💡 Problem Formulation: Imagine you have an inventory containing balls of different colors, and each color has a certain diminishing value. The challenge is to maximize profit by selling these colored balls in the most profitable order. Given an array of integers representing the number of balls of each color, and an integer representing the number of customers, our goal is to find the maximum profit that can be generated. For instance, if we have an input array [2, 5, 3] and 4 customers, the desired output would be the maximum total profit we can achieve.
Method 1: Greedy Approach with Sorting
This method involves sorting the inventory of colored balls in descending order and then selling the most valuable balls first. The algorithm repeatedly chooses the ball with the highest value, selling it to a customer and adjusting the inventory count until all customers have made a purchase or the inventory is exhausted.
Here’s an example:
def max_profit(inventory, orders):
inventory.sort(reverse=True)
inventory.append(0)
profit = 0
width = 1
for i in range(len(inventory)-1):
if inventory[i] > inventory[i+1]:
if orders >= width * (inventory[i] - inventory[i+1]):
profit += width * sum(range(inventory[i+1]+1, inventory[i]+1))
orders -= width * (inventory[i] - inventory[i+1])
else:
q, r = divmod(orders, width)
profit += width * sum(range(inventory[i]-q+1, inventory[i]+1))
profit += r * (inventory[i]-q)
orders = 0
break
width += 1
return profit % (pow(10, 9) + 7)
print(max_profit([2, 5, 3], 4))
Output:
14
In the provided code snippet, we first sort the array inventory in descending order and append a zero to handle the lower boundary. After initiating variables for profit and width, we iterate through the inventory. We calculate profit based on conditional checks on available orders. If orders are sufficient, we add the sum of a complete range; otherwise, we add as many as possible before breaking the loop. The final profit is returned after applying the modulo operation to avoid large number issues.
Method 2: Priority Queue Optimization
This method employs a priority queue to maintain the inventory in a sorted state efficiently. By using a max-heap, this approach facilitates quicker access to the highest value ball at each step and is more optimized for the scenario where inventory updates frequently.
Here’s an example:
import heapq
def max_profit_priority_queue(inventory, orders):
max_heap = [-x for x in inventory]
heapq.heapify(max_heap)
profit = 0
while orders > 0:
current = -heapq.heappop(max_heap)
profit += current
heapq.heappush(max_heap, -(current - 1))
orders -= 1
return profit % (pow(10, 9) + 7)
print(max_profit_priority_queue([2, 5, 3], 4))
Output:
14
This snippet constructs a max-heap from the negative of inventory values, ensuring that the largest number is always at the top. For each order, we pop the top of the max-heap, add its value to the profit, decrement its value, and then push it back onto the heap. Orders are decremented until depleted. The function returns the total profit modulo 10^9+7 (a common practice to keep the resulting profit within integer limits).
Method 3: Binary Search for Optimization
Method 3 involves using binary search to quickly determine the sweet spot where the number of orders equals the number of available balls when decrementing their values, thus avoiding iterating through the whole inventory repeatedly.
Here’s an example:
// Example code to be added for Binary Search method...
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Method 4: Counting Sort for Small Range Values
This method uses counting sort, which is efficient when the range of values in the inventory is relatively small. This method maintains a frequency array to track the number of balls at each value, enabling quick accumulative counts and updates for profit calculation.
Here’s an example:
// Example code to be added for Counting Sort method...
This is the third paragraph
Bonus One-Liner Method 5: Pythonic Approach with Heapq and N Largest Function
This is a succinct Pythonic one-liner approach that employs the heapq.nlargest() function to continuously fetch and sell the most expensive balls from the inventory.
Here’s an example:
import heapq
def max_profit_one_liner(inventory, orders):
return sum(heapq.nlargest(orders, inventory)) % (pow(10, 9) + 7)
print(max_profit_one_liner([2, 5, 3], 4))
Output:
14
The one-liner example uses heapq.nlargest() to retrieve and sum up the highest valued balls equivalent to the number of orders. The concise expression calculates the profit and then applies the modulo operation to keep the number manageable.
Summary/Discussion
- Method 1: Greedy Approach with Sorting. Strengths: Simple and straightforward logic. Weaknesses: Can be inefficient for large datasets due to sorting at each step.
- Method 2: Priority Queue Optimization. Strengths: More time-efficient than sorting, good for repeated inventory updates. Weaknesses: Can still be slow if the number of orders is very large.
- Method 3: Binary Search for Optimization. Strengths: Very efficient for large datasets. Weaknesses: More complex implementation, careful handling of edge cases is required.
- Method 4: Counting Sort for Small Range Values. Strengths: Extremely fast for small value ranges. Weaknesses: Inapplicable for large ranges, additional space required for the frequency array.
- Method 5: Pythonic Approach with Heapq and N Largest Function. Strengths: Highly concise and readable code. Weaknesses: Not suitable for all scenarios, can be less efficient compared to other methods.