Maximum Profitable Triplets With Increasing Prices I

Maximum Profitable Triplets With Increasing Prices I

1. What is this problem about?

The Maximum Profitable Triplets With Increasing Prices I interview question is an array optimization problem that focuses on finding a specific pattern within three elements. You are given two arrays: prices and profits. You need to find three indices (i, j, k) such that i < j < k and prices[i] < prices[j] < prices[k]. The goal is to maximize the sum of profits associated with these three indices: profits[i] + profits[j] + profits[k].

This is a variation of the "increasing triplet" problem, but instead of just finding if one exists, you are maximizing a secondary metric (profit).

2. Why is this asked in interviews?

Companies like IBM ask the Maximum Profitable Triplets With Increasing Prices I coding problem to test a candidate's ability to handle multi-constrained optimization. It evaluates whether you can move beyond a brute-force $O(N^3)$ approach and find more efficient solutions. It tests your ability to think about "central elements" in a triplet and how to precalculate or efficiently query information about elements to their left and right.

3. Algorithmic pattern used

The Array, Binary Indexed Tree, Segment Tree interview pattern is applicable here for optimal performance. A common approach is to iterate through each index j and treat it as the middle element of the triplet.

1. For a fixed j, you need to find the maximum profits[i] such that i < j and prices[i] < prices[j].
2. You also need to find the maximum profits[k] such that k > j and prices[k] > prices[j].
3. To do this efficiently, you can use a Segment Tree or a Binary Indexed Tree (BIT) to store the maximum profits seen so far, indexed by their price.

4. Example explanation

Prices: [10, 2, 5, 8], Profits: [50, 20, 30, 40] Let's check each j:

• j=1 (Price 2): No i < j with price < 2.
• j=2 (Price 5):
  • Left (i < 2, price < 5): Only index 1 (Price 2, Profit 20). Max Left Profit = 20.
  • Right (k > 2, price > 5): Index 3 (Price 8, Profit 40). Max Right Profit = 40.
  • Total: 20 + 30 + 40 = 90.
• j=3 (Price 8):
  • Left (i < 3, price < 8): Price 2 (Profit 20) and Price 5 (Profit 30). Max Left = 30.
  • Right (k > 3): None.

Max Profitable Triplet is 90.

5. Common mistakes candidates make

A common error in the Maximum Profitable Triplets With Increasing Prices I coding problem is sticking with the $O(N^3)$ approach, which will fail for larger inputs. Another mistake is forgetting that both the index order (i < j < k) and the price order (prices[i] < prices[j] < prices[k]) must be satisfied simultaneously. Some candidates also struggle with coordinate compression if the prices are very large but the number of items is small.

6. Interview preparation tip

When you need to find "something to the left" and "something to the right" for every element in an array, think about precalculation passes (one from left-to-right, one from right-to-left). If the condition depends on the values themselves (like prices[i] < prices[j]), data structures like Segment Trees or Fenwick Trees are your best friends.