Start by defining a DP table dp[i][s] representing the number of monotonic pairs of length i ending with value s. Iterate through the array and update counts using previous states, ensuring transitions maintain monotonicity. Sum the final states to obtain the total count, leveraging combinatorial optimization and prefix sums for efficiency.
Problem Statement
Given an array nums of n positive integers, count all pairs of non-negative integer arrays (arr1, arr2) that satisfy monotonic constraints. A pair is considered monotonic if each element of arr1 and arr2 maintains the required ordering, as defined by the problem.
Return the total number of monotonic pairs possible for the given nums array. The constraints are 1 <= n <= 2000 and 1 <= nums[i] <= 50, and results must account for all valid combinations efficiently.
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] <= 50
Solution Approach
Dynamic Programming State Definition
Define dp[i][s] as the number of monotonic pairs of length i where the last element of arr1 is s. This captures all valid transitions and allows building solutions incrementally while enforcing monotonicity.
Iterative Transition
For each element in nums, iterate over possible states s and update dp[i][s] using previous dp[i-1][prev] values where prev <= s. Use prefix sums to accelerate cumulative counts and avoid recomputation.
Result Aggregation
After processing all elements, sum dp[n][s] over all valid s to get the total number of monotonic pairs. Ensure boundary conditions and combinatorial counts are applied correctly to handle repeated numbers.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity depends on n and the maximum nums[i] value due to DP iterations and prefix sums, roughly O(n * maxValue). Space complexity is O(n * maxValue) for the DP table.
What Interviewers Usually Probe
- Check if candidate correctly defines dp[i][s] and understands state transitions.
- Observe whether prefix sums are used to optimize the cumulative counts in DP.
- Watch for correct handling of repeated elements and monotonic constraints.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to enforce monotonicity when updating DP states.
- Inefficiently iterating without prefix sums, leading to timeouts.
- Incorrectly summing final DP values, missing some valid pairs.
Follow-up variants
- Counting strictly increasing monotonic pairs instead of non-decreasing.
- Finding monotonic triples or larger tuples with similar DP approach.
- Applying the DP state transition pattern to other bounded integer arrays with combinatorial constraints.
How GhostInterview Helps
- Automatically generates the correct DP state definition and guides through transitions for monotonic constraints.
- Highlights the optimal prefix sum optimizations to reduce redundant computation in counting pairs.
- Validates final aggregation and ensures all edge cases with repeated numbers are handled correctly.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the main idea behind counting monotonic pairs in this problem?
The core idea is using a DP table dp[i][s] where each state represents the number of monotonic pairs ending with a specific value, ensuring transitions maintain monotonicity.
Can this problem handle arrays with maximum length 2000 efficiently?
Yes, with state transition DP and prefix sums, the algorithm scales to n = 2000 without timing out.
Why do we use prefix sums in the DP solution?
Prefix sums allow cumulative updates efficiently, avoiding O(n^2) inner loops when summing valid previous states.
Are repeated numbers in nums handled differently?
Repeated numbers are naturally handled in the DP state; the transitions include counts from previous identical values without violating monotonicity.
Is the state transition dynamic programming pattern common in combinatorial array problems?
Yes, this pattern efficiently counts sequences or pairs under constraints and is a key approach for monotonic pair counting like in this problem.
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This problem involves finding the count of monotonic pairs in an array using dynamic programming and combinatorics techniques.
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