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Leetcode 2481. Minimum Cuts to Divide a Circle

Problem Statement

You are given a perfect circle. You need to determine the minimum number of cuts required to divide this circle into exactly n equal segments. A cut is defined as a straight line that passes through the center of the circle.

Example

Clarifying Questions

  1. Can n be less than or equal to 0?
    • Since the problem expects dividing the circle into segments, the smallest valid n should be 1. Considering lesser values would be nonsensical in this context.
  2. What is the maximum possible value for n?
    • This details the constraints needed to optimize the solution, but for most common purposes, we can assume it to be within the bounds of typical integer values.
  3. Is n guaranteed to be a positive integer?
    • Yes, the value of n should be a positive integer given the problem context.

Strategy

To identify the minimum number of cuts needed:

  1. If n is 1:
    • No cuts are needed since the circle itself is already one segment.
  2. If n is 2:
    • One cut through the center suffices to split the circle into two equal segments.
  3. For values of n greater than 2:
    • If n is even: Each cut through the center can divide the circle into two additional segments. Thus, n / 2 straight cuts will generate n segments.
    • If n is odd: Similarly, n odd segments can be achieved optimally with n cuts, as one cut whenever placed still requires a separate consideration.

Based on these observations, the formula can be generalized:

Code

public class MinimumCutsCircle {
    public static int minimumCuts(int n) {
        if (n == 1) {
            return 0;
        }
        return (n + 1) / 2; // This handles both odd and even n appropriately
    }
    
    public static void main(String[] args) {
        System.out.println(minimumCuts(1)); // Output: 0
        System.out.println(minimumCuts(2)); // Output: 1
        System.out.println(minimumCuts(3)); // Output: 2
        System.out.println(minimumCuts(4)); // Output: 2
        System.out.println(minimumCuts(5)); // Output: 3
    }
}

Time Complexity

The time complexity of this solution is O(1) as we are performing a constant-time calculation regardless of the input size. It’s a direct arithmetic operation with a conditional check, thus very efficient in terms of computation.

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