keep track of the count of valid substrings, and the lengths of the strings s and t, denoted as m and n respectively.

2. Create DP Tables: Two DP tables f and g are created with dimensions (m+1) x (n+1), initializing all their entries to 0. Each cell f[i][j] will hold the length of the matching substring pair ending with s[i-1] and t[j-1]. Similarly, g[i][j] will hold the length of the matching substring pair starting with s[i] and t[j].

3. Fill the f Table: The nested loop over i and j fills the f table. For each pair (i, j), if s[i-1] is equal to t[j-1], then f[i][j] is set to the value of f[i-1][j-1] + 1, which extends the length of the matching substring by one. Otherwise, f[i][j] is already 0, indicating no match.

4. Fill the g Table and Calculate ans: Another nested loop, counting downwards from m-1 to 0 for i and from n-1 to 0 for j, fills the g table. If s[i] is equal to t[j], then g[i][j] is set to g[i+1][j+1] + 1. But when s[i] does not match t[j], a mismatch has been found. The product (f[i][j] + 1) * (g[i+1][j+1] + 1) calculates the number of valid ways considering the mismatch as the single difference point, and this number is added to the accumulator ans.

5. Return the Result: After iterating through all possible substrings, the ans variable will have accumulated the total number of valid substrings where a single character replacement in s can result in a substring in t. This accumulated result is returned as the final answer.

In this solution, dynamic programming tables f and g efficiently store useful computed values which are used to keep track of matches and calculate the number of possible valid substrings dynamically for each pair of indices (i,j). By avoiding redundant comparisons and reusing previously computed values, the algorithm avoids the naive approach's extensive computation, thus improving efficiency.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach using strings s = "acdb" and t = "abc".

1. Initialize Variables: We set ans to 0. The lengths of the strings m and n are 4 and 3 respectively.

2. Create DP Tables: We initialize the DP tables f and g as 5x4 matrices, as s has length 4, and t has length 3.

3. Fill the f Table: We iterate through strings s and t with indices i and j. When characters match, we set f[i][j] = f[i-1][j-1] + 1. Here's how f looks like after filling:

   		a	b	c
   	0	0	0	0
   a	0	1	0	0
   c	0	0	0	1
   d	0	0	0	0
   b	0	0	1	0

4. Fill the g Table and Calculate ans: Next, we fill in the g table by iterating backwards. On mismatches, we calculate (f[i][j] + 1) * (g[i+1][j+1] + 1) and add it to ans.

   Steps while filling g:

   • Start at s[3] = 'b' and t[2] = 'c', no match, set g[3][2] = 0.
   • Move to s[3] = 'b' and t[1] = 'b', they match, so set g[3][1] = g[4][2] + 1 which is 1.
   • Continue for other elements. Discrepancies are found at s[2] = 'd' and t[0] = 'a', as well as s[0] = 'a' and t[1] = 'b'. We calculate the valid ways for these mismatches, adding the results to ans.

   Here's the g matrix:

   		a	b	c
   	1	0	0
   a	0	1	0
   c	0	0	1
   d	0	0	0
   b	0	1	0

   In this case, ans will be calculated as follows:

   • For s[2] != t[0], (f[2][0] + 1) * (g[3][1] + 1) = (0 + 1) * (0 + 1) = 1.
   • For s[0] != t[1], (f[0][1] + 1) * (g[1][2] + 1) = (0 + 1) * (0 + 1) = 1.

   We add these to ans, getting us an ans = 2.

5. Return the Result: The variable ans now has the value 2, which is the number of valid substrings in s that can match substrings in t by changing exactly one character. We return this value as the answer.

Solution Implementation

Time and Space Complexity

Time Complexity

The given code consists of a nested loop structure, where two independent loops iterate over the length of s and t. The outer loop runs for m + n times and the inner nested loops run m * n times separately for the loops that build the f and g 2D arrays. The computation within the inner loops operates in O(1) time. Therefore, the total time complexity combines the O(m + n) for the outer loops and O(m * n) for the inner nested loops, resulting in O(m * n) overall.

Space Complexity

The space complexity is determined by the size of the two-dimensional arrays f and g, each of which has a size of (m + 1) * (n + 1). Hence, the space used by these data structures is O(m * n). No other data structures are used that grow with the input size, so the total space complexity is O(m * n).

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