Problem Description

You are given a binary string s that is 0-indexed and has an even length. The concept of a binary string is that it contains only two characters, '0' and '1'. A binary string is considered beautiful if it can be divided into one or more substrings where each substring satisfies two conditions. Firstly, it must have an even length, and secondly, it should only contain the character '1' or the character '0', not both. Your task is to make the string beautiful by possibly changing any of the characters in the string to either '0' or '1'. The goal is to achieve this with the least number of changes.

To assist in finding a solution, think about the characteristics of a beautiful substring. Since a beautiful substring can only have one type of character, it implies that we cannot have alternating characters. Therefore, we need to find a way to ensure that characters at specific positions are the same so that the substrings of even length have only '0's or only '1's.

Intuition

The solution is centered around dividing the string into even-length substrings that only have identical characters. To achieve the minimum number of changes, we try to identify where the string already conforms to the beautiful criteria and only change the characters that disrupt the pattern. Since every two consecutive characters must be the same, the strategy is to compare characters at all odd indices with their preceding even index neighbors.

We iterate through the string, checking characters at odd indexes, i.e., 1, 3, 5, ..., comparing each to its previous even index character. If these two consecutive characters differ, s[i] != s[i - 1], we recognize that we'd need to change the character at the odd index for the substring to be beautiful. For each necessary change, we increment the count.

This approach allows us to directly count the minimum number of changes required to make the string beautiful because it ensures that each pair of letters in the string (that would form the even-length substrings) are identical without worrying about the overall pattern of the 0's and 1's in the entire string.

Solution Approach

The implementation of the solution provided in the reference approach is straightforward and elegant due to its simplicity. It leverages a single pass counting method, which is an efficient algorithm for this problem. The choice of data structures or patterns used here is minimal as the solution primarily relies on accessing the characters of the string and performing comparisons.

The process begins by initializing a counter set to zero. This counter will keep track of the minimum number of changes required to make the binary string beautiful.

The core of the solution involves iterating through the string, but instead of looking at every character, we only examine characters at every odd index, which are $1, 3, 5, \cdots$. This approach works because beautiful substrings need to be in pairs (00 or 11), and any discrepancy in these pairs would be caught when comparing the odd-indexed character with its directly preceding even-indexed character.

During each iteration of the loop, we compare the character at the current index i to the one before it s[i - 1]. If the characters are different, s[i] != s[i - 1], it signals that we would need to change the character at index i to make the pair of characters the same, thus contributing towards a beautiful substring.

We implement the condition described above using a generator expression summed up to accumulate the total:

sum(s[i] != s[i - 1] for i in range(1, len(s), 2))

Here’s a breakdown of this expression:

• range(1, len(s), 2): Creates a sequence of numbers starting from 1 up to the length of the string s in steps of 2. This sequence represents all odd indices up to the end of the string.
• s[i] != s[i - 1]: The logic to determine if a change is required. It evaluates to True (which is equivalent to 1) when a change is needed, and False (which is equivalent to 0) when it's not.
• sum(...): Sums up the values generated by the expression for each i. Since True is equivalent to 1, summing these values gives us the total number of changes required.

The result of the sum is the total count of changes needed, which we return as the solution to the problem. This approach ensures that we only change the minimum number of characters necessary for all pairs of characters in the string to become identical, thereby making the string beautiful with the least amount of effort.

Example Walkthrough

Let's consider a small binary string as an example to illustrate the solution approach: s = "010101". We aim to find the minimum number of changes needed to make s beautiful.

With the solution approach in mind, we inspect the characters at the odd indices of s and compare each with the preceding even index character to identify where changes are needed.

1. i = 1: We compare s[1] with s[0]. Here, s[1] = '1' and s[0] = '0'. Since s[1] != s[0], this indicates that we need to make one change to make the substring s[0:2] beautiful.

2. i = 3: We compare s[3] with s[2]. Here, s[3] = '1' and s[2] = '0'. Once again, we see that s[3] != s[2], signaling the need for another change for the substring s[2:4] to be beautiful.

3. i = 5: Finally, we compare s[5] with s[4]. We find that s[5] = '1' and s[4] = '0', which means s[5] != s[4]. This means a change is required for the substring s[4:6] to conform to the beautiful criteria.

In every comparison for this specific string, a change is needed. As such, the counter would increment by 1 for each of the three steps, resulting in a total of 3 changes necessary to transform s into a beautiful binary string. Specifically, we could change s to either "000000" or "111111". Either way, the minimum number of changes required is 3.

Using the solution approach's sum generator expression:

sum(s[i] != s[i - 1] for i in range(1, len(s), 2))

We calculate the result as:

• At i = 1, s[1] != s[0] yields True (1).
• At i = 3, s[3] != s[2] yields True (1).
• At i = 5, s[5] != s[4] yields True (1).

Therefore, the sum is 1 + 1 + 1 = 3. This confirms that 3 is the minimum number of changes needed to make the string s beautiful, aligning with our step-by-step analysis.

Solution Implementation

Time and Space Complexity

The time complexity of the function minChanges is O(n), where n is the length of the string s. This is because the function includes a loop that iterates over the length of the string, starting from the second character (index 1) to the end of the string, with increments of 2. In each iteration, the function calculates whether the current character is different from its predecessor, which is an O(1) operation. Therefore, the loop runs approximately n/2 times, which still yields a linear time complexity of O(n).

The space complexity is O(1) because the function only uses a fixed amount of extra space. It maintains a running total of the changes needed, and this does not depend on the input size. Hence, it uses constant additional space regardless of the length of the input string s.