There are N workers in a factory. Each worker packs the same type of product, and for each of them we know the time it takes them to pack one product.
Find out the minimum amount of time the workers need to pack at least K products.
Example:
Input: times = [4, 7, 3, 6, 7, 1], K = 60
Output:30
Explanation:
1st worker packs 7 products in 4 * 7 = 28 units of time
2nd worker packs 4 products in 7 * 4 = 28 units of time
3rd worker packs 10 products in 3 * 10 = 30 units of time
4th worker packs 5 products in 6 * 5 = 30 units of time
5th worker packs 4 products in 7 * 4 = 28 units of time
6th worker packs 30 products in 1 * 30 = 30 units of time
We have 7 + 4 + 10 + 5 + 4 + 30 = 60 total products
in max(28, 28, 30, 30, 28, 30) = 30 units of time
and this is the minimum amount of time
since in 29 units of time we can pack only
7 + 4 + 9 + 4 + 4 + 29 = 57 products
The core challenge of this problem is to determine the minimum time required for the workers to pack at least K products. This problem is significant in scenarios where optimizing production time is crucial, such as in manufacturing and logistics.
Potential pitfalls include misunderstanding the relationship between time and the number of products packed by each worker, and not considering the maximum time taken by any worker as the limiting factor.
To solve this problem, we can use a binary search approach on the time. The idea is to find the minimum time required such that the total number of products packed by all workers is at least K.
A naive solution would involve iterating through each possible time and checking if the workers can pack at least K products within that time. This approach is not optimal due to its high time complexity.
We can optimize the solution using binary search. The key idea is to search for the minimum time in a range from 1 to the maximum possible time (K * max(time per product)). For each midpoint in this range, we calculate the total number of products packed by all workers and adjust our search range accordingly.
1. Initialize the search range: `low = 1` and `high = K * max(times)`.
2. Perform binary search:
3. The minimum time required is the value of `low` after the binary search completes.
Here is a step-by-step breakdown of the binary search algorithm:
def min_time_to_pack_products(times, K):
# Initialize the search range
low, high = 1, K * max(times)
# Binary search for the minimum time
while low < high:
mid = (low + high) // 2
total_products = sum(mid // time for time in times)
if total_products >= K:
high = mid
else:
low = mid + 1
return low
# Example usage
times = [4, 7, 3, 6, 7, 1]
K = 60
print(min_time_to_pack_products(times, K)) # Output: 30
The code implementation of the binary search approach is provided below:
def min_time_to_pack_products(times, K):
# Initialize the search range
low, high = 1, K * max(times)
# Binary search for the minimum time
while low < high:
mid = (low + high) // 2
total_products = sum(mid // time for time in times)
if total_products >= K:
high = mid
else:
low = mid + 1
return low
# Example usage
times = [4, 7, 3, 6, 7, 1]
K = 60
print(min_time_to_pack_products(times, K)) # Output: 30
The time complexity of the binary search approach is O(n * log(k * T)), where n is the number of workers, k is the number of products, and T is the maximum time per product. The space complexity is O(1) as we are using a constant amount of extra space.
Potential edge cases include:
Each algorithm handles these edge cases by ensuring the binary search range is correctly initialized and adjusted.
To test the solution comprehensively, consider the following test cases:
Using a testing framework like `unittest` in Python can help automate and validate these test cases.
When approaching such problems, consider the following tips:
Practice solving similar problems and studying algorithms to improve problem-solving skills.
In this blog post, we discussed the problem of finding the minimum time required for workers to pack at least K products. We explored a binary search approach to optimize the solution and provided a detailed explanation of the algorithm and its implementation. Understanding and solving such problems is crucial for optimizing production processes in various industries.
We encourage readers to practice and explore further to enhance their problem-solving skills.
For further reading and practice problems related to this topic, consider the following resources: