Maximum Increasing Triplet Value

Maximum Increasing Triplet Value

What is this problem about?

The Maximum Increasing Triplet Value coding problem asks you to find three indices i < j < k in an array such that nums[i] < nums[j] < nums[k] and the value nums[i] * nums[j] * nums[k] is maximized (or some variant combining the elements). This is an extension of the classic "does a 132 pattern exist" question, now requiring optimization of a value over all valid increasing triplets.

Why is this asked in interviews?

Uber uses this problem to test candidates' ability to efficiently scan for constrained subsequences. A naive O(n³) triple loop is easy to write but doesn't scale. Candidates who know ordered sets or sorted structures can query "the best left value for each middle element" and "the best right value for each middle element" in O(log n), yielding an O(n log n) solution. This demonstrates algorithmic maturity.

Algorithmic pattern used

The solution uses an ordered set (balanced BST) approach. For each index j (the middle element):

• Maintain a sorted structure of all elements to the left of j that are less than nums[j].
• Maintain a sorted structure of all elements to the right of j that are greater than nums[j].
• For each j, query the best left element (e.g., maximum < nums[j]) and best right element (e.g., maximum > nums[j]) to maximize the product.

Pre-processing right-side maxima with a suffix pass can also work for certain variants, reducing complexity further.

Example explanation

Array: [5, 1, 4, 2, 3] Triplets i < j < k with nums[i] < nums[j] < nums[k]:

• (1, 4, _): No right element > 4 exists after index 2. Skip.
• (1, 2, 3): Value = 1 * 2 * 3 = 6.
• (1, 4, _): not valid.
• (2, 3, _): No element after index 4 > 3.

Best valid triplet: (1, 2, 3) → product = 6.

With ordered set: at j=3 (value 2), left ordered set has {1} (elements < 2 seen so far). Right pass gives max > 2 after index 3 is 3. Product = 1 * 2 * 3 = 6.

Common mistakes candidates make

• Brute-forcing with O(n³): Always fails on large inputs. Mention it as a starting point but immediately optimize.
• Not considering all combinations for the left/right element: The maximum product may come from a non-obvious combination of left and right elements.
• Confusing "maximum" vs "any valid" triplet: The problem asks for maximum value, not just existence.

Interview preparation tip

For the Array Ordered Set interview pattern, practice using a sorted container (like a multiset in C++ or SortedList in Python) to maintain running candidates. Triplet subsequence problems almost always benefit from separating left and right preprocessing passes. Think: "for each middle element, what's the best left and right?"