# 2599. Make the Prefix Sum Non-negative

## Description

You are given a 0-indexed integer array nums. You can apply the following operation any number of times:

• Pick any element from nums and put it at the end of nums.

The prefix sum array of nums is an array prefix of the same length as nums such that prefix[i] is the sum of all the integers nums[j] where j is in the inclusive range [0, i].

Return the minimum number of operations such that the prefix sum array does not contain negative integers. The test cases are generated such that it is always possible to make the prefix sum array non-negative.

Example 1:

Input: nums = [2,3,-5,4]
Output: 0
Explanation: we do not need to do any operations.
The array is [2,3,-5,4]. The prefix sum array is [2, 5, 0, 4].


Example 2:

Input: nums = [3,-5,-2,6]
Output: 1
Explanation: we can do one operation on index 1.
The array after the operation is [3,-2,6,-5]. The prefix sum array is [3, 1, 7, 2].


Constraints:

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

## Solutions

Solution 1: Greedy + Priority Queue (Min Heap)

We use a variable $s$ to record the prefix sum of the current array.

Traverse the array $nums$, add the current element $x$ to the prefix sum $s$. If $x$ is a negative number, add $x$ to the min heap. If $s$ is negative at this time, greedily take out the smallest negative number and subtract it from $s$, and add one to the answer. Finally, return the answer.

The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$, where $n$ is the length of the array $nums$.

• class Solution {
public int makePrefSumNonNegative(int[] nums) {
PriorityQueue<Integer> pq = new PriorityQueue<>();
int ans = 0;
long s = 0;
for (int x : nums) {
s += x;
if (x < 0) {
pq.offer(x);
}
while (s < 0) {
s -= pq.poll();
++ans;
}
}
return ans;
}
}

• class Solution {
public:
int makePrefSumNonNegative(vector<int>& nums) {
priority_queue<int, vector<int>, greater<int>> pq;
int ans = 0;
long long s = 0;
for (int& x : nums) {
s += x;
if (x < 0) {
pq.push(x);
}
while (s < 0) {
s -= pq.top();
pq.pop();
++ans;
}
}
return ans;
}
};

• class Solution:
def makePrefSumNonNegative(self, nums: List[int]) -> int:
h = []
ans = s = 0
for x in nums:
s += x
if x < 0:
heappush(h, x)
while s < 0:
s -= heappop(h)
ans += 1
return ans


• func makePrefSumNonNegative(nums []int) (ans int) {
pq := hp{}
s := 0
for _, x := range nums {
s += x
if x < 0 {
heap.Push(&pq, x)
}
for s < 0 {
s -= heap.Pop(&pq).(int)
ans++
}
}
return ans
}

type hp struct{ sort.IntSlice }

func (h hp) Less(i, j int) bool { return h.IntSlice[i] < h.IntSlice[j] }
func (h *hp) Push(v any)        { h.IntSlice = append(h.IntSlice, v.(int)) }
func (h *hp) Pop() any {
a := h.IntSlice
v := a[len(a)-1]
h.IntSlice = a[:len(a)-1]
return v
}

• function makePrefSumNonNegative(nums: number[]): number {
const pq = new MinPriorityQueue();
let ans = 0;
let s = 0;
for (const x of nums) {
s += x;
if (x < 0) {
pq.enqueue(x);
}
while (s < 0) {
s -= pq.dequeue().element;
++ans;
}
}
return ans;
}