Problem Description

In this problem, we're given two 0-indexed integer arrays: prices and profits. Both arrays have a length of n, and they represent items in a store where the ith item has a price of prices[i] and a potential profit of profits[i].

The task is to select three different items to maximize our profit under a specific condition: we must pick items in a manner that their indices i, j, and k follow i < j < k, and their prices also follow an increasing order, prices[i] < prices[j] < prices[k]. In simpler terms, the second item's price and index should be higher than the first, and the third item's price and index should be higher than the second.

Our profit from such a selection would be the sum of the corresponding profits of these items: profits[i] + profits[j] + profits[k].

The output should be the maximum achievable profit following these rules. However, if it's impossible to find such a triplet of items, the function should return -1.

Intuition

The intuitive approach to solve this problem is by trying to break it down and identify subproblems that make up the whole. We do this by picking one piece of the problem to focus on and then solving the remaining aspects around it.

In this case, we can streamline our search by fixing one of the elements as the pivotal middle element - that's the j index or our second item's position. The aim is to select this middle element such that we can find one i and one k where both the index and price restrictions are satisfied (i < j < k with prices[i] < prices[j] < prices[k]).

Once j is fixed, we look for the maximum profit on the left (profits[i]) and the maximum profit on the right (profits[k]) that fulfill our conditions related to both price and index ordering. By doing this for every possible middle element, we ensure that no possible profitable combination is missed.

The variables left and right in the solution help us keep track of the maximum profits found to the left and right of our fixed middle element. If we find valid left and right profits, we calculate the total profit for the current selection and compare it with the maximum found so far (ans). Repeating this process and updating ans when we find a greater total profit ensures that we arrive at the highest possible profit by the end of the function.

This strategy reduces the problem from a three-dimensional one (where we could naively check every possible triplet of items) to a one-dimensional problem focused around the chosen mid-point, which is much more efficient.

Learn more about Dynamic Programming patterns.

Solution Approach

The implementation of the solution begins by first understanding that the problem can be approached by iterating through the list of items and fixing the middle element of the triplet to simplify the process. This approach falls under the pattern of iteration combined with local optimization.

Here's how the solution works:

1. Initialization: The variable ans is initialized with -1, this will store the maximum profit or remain -1 if no valid trio is found.

2. Outer Loop: We start by creating an outer loop to iterate over each potential middle item j:

   for j, x in enumerate(profits):

3. Initialize Left and Right Profit: For each j, initialize two variables left and right both to 0. These will store the maximum profits to the left and to the right of j respectively.

4. Find Left Maximum Profit: We iterate from the start of the list to the element just before j to find the maximum profit (profits[i]) such that the price at i is less than the price at j:

   for i in range(j):
       if prices[i] < prices[j] and left < profits[i]:
           left = profits[i]

5. Find Right Maximum Profit: Similarly, we iterate from the element after j to the end of the list to find the maximum profit (profits[k]) such that the price at j is less than the price at k:

   for k in range(j + 1, n):
       if prices[j] < prices[k] and right < profits[k]:
           right = profits[k]

6. Calculate and Compare Profit: If we found valid left and right profits (meaning that left and right are not 0), we calculate the total profit for this triplet, which is left + x + right, where x is profits[j]. We then check if this total profit is greater than the current maximum profit stored in ans:

   if left and right:
       ans = max(ans, left + x + right)

7. Return Result: After iterating through all possible middle elements, ans contains the maximum profit. The function finally returns ans which is either the maximum profit found or -1 if no valid triplet was found.

The algorithm efficiently updates its best-known solution with each iteration, employing an iterative approach, while breaking the problem down into smaller sub-tasks (finding left and right maximums) that can be solved independently of each other.

This algorithm runs in O(n^2) time, because for each of the n elements chosen as a middle element, it performs potentially O(n) work to find left and O(n) work to find right. While not the most efficient for very large numbers of items, it's a reasonable approach that improves over a naive O(n^3) solution that would examine all possible triplets one by one.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Suppose we have the following prices and profits arrays:

• prices = [3, 1, 4, 2]
• profits = [4, 1, 5, 3]

And we wish to maximize our profit by selecting three items such that their prices and indices are in strictly increasing order.

The process will be as follows:

1. Initialize ans = -1, as we have not found any triplet yet.

2. The outer loop iterates with j starting from 0 to n-1. In this case, n is 4. So j will take on values 0, 1, 2, and 3 one by one.

3. For each j, initialize left = 0 and right = 0.

4. For j = 0, there is no element to the left of profits[j], so we can skip and continue with j = 1.

5. For j = 1, we have:

   • left = 0: No profit picked yet.
   • Start i loop from 0 to (j-1), i.e., i = 0.
   • Check if prices[0] < prices[1], which is 3 < 1. This is false, so we skip.

6. Move to j = 2:

   • left = 0: No profit picked yet.

   • Start i loop:

     • i = 0: prices[0] < prices[2] is 3 < 4, true, and profits[0] is 4, which is greater than left.

     • Update left = 4.

     • i = 1: prices[1] < prices[2] is 1 < 4, true, but profits[1] is 1, which is not greater than the current left.

     • Keep left = 4.

   • Start k loop from (j+1) to n, i.e., k = 3.

   • prices[2] < prices[3] is 4 < 2. This is false, so no right profit is updated.

7. Continue to j = 3:

   • left = 0 again.

   • Start i loop:

     • i = 0: prices[0] < prices[3] is 3 < 2. This is false, so continue.
     • i = 1: prices[1] < prices[3] is 1 < 2. This is true and profits[1] is 1.
     • Update left = 1.
     • i = 2: prices[2] < prices[3]. This is false, so keep left = 1.

   • Since we are at the last element, there are no elements to the right for comparison; hence, no update to right.

Final step:

• The maximum profit calculated with j=2 was left + profits[j] + right, but because we did not find a valid right, we do not update ans.
• Therefore, the output of the algorithm would be -1 since we never updated our initial ans value from -1 given that no valid triplet could satisfy the conditions of the problem with the provided example arrays.

Solution Implementation

Time and Space Complexity

The time complexity of this function maxProfit is O(n^2). This is because there are two nested loops. The outer loop runs n times, where n is the length of the input lists prices and profits. For each iteration of the outer loop, the inner loops (one for calculating left and another for calculating right) in total can iterate up to n times in the worst case (when j is at the center of the array). Since these inner loops are nested within the outer loop, the total number of operations can be up to n * n, which simplifies to O(n^2).

The space complexity of this function is O(1). This is because the extra space used by the function does not depend on the input size n. Only a fixed number of variables (ans, left, right, n, j, x, i, and k) are used to keep track of the maximum profit as well as loop counters and values. Therefore, the space used remains constant regardless of the input size.

Learn more about how to find time and space complexity quickly using problem constraints.