#3251

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Find the Count of Monotonic Pairs II

This problem involves finding the count of monotonic pairs in an array using dynamic programming and combinatorics techniques.

array math dynamic programming combinatorics prefix sum

Problem Statement

You are given an array nums of length n, where each element is a positive integer. A pair of indices (i, j) is called a monotonic pair if nums[i] <= nums[j] when i < j, or nums[i] >= nums[j] when i < j. Your task is to return the count of all such monotonic pairs in the array.

You must consider all valid pairs (i, j) where i < j. The number of monotonic pairs is crucial to the solution of this problem and requires leveraging dynamic programming and combinatorics techniques to efficiently compute the result. Note that brute-force approaches will not scale well due to the constraints.

Examples

Example 1

Input: nums = [2,3,2]

Output: 4

The good pairs are:

Example 2

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

Output: 126

Example details omitted.

Constraints

1 <= n == nums.length <= 2000
1 <= nums[i] <= 1000

Solution Approach

Dynamic Programming with State Transitions

The first step in solving this problem is using dynamic programming (DP). A state transition DP approach can help efficiently track and compute the number of monotonic pairs as we iterate through the array. By maintaining arrays that track the number of valid pairs for each element in the array, we can accumulate results without checking every pair.

Combinatorics for Pair Counting

Another key approach is to use combinatorics. Given that the numbers can repeat, counting combinations of elements that form valid pairs is more efficient than individually checking every combination. This reduces the complexity of finding monotonic pairs, especially in cases where large numbers of duplicate elements exist.

Prefix Sum to Accelerate Computation

Prefix sums can be employed to quickly calculate the number of monotonic pairs up to each point in the array. This can be used to speed up the solution by reducing the need to calculate sums repeatedly for each pair, making the solution more optimal and efficient.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity of the solution depend on the exact approach chosen. A dynamic programming approach can run in O(n^2) time, where n is the length of the array, while using combinatorics and prefix sums can help optimize the solution, potentially reducing the time complexity to O(n log n) or better in some cases.

What Interviewers Usually Probe

The candidate demonstrates an understanding of dynamic programming and combinatorics as applied to array problems.
The candidate is able to optimize brute-force solutions using more efficient techniques like prefix sums.
The candidate's explanation of complexity and trade-offs shows awareness of both time and space optimization.

Common Pitfalls or Variants

Common pitfalls

Failing to optimize the brute force approach leads to excessive time complexity.
Incorrectly handling duplicate elements in the array, which may lead to overcounting or undercounting pairs.
Misunderstanding the monotonic condition, which could lead to counting invalid pairs.

Follow-up variants

Handling arrays with duplicates more efficiently.
Optimizing for arrays with a small number of elements.
Applying a divide-and-conquer approach to improve performance on large arrays.

FAQ

What is the primary approach to solving the "Find the Count of Monotonic Pairs II" problem?

The primary approach to solving this problem involves using dynamic programming with state transitions to efficiently track valid monotonic pairs.

How do I avoid brute-force time complexity in "Find the Count of Monotonic Pairs II"?

You can avoid brute-force time complexity by leveraging dynamic programming, combinatorics, and prefix sums to optimize the counting of pairs.

What are common mistakes when solving "Find the Count of Monotonic Pairs II"?

Common mistakes include failing to optimize for time complexity, mishandling duplicate elements, and misunderstanding the conditions for monotonic pairs.

What is the complexity of the "Find the Count of Monotonic Pairs II" problem?

The complexity depends on the approach: using dynamic programming can result in O(n^2) time complexity, while combinatorics and prefix sums can optimize the solution to O(n log n) or better.

How does GhostInterview help with the "Find the Count of Monotonic Pairs II" problem?

GhostInterview offers structured guidance on applying dynamic programming, combinatorics, and optimization techniques to solve "Find the Count of Monotonic Pairs II" efficiently.

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Solution

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

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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

Continue Practicing

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