Find the Count of Monotonic Pairs I

Find the Count of Monotonic Pairs I

What is this problem about?

The Find the Count of Monotonic Pairs I interview question asks you to count the number of pairs of arrays $(arr1, arr2)$ such that:

1. Both arrays have the same length as a given array nums.
2. $arr1[i] + arr2[i] = nums[i]$ for all $i$.
3. $arr1$ is monotonically non-decreasing.
4. $arr2$ is monotonically non-increasing. Your goal is to find the total number of such valid pairs modulo $10^9 + 7$.

Why is this asked in interviews?

Companies like Google ask the Find the Count of Monotonic Pairs coding problem to test your proficiency with Dynamic Programming. It’s a sophisticated counting problem where each choice at index $i$ is constrained by the choice at index $i-1$. It evaluations your ability to handle state transitions and multi-variable constraints.

Algorithmic pattern used

This problem is solved using Dynamic Programming.

1. State Definition: dp[i][v] is the number of valid pairs for the first i elements where arr1[i] = v.
2. Transitions: To find dp[i][v], we look at all valid dp[i-1][prev_v].

• Condition for arr1: $prev\_v \leq v$.
• Condition for arr2: $nums[i-1] - prev\_v \geq nums[i] - v$.

1. Logic: These two conditions define a range of valid prev_v values.
2. Summation: The total count is the sum of dp[n-1][v] for all possible $v \in [0, nums[n-1]]$.

Example explanation

nums = [2, 3]

• Index 0: arr1[0] can be 0, 1, 2. dp[0][0]=1, dp[0][1]=1, dp[0][2]=1.
• Index 1: nums[1] = 3. Try arr1[1] = 2.
• arr2[1] = 3 - 2 = 1.
• Need arr1[0] \leq 2 and arr2[0] \geq 1.
• Possible arr1[0] values: 0, 1, 2.
• Check arr2[0]: $2-0=2 \geq 1$ (OK), $2-1=1 \geq 1$ (OK), $2-2=0 < 1$ (NO).
• So dp[1][2] = dp[0][0] + dp[0][1] = 2.

Common mistakes candidates make

• $O(N \cdot Max^2)$ Complexity: Using a nested loop to sum previous DP states without optimization, which might be too slow for large values of nums[i].
• Modulo: Forgetting to apply modulo $10^9 + 7$ at each addition.
• Constraint handling: Missing one of the two monotonicity requirements during the transition phase.

Interview preparation tip

For DP problems involving sums of previous states, always think about Prefix Sums. If you can calculate the sum of dp[i-1] in $O(1)$ using a prefix sum array, you can reduce the overall complexity from $O(N \cdot M^2)$ to $O(N \cdot M)$.