Greedy_Approach_for_Minimum_Patches.md

Intuition

This problem requires efficiently patching gaps in a sequence formed by a sorted integer array nums to cover the range [1, n]. Using a greedy algorithm ensures that we always extend the covered range by adding the minimum possible number to target.

Approach

1. Initialization: Start with target initialized to 0 and patches set to 0 to count how many patches are added.

2. Greedy Strategy: Iterate through nums to ensure target covers up to n. For each number in nums:

   • If the current number is less than or equal to target + 1, extend target by adding this number.
   • If the current number is greater than target + 1, add patches to cover the gap using successive increments of target + 1.

3. Final Gap Handling: After iterating through nums, ensure there are no gaps between target and n. Add patches to fill this gap using successive increments of target + 1 until target reaches or exceeds n.

4. Return patches: Once target covers the range [1, n], return the total number of patches added.

Complexity

• Time complexity: O(m + log n), where m is the length of nums. This complexity arises from iterating through nums and adjusting target. The logarithmic component comes from the operations involving target and its growth.

• Space complexity: O(1). The algorithm uses a constant amount of extra space regardless of the input size, primarily for storing target, patches, and iterating variables.

Code

public class Solution {
    public int MinPatches(int[] nums, int n) {
        long target = 0l; 
        int patches = 0; 
        int index = 0;

        while (target < n && index < nums.Length && nums[index] <= n)
        {
            while (nums[index] > target + 1 && target < n)
            {
                target += target + 1;
                patches++;
            }

            target += nums[index];
            index++;
        }

        while(target < n)
        {
            patches++;
            target += target + 1;
        }

        return patches;
    }
}