Maximum Profitable Triplets With Increasing Prices II

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ArraysDynamic Programming

You are given a 0-indexed array prices of length n representing the prices of various items, and a positive integer k.

You are tasked with finding the maximum profit you can make by buying exactly three items such that:

You buy the items in increasing order of index.
The price of the third item is at least k times the price of the first item.

More formally, you need to find indices i, j, and l such that 0 <= i < j < l < n and prices[l] >= k * prices[i], and maximize the profit prices[i] - prices[j] + prices[l].

Return the maximum profit you can make, or -1 if it's not possible to buy exactly three items with the given conditions.

Example 1:

Input: prices = [1,2,3,4,5], k = 2
Output: 7
Explanation: We can buy items at indices 0, 1, and 4. The prices are 1, 2, and 5 respectively.
The price of the third item is at least k times the price of the first item (5 >= 2 * 1).
The profit is 1 - 2 + 5 = 4. It can be proven that 4 is the maximum profit we can make.

Example 2:

Input: prices = [1,3,2,4,5], k = 2
Output: 7
Explanation: We can buy items at indices 0, 2, and 4. The prices are 1, 2, and 5 respectively.
The price of the third item is at least k times the price of the first item (5 >= 2 * 1).
The profit is 1 - 2 + 5 = 4. It can be proven that 4 is the maximum profit we can make.

Example 3:

Input: prices = [1,2,3,4], k = 2
Output: -1
Explanation: It is not possible to buy exactly three items such that the price of the third item is at least k times the price of the first item.

Constraints:

3 <= prices.length <= 10⁵
1 <= prices[i] <= 10⁵
1 <= k <= 100

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the `prices` and `profits` values? Could they be negative or zero?
What should I return if no valid triplet exists that satisfies the conditions?
Can the input arrays `prices` and `profits` be empty, or contain null or undefined values?
Do `prices` and `profits` always have the same length, and is this length at least 3 to form a triplet?
Are duplicate price values allowed in the `prices` array, and if so, how does that affect the definition of a valid triplet?

Brute Force Solution

Approach

The brute force approach involves checking every single possible combination of three price points to find the most profitable triplet. We look at all groups of three prices, see if they're in increasing order, and then calculate the profit from each valid group. Finally, we pick the group with the highest profit.

Here's how the algorithm would work step-by-step:

Take the first price in the list.
For each remaining price after the first, consider it as the second price.
For each remaining price after the second, consider it as the third price.
Check if the first price is less than the second price and if the second price is less than the third price.
If the prices are in increasing order, calculate the profit by subtracting the first price from the second, and then subtracting the second price from the third.
Remember the profit calculated for this triplet.
Repeat this process for every possible combination of three prices from the original list.
After checking every combination, compare all the recorded profits to find the largest one. That's the maximum profit.

Code Implementation

def find_maximum_profitable_triplets(prices):
    highest_profit = 0
    number_of_prices = len(prices)

    # Iterate through all possible combinations of three prices
    for first_index in range(number_of_prices):
        for second_index in range(first_index + 1, number_of_prices):
            for third_index in range(second_index + 1, number_of_prices):
                # Check if the prices are in increasing order
                if prices[first_index] < prices[second_index] < prices[third_index]:
                    current_profit = prices[first_index] + prices[second_index] + prices[third_index]

                    # Keep track of the highest profit
                    if current_profit > highest_profit:
                        highest_profit = current_profit

    return highest_profit

Big(O) Analysis

Time Complexity

O(n^3) – The brute force approach iterates through all possible triplets in the input array of size n. We have three nested loops to choose the first, second, and third prices, respectively. The outer loop iterates n times, the second loop iterates up to n times, and the innermost loop also iterates up to n times. Therefore, the total number of operations is proportional to n * n * n, resulting in a time complexity of O(n^3).

Space Complexity

O(1) – The brute force approach, as described, only uses a few constant space variables to keep track of the indices of the prices being considered in the triplets and the maximum profit found so far. No auxiliary data structures that scale with the input size N (number of prices) are used. Therefore, the space complexity remains constant, regardless of the number of prices.

Optimal Solution

Approach

To find the most profitable set of three increasing prices, we efficiently determine the best 'middle' price. We find the maximum profit possible before and after each price point, then combine them to find the highest total profit for any valid triplet.

Here's how the algorithm would work step-by-step:

For each price, we need to find the best buying price that comes before it.
To do this, we go through the prices from beginning to end, keeping track of the lowest buying price we've seen so far.
For each price, we subtract the lowest buying price we've seen to calculate the maximum profit possible if we bought at the lowest earlier price and sold at the current price.
Next, we need to find the best selling price that comes after the middle price.
We go through the prices from end to beginning, keeping track of the highest selling price we've seen so far.
For each price, we calculate the maximum profit possible if we bought at the current price and sold at the highest later price.
Finally, for each price, we add the maximum profit from buying earlier to the maximum profit from selling later, along with the value of the current middle price.
We keep track of the largest combined profit we find and return it as the result.

Code Implementation

def max_profit_triplet(prices):
    number_of_prices = len(prices)
    best_buy_before = [0] * number_of_prices
    best_sell_after = [0] * number_of_prices

    # Find best buying price before each day
    min_price_so_far = prices[0]
    best_buy_before[0] = 0
    for i in range(1, number_of_prices):
        best_buy_before[i] = min_price_so_far
        min_price_so_far = min(min_price_so_far, prices[i])

    # Find best selling price after each day
    max_price_so_far = prices[-1]
    best_sell_after[-1] = 0
    for i in range(number_of_prices - 2, -1, -1):
        best_sell_after[i] = max_price_so_far

        max_price_so_far = max(max_price_so_far, prices[i])

    max_profit = 0

    # Calculate the profit for each triplet
    for i in range(1, number_of_prices - 1):

        if best_buy_before[i] < prices[i] and prices[i] < best_sell_after[i]:

            profit = best_sell_after[i] - prices[i] + prices[i] - best_buy_before[i]

            max_profit = max(max_profit, profit)

    return max_profit

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the prices array three times. The first iteration (steps 2 and 3) finds the best buying price before each price. The second iteration (steps 5 and 6) finds the best selling price after each price. The final iteration (step 7) combines these profits to find the maximum total profit. Each of these iterations depends on the input size 'n', the number of prices, and each element is visited once per iteration. Since the number of operations scales linearly with the input size, the time complexity is O(n).

Space Complexity

O(N) – The algorithm uses two auxiliary lists, `buy_profits` and `sell_profits`, to store the maximum profit possible before and after each price, respectively. Both lists have the same length as the input `prices` array, which we denote as N. Therefore, the space complexity is directly proportional to the input size N because two arrays of size N are created to hold intermediate profit values.

Edge Cases

Case	How to Handle
Empty prices or profits array	Return 0 since no triplet can be formed and thus no profit exists.
prices or profits array has less than 3 elements	Return 0 since a triplet i < j < k cannot be formed.
prices or profits are null	Return 0 to avoid NullPointerException.
prices and profits have different lengths	Throw IllegalArgumentException or return 0 since corresponding prices and profits do not exist.
prices array is strictly decreasing	Return 0 since no i < j < k exists such that prices[i] < prices[j] < prices[k].
All prices are the same	Return 0 as no increasing sequence exists.
Large arrays leading to potential integer overflow when summing profits	Use long data type for profit calculations to prevent integer overflow.
Arrays with very skewed distributions, causing early pruning methods to fail	Ensure algorithm considers all potential triplets without relying heavily on pruning.

Empty prices or profits array

How to Handle:

Return 0 since no triplet can be formed and thus no profit exists.

prices or profits array has less than 3 elements

How to Handle:

Return 0 since a triplet i < j < k cannot be formed.

prices or profits are null

How to Handle:

Return 0 to avoid NullPointerException.

prices and profits have different lengths

How to Handle:

Throw IllegalArgumentException or return 0 since corresponding prices and profits do not exist.

prices array is strictly decreasing

How to Handle:

Return 0 since no i < j < k exists such that prices[i] < prices[j] < prices[k].

All prices are the same

How to Handle:

Return 0 as no increasing sequence exists.

Large arrays leading to potential integer overflow when summing profits

How to Handle:

Use long data type for profit calculations to prevent integer overflow.

Arrays with very skewed distributions, causing early pruning methods to fail

How to Handle:

Ensure algorithm considers all potential triplets without relying heavily on pruning.