# 2025. Maximum Number of Ways to Partition an Array

## Description

You are given a 0-indexed integer array nums of length n. The number of ways to partition nums is the number of pivot indices that satisfy both conditions:

• 1 <= pivot < n
• nums[0] + nums[1] + ... + nums[pivot - 1] == nums[pivot] + nums[pivot + 1] + ... + nums[n - 1]

You are also given an integer k. You can choose to change the value of one element of nums to k, or to leave the array unchanged.

Return the maximum possible number of ways to partition nums to satisfy both conditions after changing at most one element.

Example 1:

Input: nums = [2,-1,2], k = 3
Output: 1
Explanation: One optimal approach is to change nums[0] to k. The array becomes [3,-1,2].
There is one way to partition the array:
- For pivot = 2, we have the partition [3,-1 | 2]: 3 + -1 == 2.


Example 2:

Input: nums = [0,0,0], k = 1
Output: 2
Explanation: The optimal approach is to leave the array unchanged.
There are two ways to partition the array:
- For pivot = 1, we have the partition [0 | 0,0]: 0 == 0 + 0.
- For pivot = 2, we have the partition [0,0 | 0]: 0 + 0 == 0.


Example 3:

Input: nums = [22,4,-25,-20,-15,15,-16,7,19,-10,0,-13,-14], k = -33
Output: 4
Explanation: One optimal approach is to change nums[2] to k. The array becomes [22,4,-33,-20,-15,15,-16,7,19,-10,0,-13,-14].
There are four ways to partition the array.


Constraints:

• n == nums.length
• 2 <= n <= 105
• -105 <= k, nums[i] <= 105

## Solutions

Solution 1: Prefix Sum + Hash Table

We can preprocess to get the prefix sum array $s$ corresponding to the array $nums$, where $s[i]$ represents the sum of the array $nums[0,…i-1]$. Therefore, the sum of all elements in the array is $s[n - 1]$.

If we do not modify the array $nums$, the condition for the sums of the two subarrays to be equal is that $s[n - 1]$ must be even. If $s[n - 1]$ is even, then we calculate $ans = \frac{right[s[n - 1] / 2]}{2}$.

If we modify the array $nums$, we can enumerate each modification position $i$, change $nums[i]$ to $k$, then the change in the total sum of the array is $d = k - nums[i]$. At this time, the sum of the left part of $i$ remains unchanged, so the legal split must satisfy $s[i] = s[n - 1] + d - s[i]$, that is, $s[i] = \frac{s[n - 1] + d}{2}$. Each prefix sum of the right part has increased by $d$, so the legal split must satisfy $s[i] + d = s[n - 1] + d - (s[i] + d)$, that is, $s[i] = \frac{s[n - 1] - d}{2}$. We use hash tables $left$ and $right$ to record the number of times each prefix sum appears in the left and right parts, respectively. Then we can calculate $ans = max(ans, left[\frac{s[n - 1] + d}{2}]) + right[\frac{s[n - 1] - d}{2}]$.

Finally, we return $ans$.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array $nums$.

• class Solution {
public int waysToPartition(int[] nums, int k) {
int n = nums.length;
int[] s = new int[n];
s[0] = nums[0];
Map<Integer, Integer> right = new HashMap<>();
for (int i = 0; i < n - 1; ++i) {
right.merge(s[i], 1, Integer::sum);
s[i + 1] = s[i] + nums[i + 1];
}
int ans = 0;
if (s[n - 1] % 2 == 0) {
ans = right.getOrDefault(s[n - 1] / 2, 0);
}
Map<Integer, Integer> left = new HashMap<>();
for (int i = 0; i < n; ++i) {
int d = k - nums[i];
if ((s[n - 1] + d) % 2 == 0) {
int t = left.getOrDefault((s[n - 1] + d) / 2, 0)
+ right.getOrDefault((s[n - 1] - d) / 2, 0);
ans = Math.max(ans, t);
}
left.merge(s[i], 1, Integer::sum);
right.merge(s[i], -1, Integer::sum);
}
return ans;
}
}

• class Solution {
public:
int waysToPartition(vector<int>& nums, int k) {
int n = nums.size();
long long s[n];
s[0] = nums[0];
unordered_map<long long, int> right;
for (int i = 0; i < n - 1; ++i) {
right[s[i]]++;
s[i + 1] = s[i] + nums[i + 1];
}
int ans = 0;
if (s[n - 1] % 2 == 0) {
ans = right[s[n - 1] / 2];
}
unordered_map<long long, int> left;
for (int i = 0; i < n; ++i) {
int d = k - nums[i];
if ((s[n - 1] + d) % 2 == 0) {
int t = left[(s[n - 1] + d) / 2] + right[(s[n - 1] - d) / 2];
ans = max(ans, t);
}
left[s[i]]++;
right[s[i]]--;
}
return ans;
}
};

• class Solution:
def waysToPartition(self, nums: List[int], k: int) -> int:
n = len(nums)
s = [nums[0]] * n
right = defaultdict(int)
for i in range(1, n):
s[i] = s[i - 1] + nums[i]
right[s[i - 1]] += 1

ans = 0
if s[-1] % 2 == 0:
ans = right[s[-1] // 2]

left = defaultdict(int)
for v, x in zip(s, nums):
d = k - x
if (s[-1] + d) % 2 == 0:
t = left[(s[-1] + d) // 2] + right[(s[-1] - d) // 2]
if ans < t:
ans = t
left[v] += 1
right[v] -= 1
return ans


• func waysToPartition(nums []int, k int) (ans int) {
n := len(nums)
s := make([]int, n)
s[0] = nums[0]
right := map[int]int{}
for i := range nums[:n-1] {
right[s[i]]++
s[i+1] = s[i] + nums[i+1]
}
if s[n-1]%2 == 0 {
ans = right[s[n-1]/2]
}
left := map[int]int{}
for i, x := range nums {
d := k - x
if (s[n-1]+d)%2 == 0 {
t := left[(s[n-1]+d)/2] + right[(s[n-1]-d)/2]
if ans < t {
ans = t
}
}
left[s[i]]++
right[s[i]]--
}
return
}