Maximize Total Tastiness of Purchased Fruits

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Dynamic Programming

You are in a fruit market where you want to buy some fruits. You are given two 0-indexed integer arrays price and tastiness of length n representing the price and tastiness of the i^th fruit, respectively.

You have a total budget of maxAmount representing the maximum amount of money you can spend. You want to buy fruits in such a way that you maximize the total tastiness of the fruits you buy.

Return the maximum total tastiness of the fruits you can buy with the given maxAmount.

Example 1:

Input: price = [13,5,17,2,4,15], tastiness = [12,12,13,15,1,13], maxAmount = 42
Output: 51
Explanation: You can buy fruits with indices 0, 1, 3, and 5. The total price is 13 + 5 + 2 + 15 = 35, which is less than or equal to maxAmount = 42.
The total tastiness is 12 + 12 + 15 + 13 = 52. It can be shown that 52 is the maximum tastiness.

Example 2:

Input: price = [3,6,1,4,6], tastiness = [6,1,7,1,9], maxAmount = 26
Output: 23
Explanation: You can buy fruits with indices 0, 2, 3, and 4. The total price is 3 + 1 + 4 + 6 = 14, which is less than or equal to maxAmount = 26.
The total tastiness is 6 + 7 + 1 + 9 = 23. It can be shown that 23 is the maximum tastiness.

Example 3:

Input: price = [1,10,3,1,10,3], tastiness = [7,8,2,3,3,3], maxAmount = 17
Output: 20
Explanation: You can buy fruits with indices 0, 1, and 3. The total price is 1 + 10 + 1 = 12, which is less than or equal to maxAmount = 17.
The total tastiness is 7 + 8 + 3 = 18. It can be shown that 20 is the maximum tastiness.

Constraints:

1 <= n == price.length == tastiness.length <= 1000
1 <= price[i] <= 1000
1 <= tastiness[i] <= 1000
1 <= maxAmount <= 10000

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, tastiness values, and the budget? Can they be zero or negative?
If the budget is zero or insufficient to purchase any fruit, what should be the return value?
Are the prices and tastiness arrays guaranteed to have the same length? What should I do if they don't?
Can I purchase multiple quantities of the same fruit, or can I only purchase at most one of each fruit?
If there are multiple combinations of fruits that yield the same maximum tastiness, is any one of them acceptable?

Brute Force Solution

Approach

The brute force method for maximizing tastiness involves trying every conceivable combination of fruits you could buy. We meticulously evaluate the total tastiness of each combination to identify the absolute best one. It's like exploring every single shopping cart before settling on the tastiest.

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

First, imagine you buy no fruits at all and calculate the total tastiness (which would be zero). Keep this in mind as a starting point.
Next, consider buying only the first fruit and calculate its tastiness. Compare this to the previous tastiness (zero) and remember the better option.
Then, consider buying only the second fruit, and again, calculate its tastiness and compare it to the best tastiness you've seen so far.
Continue doing this for each fruit individually.
Now, consider buying combinations of two fruits. Try every possible pair of fruits, calculate the combined tastiness for each pair, and update the best tastiness if you find a better one.
After that, consider all combinations of three fruits. Calculate the total tastiness for each of these groups of three and update your best tastiness if needed.
Keep going, increasing the number of fruits in each combination, until you've considered buying all the fruits at once.
Finally, after checking all possible combinations of fruits, the best tastiness you've recorded will be the maximum possible tastiness you can achieve.

Code Implementation

def maximize_tastiness_brute_force(costs, tastiness_values, budget):
    number_of_fruits = len(costs)
    maximum_tastiness = 0

    for i in range(1 << number_of_fruits):
        current_cost = 0
        current_tastiness = 0

        for j in range(number_of_fruits):
            # Check if the j-th fruit is included in the current combination
            if (i >> j) & 1:
                current_cost += costs[j]
                current_tastiness += tastiness_values[j]

        # We proceed only if the current combination is within budget
        if current_cost <= budget:

            # Update maximum_tastiness if current is better
            if current_tastiness > maximum_tastiness:
                maximum_tastiness = current_tastiness

    return maximum_tastiness

Big(O) Analysis

Time Complexity

O(2^n) – The described brute force method involves generating all possible subsets of the input fruits. For an input of size n, the number of possible subsets (including the empty set) is 2^n. Each subset needs to be evaluated to calculate its total tastiness. Therefore, the time complexity is directly proportional to the number of subsets, resulting in O(2^n).

Space Complexity

O(1) – The provided brute force method primarily involves iterative calculations of tastiness for various fruit combinations. It maintains only a single variable to store the maximum tastiness found so far. No auxiliary data structures like arrays, hash maps, or recursion stacks are used to store intermediate combinations or results. Therefore, the space complexity remains constant, irrespective of the number of fruits (N).

Optimal Solution

Approach

We want to pick fruits to maximize tastiness while staying within a budget. Instead of checking every possible fruit combination, we will sort the fruits by their tastiness-to-cost ratio and strategically pick the best ones first. This gives us the most tastiness for our money without exhausting all the options.

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

Calculate the tastiness-to-cost ratio for each fruit. This tells us how much tastiness we get per unit of cost.
Sort the fruits in descending order based on their tastiness-to-cost ratio. The fruits with the highest ratio will be at the beginning.
Start selecting fruits from the beginning of the sorted list (the ones with the highest ratio).
Keep adding fruits to your selection as long as you don't exceed your total budget.
If adding the next fruit would exceed your budget, skip it and move to the next fruit in the sorted list.
Continue this process until you have either exhausted all the fruits or you have reached your budget limit.
The total tastiness of the fruits you have selected is the maximum tastiness you can achieve within your budget.

Code Implementation

def maximize_total_tastiness(fruit_tastiness_values, fruit_costs, budget):
    number_of_fruits = len(fruit_tastiness_values)
    tastiness_per_dollar = []
    for i in range(number_of_fruits):
        tastiness_per_dollar.append((fruit_tastiness_values[i] / fruit_costs[i], i))

    # Sort by tastiness per dollar in descending order.
    tastiness_per_dollar.sort(reverse=True)

    total_tastiness = 0
    remaining_budget = budget

    for tastiness_ratio, fruit_index in tastiness_per_dollar:
        # Prioritize highest tastiness per dollar.
        if fruit_costs[fruit_index] <= remaining_budget:

            total_tastiness += fruit_tastiness_values[fruit_index]

            # Reduce budget by cost of selected fruit.
            remaining_budget -= fruit_costs[fruit_index]

    return total_tastiness

Big(O) Analysis

Time Complexity

O(n log n) – The solution's time complexity is primarily determined by the sorting step. Calculating the tastiness-to-cost ratio for each of the n fruits takes O(n) time. Sorting the fruits based on this ratio using an efficient algorithm like merge sort or quicksort takes O(n log n) time. The subsequent selection process iterates through the sorted list of fruits, which takes at most O(n) time. Therefore, the dominant factor is the sorting step, resulting in an overall time complexity of O(n log n).

Space Complexity

O(N) – The primary auxiliary space usage comes from sorting the fruits based on their tastiness-to-cost ratio. This sorting operation, depending on the algorithm used (e.g., merge sort, quicksort in certain implementations), generally requires creating a copy of the fruit data or using additional space for managing the sorting process. Therefore, an auxiliary data structure of size N (where N is the number of fruits) is created to facilitate sorting. This results in O(N) auxiliary space.

Edge Cases

Case	How to Handle
prices or tastiness is null or empty	Return 0, as no fruits can be purchased.
prices and tastiness have different lengths	Return 0 or throw an IllegalArgumentException because the inputs are invalid.
budget is 0	Return 0 because no fruit can be bought.
prices contains zero values	Handle it normally; a zero-priced fruit can be included if budget allows.
tastiness contains negative values	Handle this normally; a negative tastiness will lower the total tastiness.
All fruits are more expensive than the budget	Return 0, as no fruit can be bought.
Large input size with large budget	Dynamic programming solution may lead to integer overflow so use long data type or consider other optimization techniques.
Prices or tastiness values are extremely large, potentially causing overflow during calculations	Use long data type to prevent overflow, or consider modulo if appropriate.

prices or tastiness is null or empty

How to Handle:

Return 0, as no fruits can be purchased.

prices and tastiness have different lengths

How to Handle:

Return 0 or throw an IllegalArgumentException because the inputs are invalid.

budget is 0

How to Handle:

Return 0 because no fruit can be bought.

prices contains zero values

How to Handle:

Handle it normally; a zero-priced fruit can be included if budget allows.

tastiness contains negative values

How to Handle:

Handle this normally; a negative tastiness will lower the total tastiness.

All fruits are more expensive than the budget

How to Handle:

Return 0, as no fruit can be bought.

Large input size with large budget

How to Handle:

Dynamic programming solution may lead to integer overflow so use long data type or consider other optimization techniques.

Prices or tastiness values are extremely large, potentially causing overflow during calculations

How to Handle:

Use long data type to prevent overflow, or consider modulo if appropriate.