Given a 0-indexed integer array nums of size n containing all numbers from 1 to n, return the number of increasing quadruplets.
A quadruplet (i, j, k, l) is increasing if:
0 <= i < j < k < l < n, andnums[i] < nums[k] < nums[j] < nums[l].Example 1:
Input: nums = [1,3,2,4,5] Output: 2 Explanation: - When i = 0, j = 1, k = 2, and l = 3, nums[i] < nums[k] < nums[j] < nums[l]. - When i = 0, j = 1, k = 2, and l = 4, nums[i] < nums[k] < nums[j] < nums[l]. There are no other quadruplets, so we return 2.
Example 2:
Input: nums = [1,2,3,4] Output: 0 Explanation: There exists only one quadruplet with i = 0, j = 1, k = 2, l = 3, but since nums[j] < nums[k], we return 0.
Constraints:
4 <= nums.length <= 40001 <= nums[i] <= nums.lengthnums are unique. nums is a permutation.The goal of #2552 Count Increasing Quadruplets is to count index tuples (i, j, k, l) such that i < j < k < l and the values satisfy nums[i] < nums[k] < nums[j] < nums[l]. A brute‑force enumeration of all quadruplets would take O(n^4), which is infeasible for typical input sizes.
A more efficient strategy observes that the middle pair (j, k) determines the structure of valid quadruplets. We can precompute counts of elements smaller or larger on each side using prefix counting or dynamic programming. For each pair j < k where nums[j] > nums[k], we combine the number of valid i positions to the left and l positions to the right that satisfy the required inequalities.
To further optimize counting queries, a Binary Indexed Tree (Fenwick Tree) can be used to maintain prefix frequencies while iterating through the array. These optimizations reduce the complexity to roughly O(n^2) (or O(n^2 log n) with a BIT), making the problem manageable for larger arrays.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Dynamic Programming with Prefix Counts | O(n^2) | O(n) |
| Binary Indexed Tree (Fenwick Tree) | O(n^2 log n) | O(n) |
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Use these hints if you're stuck. Try solving on your own first.
Can you loop over all possible (j, k) and find the answer?
We can pre-compute all possible (i, j) and (k, l) and store them in 2 matrices.
The answer will the sum of prefix[j][k] * suffix[k][j].
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Yes, variations of counting patterns in arrays using prefix counts, dynamic programming, or Fenwick Trees are common in FAANG-style interviews. This problem tests your ability to transform brute-force enumeration into an efficient counting approach.
The optimal approach analyzes the middle pair of indices and counts valid elements on both sides using prefix statistics. By combining dynamic programming with prefix counts, the problem can be solved in about O(n^2) time instead of checking all quadruplets.
A Binary Indexed Tree (Fenwick Tree) is commonly used to maintain prefix frequencies efficiently. It helps answer how many numbers are smaller or larger than a given value while iterating through the array.
A brute-force solution checks all possible quadruplets, which requires O(n^4) time. With larger arrays, this becomes extremely slow, so optimized counting strategies using prefix information are necessary.