WhiteWall Problem (Linear time and space complexity)

editorial
1

Minimum Operations to Transform a String

Problem Statement

Given a string, we need to determine the minimum number of operations required to transform it into a new string that satisfies the following condition:

  • Any substring of length that is a multiple of 3 (i.e., 3, 6, 9, 12, etc.) must have an equal number of ‘R’, ‘G’, and ‘B’ characters.
  • The number of character changes should be minimized.

Observations

From analyzing examples, we can deduce that solving the problem for substrings of length 3 automatically extends to larger multiples of 3. The transformed string will have a repeating pattern of 3 characters.

For example, given the input string:

GBRRRBG

The transformed string that satisfies the condition would be:

GBRGBRG

where the repeating pattern is GBR.

To find the minimum number of operations, we count the number of character mismatches between the given string and the transformed string.

Handling Edge Cases

In cases where the first three characters have redundant characters (e.g., RRG), we cannot directly determine the best repeating sequence. To resolve this, we iterate through all permutations of “RGB” (RGB, GBR, RBG, GRB, BRG, BGR) and choose the one that results in the least number of mismatches.

Python Solution

# Number of test cases
t = int(input())
for _ in range(t):
    n = int(input())  # Length of the original string
    s = list(input())  # Original string
    res = float("inf") 
    
    # Possible repeating patterns of "RGB"
    patterns = ["RGB", "GBR", "RBG", "GRB", "BRG", "BGR"]
    
    for pattern in patterns:
        # Generate the transformed string based on the repeating pattern
        transformed_string = pattern * (n // 3) + pattern[: n % 3]
        
        # Count mismatches between original and transformed string
        count = sum(1 for i in range(n) if s[i] != transformed_string[i])
        
        # Update minimum operations required
        res = min(res, count) 
    
    print(res)

Explanation of the Code

  1. Read the number of test cases.
  2. Iterate through each test case:
  • Read the length of the original string.
  • Read the original string.
  • Initialize res to a large number to store the minimum operations.
  • Iterate through all six possible repeating patterns (RGB permutations).
  • Construct a transformed string based on the repeating pattern.
  • Count mismatches between the original and transformed strings.
  • Update res with the minimum mismatch count found.
  1. Print the minimum number of operations required for each test case.

Complexity Analysis

Since we are iterating through 6 possible patterns and performing a comparison for each character in the string, the time complexity is O(6 * n) = O(n), making the approach efficient for large values of n.

Example Input & Output

Input:

1
7
GBRRRBG

Output:

3

This means that at least 3 character modifications are required to meet the condition.

This approach ensures that we efficiently transform the string while keeping changes to a minimum.