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Formatted question description: https://leetcode.ca/all/2195.html

# 2195. Append K Integers With Minimal Sum (Medium)

You are given an integer array nums and an integer k. Append k unique positive integers that do not appear in nums to nums such that the resulting total sum is minimum.

Return the sum of the k integers appended to nums.

Example 1:

Input: nums = [1,4,25,10,25], k = 2
Output: 5
Explanation: The two unique positive integers that do not appear in nums which we append are 2 and 3.
The resulting sum of nums is 1 + 4 + 25 + 10 + 25 + 2 + 3 = 70, which is the minimum.
The sum of the two integers appended is 2 + 3 = 5, so we return 5.

Example 2:

Input: nums = [5,6], k = 6
Output: 25
Explanation: The six unique positive integers that do not appear in nums which we append are 1, 2, 3, 4, 7, and 8.
The resulting sum of nums is 5 + 6 + 1 + 2 + 3 + 4 + 7 + 8 = 36, which is the minimum.
The sum of the six integers appended is 1 + 2 + 3 + 4 + 7 + 8 = 25, so we return 25.


Constraints:

• 1 <= nums.length <= 105
• 1 <= nums[i], k <= 109

Similar Questions:

## Solution 1. Sorting

Intuition: Sort the array. Traverse from left to right, sum up the missing numbers between A[i-1] and A[i] until we’ve used k missing numbers.

Algorithm:

For a given A[i], the previous number prev is A[i-1] or 0 if A[i-1] doesn’t exist.

There are cnt = min(k, max(0, A[i] - prev - 1)) missing numbers inbetween, i.e. from prev+1 to prev+cnt. The sum of these numbers is (prev+1 + prev+cnt) * cnt / 2.

If there are still k missing numbers after traversing the array, the rest of the missing numbers are A[N-1]+1 to A[N-1]+k. The sum of them is (A[N-1]+1 + A[N-1]+k) * k / 2.

• // OJ: https://leetcode.com/problems/append-k-integers-with-minimal-sum/
// Time: O(NlogN)
// Space: O(1)
class Solution {
public:
long long minimalKSum(vector<int>& A, int k) {
sort(begin(A), end(A));
long ans = 0, N = A.size();
for (int i = 0; i < N && k; ++i) {
long prev = i == 0 ? 0 : A[i - 1]; // the previous number
long cnt = min((long)k, max((long)0, A[i] - prev - 1)); // the count of missing numbers between prev and A[i]
k -= cnt; // use these cnt missing numbers
ans += (long)(prev + 1 + prev + cnt) * cnt / 2; // sum of these cnt missing numbers [prev+1, prev+cnt].
}
if (k > 0) ans += ((long)A.back() + 1 + A.back() + k) * k / 2; // If there are still missing numbers, add the sum of numbers[A.back()+1, A.back()+k] to answer
return ans;
}
};

• class Solution:
def minimalKSum(self, nums: List[int], k: int) -> int:
nums.append(0)
nums.append(2 * 10**9)
nums.sort()
ans = 0
for a, b in pairwise(nums):
n = min(k, b - a - 1)
if n <= 0:
continue
k -= n
ans += (a + 1 + a + n) * n // 2
if k == 0:
break
return ans

############

# 2195. Append K Integers With Minimal Sum
# https://leetcode.com/problems/append-k-integers-with-minimal-sum/

class Solution:
def minimalKSum(self, nums: List[int], k: int) -> int:
res = 0
nums.sort()
prev = 0

for i, x in enumerate(nums):
if (i == 0 and x == 1) or (i > 0 and x == nums[i - 1]):
prev = x
continue

xx = x - 1
currentDelta = min(k, xx - prev)
if k <= currentDelta:
for j in range(prev + 1, prev + k + 1):
res += j

return res
else:
delta = xx

if currentDelta == 0:
prev = x
continue

upper = delta * (delta + 1) // 2
lower = prev * (prev + 1) // 2

res += upper - lower

if currentDelta >= k:
return res
k -= currentDelta
prev = x

if k > 0:
delta = prev + k
upper = delta * (delta + 1) // 2
lower = prev * (prev + 1) // 2
res += upper - lower

return res


• class Solution {
public long minimalKSum(int[] nums, int k) {
int[] arr = new int[nums.length + 2];
arr[arr.length - 1] = (int) 2e9;
for (int i = 0; i < nums.length; ++i) {
arr[i + 1] = nums[i];
}
Arrays.sort(arr);
long ans = 0;
for (int i = 1; i < arr.length; ++i) {
int a = arr[i - 1], b = arr[i];
int n = Math.min(k, b - a - 1);
if (n <= 0) {
continue;
}
k -= n;
ans += (long) (a + 1 + a + n) * n / 2;
if (k == 0) {
break;
}
}
return ans;
}
}

• func minimalKSum(nums []int, k int) int64 {
nums = append(nums, 0, 2e9)
sort.Ints(nums)
ans := 0
for i := 1; i < len(nums); i++ {
a, b := nums[i-1], nums[i]
n := min(k, b-a-1)
if n <= 0 {
continue
}
k -= n
ans += (a + 1 + a + n) * n / 2
if k == 0 {
break
}
}
return int64(ans)
}

func min(a, b int) int {
if a < b {
return a
}
return b
}


## Discuss

https://leetcode.com/problems/append-k-integers-with-minimal-sum/discuss/1823619/