3250. Find the Count of Monotonic Pairs I

Description

You are given an array of positive integers nums of length n.

We call a pair of non-negative integer arrays (arr1, arr2) monotonic if:

• The lengths of both arrays are n.
• arr1 is monotonically non-decreasing, in other words, arr1[0] <= arr1[1] <= ... <= arr1[n - 1].
• arr2 is monotonically non-increasing, in other words, arr2[0] >= arr2[1] >= ... >= arr2[n - 1].
• arr1[i] + arr2[i] == nums[i] for all 0 <= i <= n - 1.

Return the count of monotonic pairs.

Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: nums = [2,3,2]

Output: 4

Explanation:

The good pairs are:

1. ([0, 1, 1], [2, 2, 1])
2. ([0, 1, 2], [2, 2, 0])
3. ([0, 2, 2], [2, 1, 0])
4. ([1, 2, 2], [1, 1, 0])

Example 2:

Input: nums = [5,5,5,5]

Output: 126

Constraints:

• 1 <= n == nums.length <= 2000
• 1 <= nums[i] <= 50

Solutions

Solution 1: Dynamic Programming + Prefix Sum Optimization

We define $f[i][j]$ to represent the number of monotonic array pairs for the subarray $[0, \ldots, i]$ where $arr1[i] = j$. Initially, $f[i][j] = 0$, and the answer is $\sum_{j=0}^{\textit{nums}[n-1]} f[n-1][j]$.

When $i = 0$, we have $f[0][j] = 1$ for $0 \leq j \leq \textit{nums}[0]$.

When $i > 0$, we can calculate $f[i][j]$ based on $f[i-1][j’]$. Since $\textit{arr1}$ is non-decreasing, $j’ \leq j$. Additionally, since $\textit{arr2}$ is non-increasing, $\textit{nums}[i] - j \leq \textit{nums}[i - 1] - j’$. Thus, $j’ \leq \min(j, j + \textit{nums}[i - 1] - \textit{nums}[i])$.

The answer is $\sum_{j=0}^{\textit{nums}[n-1]} f[n-1][j]$.

The time complexity is $O(n \times m)$, and the space complexity is $O(n \times m)$. Here, $n$ represents the length of the array $\textit{nums}$, and $m$ represents the maximum value in the array $\textit{nums}$.

• class Solution {
public int countOfPairs(int[] nums) {
final int mod = (int) 1e9 + 7;
int n = nums.length;
int m = Arrays.stream(nums).max().getAsInt();
int[][] f = new int[n][m + 1];
for (int j = 0; j <= nums[0]; ++j) {
f[0][j] = 1;
}
int[] g = new int[m + 1];
for (int i = 1; i < n; ++i) {
g[0] = f[i - 1][0];
for (int j = 1; j <= m; ++j) {
g[j] = (g[j - 1] + f[i - 1][j]) % mod;
}
for (int j = 0; j <= nums[i]; ++j) {
int k = Math.min(j, j + nums[i - 1] - nums[i]);
if (k >= 0) {
f[i][j] = g[k];
}
}
}
int ans = 0;
for (int j = 0; j <= nums[n - 1]; ++j) {
ans = (ans + f[n - 1][j]) % mod;
}
return ans;
}
}


• class Solution {
public:
int countOfPairs(vector<int>& nums) {
const int mod = 1e9 + 7;
int n = nums.size();
int m = *max_element(nums.begin(), nums.end());
vector<vector<int>> f(n, vector<int>(m + 1));
for (int j = 0; j <= nums[0]; ++j) {
f[0][j] = 1;
}
vector<int> g(m + 1);
for (int i = 1; i < n; ++i) {
g[0] = f[i - 1][0];
for (int j = 1; j <= m; ++j) {
g[j] = (g[j - 1] + f[i - 1][j]) % mod;
}
for (int j = 0; j <= nums[i]; ++j) {
int k = min(j, j + nums[i - 1] - nums[i]);
if (k >= 0) {
f[i][j] = g[k];
}
}
}
int ans = 0;
for (int j = 0; j <= nums[n - 1]; ++j) {
ans = (ans + f[n - 1][j]) % mod;
}
return ans;
}
};


• class Solution:
def countOfPairs(self, nums: List[int]) -> int:
mod = 10**9 + 7
n, m = len(nums), max(nums)
f = [[0] * (m + 1) for _ in range(n)]
for j in range(nums[0] + 1):
f[0][j] = 1
for i in range(1, n):
s = list(accumulate(f[i - 1]))
for j in range(nums[i] + 1):
k = min(j, j + nums[i - 1] - nums[i])
if k >= 0:
f[i][j] = s[k] % mod
return sum(f[-1][: nums[-1] + 1]) % mod


• func countOfPairs(nums []int) (ans int) {
const mod int = 1e9 + 7
n := len(nums)
m := slices.Max(nums)
f := make([][]int, n)
for i := range f {
f[i] = make([]int, m+1)
}
for j := 0; j <= nums[0]; j++ {
f[0][j] = 1
}
g := make([]int, m+1)
for i := 1; i < n; i++ {
g[0] = f[i-1][0]
for j := 1; j <= m; j++ {
g[j] = (g[j-1] + f[i-1][j]) % mod
}
for j := 0; j <= nums[i]; j++ {
k := min(j, j+nums[i-1]-nums[i])
if k >= 0 {
f[i][j] = g[k]
}
}
}
for j := 0; j <= nums[n-1]; j++ {
ans = (ans + f[n-1][j]) % mod
}
return
}


• function countOfPairs(nums: number[]): number {
const mod = 1e9 + 7;
const n = nums.length;
const m = Math.max(...nums);
const f: number[][] = Array.from({ length: n }, () => Array(m + 1).fill(0));
for (let j = 0; j <= nums[0]; j++) {
f[0][j] = 1;
}
const g: number[] = Array(m + 1).fill(0);
for (let i = 1; i < n; i++) {
g[0] = f[i - 1][0];
for (let j = 1; j <= m; j++) {
g[j] = (g[j - 1] + f[i - 1][j]) % mod;
}
for (let j = 0; j <= nums[i]; j++) {
const k = Math.min(j, j + nums[i - 1] - nums[i]);
if (k >= 0) {
f[i][j] = g[k];
}
}
}
let ans = 0;
for (let j = 0; j <= nums[n - 1]; j++) {
ans = (ans + f[n - 1][j]) % mod;
}
return ans;
}