Find the Count of Monotonic Pairs II

Find the Count of Monotonic Pairs II

What is this problem about?

The Find the Count of Monotonic Pairs II interview question is the optimized version of the monotonic pairs challenge. The rules are identical: find pairs $(arr1, arr2)$ where $arr1$ increases, $arr2$ decreases, and $arr1[i] + arr2[i] = nums[i]$. However, the constraints on $N$ and $nums[i]$ are significantly larger ($10^5$ and $1000$ respectively), requiring a highly efficient solution.

Why is this asked in interviews?

BNY Mellon and other financial firms use the Find the Count of Monotonic Pairs II coding problem to test a candidate's ability to optimize Dynamic Programming transitions. It separates candidates who can write a basic DP from those who can use advanced techniques like Prefix Sum optimization or Difference Arrays to achieve $O(N \cdot M)$ time complexity.

Algorithmic pattern used

This problem relies on DP with Prefix Sum Optimization.

1. Inequality Logic: The conditions $prev\_v \leq v$ and $nums[i-1] - prev\_v \geq nums[i] - v$ can be combined into: prev_v <= min(v, v - (nums[i] - nums[i-1])).
2. Simplified Range: Let $diff = max(0, nums[i] - nums[i-1])$. The condition becomes prev_v <= v - diff.
3. Prefix Sums:

• Calculate dp[i-1] for all values.
• Compute the prefix sums of dp[i-1].
• The value dp[i][v] is simply the prefix sum of dp[i-1] up to the index v - diff.

1. Result: This allows you to fill the DP table in $O(N imes ext{MaxValue})$.

Example explanation

nums = [2, 5]. $diff = 5 - 2 = 3$.

• dp[0] prefix sums: [1, 2, 3] (for values 0, 1, 2).
• For dp[1][4]:
• v = 4, diff = 3.
• prev_v \leq 4 - 3 = 1.
• dp[1][4] = prefixSum[1] = 2. Pairs: (arr1:[0, 4], arr2:[2, 1]) and (arr1:[1, 4], arr2:[1, 1]).

Common mistakes candidates make

• Inefficient loops: Failing to use prefix sums, resulting in a cubic time complexity that times out on large test cases.
• Negative Indices: Not checking if v - diff is less than zero before accessing the prefix sum array.
• Memory Limit: Using a full $N imes M$ 2D array. You only need the previous row, so space can be optimized to $O(M)$.

Interview preparation tip

Master DP state optimization. Whenever your DP transition involves summing a range of values from the previous step, a prefix sum array is almost always the intended way to optimize it. This is a core Prefix Sum interview pattern.