Patching Array Problem


Description

LeetCode Problem 330.

Given a sorted integer array nums and an integer n, add/patch elements to the array such that any number in the range [1, n] inclusive can be formed by the sum of some elements in the array. Return the minimum number of patches required. Example 1:

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Input: nums = [1,3], n = 6
Output: 1
Explanation:
Combinations of nums are [1], [3], [1,3], which form possible sums of: 1, 3, 4.
Now if we add/patch 2 to nums, the combinations are: [1], [2], [3], [1,3], [2,3], [1,2,3].
Possible sums are 1, 2, 3, 4, 5, 6, which now covers the range [1, 6].
So we only need 1 patch.

Example 2:

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Input: nums = [1,5,10], n = 20
Output: 2
Explanation: The two patches can be [2, 4].

Example 3:

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Input: nums = [1,2,2], n = 5
Output: 0

Constraints:

  • 1 <= nums.length <= 1000
  • 1 <= nums[i] <= 10^4
  • nums is sorted in ascending order.
  • 1 <= n <= 2^31 - 1


Sample C++ Code

We use a greedy approach. We first find miss which is the smallest sum in [0, n] that is missing. This means we can build all sums in [0, miss). For example, if the input is num = [1, 2, 4, 13, 43], and n = 100, then miss = 8.

Then in the given array, if we have a number num <= miss that is not used in building the sums in [0, miss), we can add it to build all sums in [0, miss+num). If we don’t have such a number, then we add miss to the array to maximize the range of sums we can reach.

In the example, we first add 8 to the array, so we can build all sums in [0, 16). As we have 13 that is not used in building sums in [0, 16), we can use it to build sums in [0, 29). Then we insert 29 in the array. This extends the range of sums to [0, 58). Next we consider 43 and extend the range to [0, 101). Then we are done.

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int minPatches(vector<int>& nums, int n) {
    long miss = 1, added = 0, i = 0;
    while (miss <= n) {
        if (i < nums.size() && nums[i] <= miss) {
            miss += nums[i++];
        } else {
            miss += miss;
            added++;
        }
    }
    return added;
}




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