Start by identifying all valid sequences of three soldiers with strictly increasing or decreasing ratings. Using state transition dynamic programming allows you to count partial sequences efficiently, avoiding the brute-force cubic approach. This method leverages prefix and suffix counts to track smaller and larger ratings at each position, enabling quick aggregation for final team counts.
Problem Statement
You are given an array representing the ratings of n soldiers standing in a line. Each soldier has a unique rating, and you want to form teams of exactly three soldiers that follow specific rating rules.
A team is valid if the ratings of the three chosen soldiers are strictly increasing or strictly decreasing according to their positions in the array. Return the total number of valid teams that can be formed. Note that a soldier may belong to multiple teams.
Examples
Example 1
Input: rating = [2,5,3,4,1]
Output: 3
We can form three teams given the conditions. (2,3,4), (5,4,1), (5,3,1).
Example 2
Input: rating = [2,1,3]
Output: 0
We can't form any team given the conditions.
Example 3
Input: rating = [1,2,3,4]
Output: 4
Example details omitted.
Constraints
- n == rating.length
- 3 <= n <= 1000
- 1 <= rating[i] <= 105
- All the integers in rating are unique.
Solution Approach
Brute-force enumeration
Iterate over all triplets of soldiers and check if their ratings form a strictly increasing or decreasing sequence. This method is simple but inefficient for larger arrays, as it has O(n^3) time complexity.
State transition dynamic programming
Use DP arrays to store the number of smaller and larger ratings before and after each soldier. For each position, calculate potential teams using the counts of preceding and succeeding soldiers with appropriate ratings. This reduces complexity significantly compared to brute-force.
Optimized counting with Binary Indexed Tree
Leverage a Binary Indexed Tree or Segment Tree to maintain prefix and suffix counts efficiently. Update and query the number of smaller and larger ratings in logarithmic time to calculate the total number of valid teams, achieving O(n log(maxRating)) time complexity.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n \cdot \log(\text{maxRating})) |
| Space | O(\text{maxRating}) |
The optimized approach uses O(n log(maxRating)) time due to BIT updates and queries, and O(maxRating) space for frequency arrays. Brute-force requires O(n^3) time, while simple DP without BIT uses O(n^2).
What Interviewers Usually Probe
- Look for strictly increasing or decreasing sequences of length three.
- Check if the candidate avoids brute-force cubic time by counting sequences efficiently.
- Verify understanding of state transitions and prefix-suffix relationships for DP.
Common Pitfalls or Variants
Common pitfalls
- Counting teams incorrectly by mixing indices instead of ratings order.
- Using nested loops without considering DP or BIT optimizations for larger arrays.
- Mismanaging prefix and suffix counts, leading to off-by-one errors in team totals.
Follow-up variants
- Count teams of length k > 3 following increasing or decreasing rules.
- Compute total teams allowing repeated ratings with adjusted DP rules.
- Find the longest strictly increasing or decreasing subsequence instead of fixed-length teams.
How GhostInterview Helps
- Automatically sets up prefix and suffix counts to track increasing and decreasing sequences for you.
- Identifies edge cases where brute-force might fail, ensuring correct total team count.
- Provides step-by-step DP or BIT computation patterns to avoid common pitfalls with state transitions.
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 approach to solve Count Number of Teams efficiently?
The most efficient method uses state transition dynamic programming combined with prefix and suffix counts to calculate all valid three-soldier teams.
Can I use a brute-force approach?
Yes, but brute-force has O(n^3) complexity and will be too slow for arrays close to 1000 elements.
Why are Binary Indexed Trees useful for this problem?
BITs allow you to maintain prefix and suffix counts of smaller or larger ratings efficiently, reducing overall computation from O(n^2) to O(n log(maxRating)).
Are soldiers allowed in multiple teams?
Yes, each soldier can be part of multiple valid teams as long as the sequence rules are satisfied.
What are common mistakes when implementing DP for this pattern?
Typical mistakes include miscounting prefix or suffix arrays, confusing index order with rating order, or forgetting to handle both increasing and decreasing sequences.
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