Find the Count of Monotonic Pairs I

Problem Description

You are given an array of positive integers nums of length n. We call a pair of non-negative integer arrays arr1 and arr2 monotonic if:

• The lengths of both arrays are n.
• arr1 is monotonically non-decreasing.
• arr2 is monotonically non-increasing.
• 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 10^9 + 7.

Key Insights

• The problem requires generating pairs of arrays that satisfy specific monotonic conditions.
• The sum of each pair's elements must equal the corresponding element in the nums array.
• The constraints on the values of nums and its length suggest using combinatorial counting techniques.
• Dynamic programming can be leveraged to efficiently count valid pairs while maintaining the monotonicity conditions.

Space and Time Complexity

Time Complexity: O(n * max_num^2), where max_num is the maximum value in nums. Space Complexity: O(max_num), used for storing counts of potential values for arr1.

Solution

We can approach this problem using dynamic programming. The idea is to maintain counts of how many ways we can form valid pairs while ensuring the monotonic conditions are met.

1. Dynamic Programming Table: Use a DP table where dp[i][j] represents the number of ways to fill arr1 and arr2 up to index i such that arr1[i] = j and the monotonicity conditions are satisfied.

2. Transition: For each index i in nums, we calculate valid pairs for arr1 and arr2:

   • Iterate through possible values of arr1[i] from 0 to nums[i].
   • For each choice of arr1[i], compute arr2[i] as nums[i] - arr1[i].
   • Update counts based on previous entries in the DP table while ensuring the non-decreasing and non-increasing properties.

3. Final Count: Sum all valid configurations from the last index of the DP table.

This approach ensures we efficiently count the valid pairs without explicitly generating all configurations.

Code Solutions