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Minimized Maximum of Products Distributed to Any Store

Difficulty: Medium

Problem Description

You are given an integer n indicating there are n specialty retail stores. There are m product types of varying amounts, which are given as a 0-indexed integer array quantities, where quantities[i] represents the number of products of the ith product type.

You need to distribute all products to the retail stores following these rules:

  • A store can only be given at most one product type but can be given any amount of it.
  • After distribution, each store will have been given some number of products (possibly 0). Let x represent the maximum number of products given to any store. You want x to be as small as possible, i.e., you want to minimize the maximum number of products that are given to any store.

Return the minimum possible x.

Key Insights

  • Each store can receive only one type of product, which introduces a constraint on the distribution.
  • The problem can be approached using binary search to determine the minimum maximum number of products per store.
  • The feasibility of a candidate maximum can be checked by calculating how many stores are needed to distribute the products without exceeding that maximum.

Space and Time Complexity

Time Complexity: O(m log(max(quantities)))
Space Complexity: O(1)

Solution

To solve the problem, we employ a binary search strategy. We define the search range from 1 (minimum possible products per store) to the maximum value in the quantities array (the worst-case scenario where one store gets all products of one type).

  1. For a midpoint value, we check if it’s feasible to distribute the products such that no store receives more than this midpoint value.
  2. We calculate how many stores are needed for each product type based on the midpoint. If the total number of stores needed is less than or equal to n, then it’s feasible.
  3. We adjust our binary search range based on whether the current midpoint was feasible or not.

This process continues until we converge on the minimum possible maximum number of products that can be assigned to any store.

Code Solutions

def minimizedMaximum(n: int, quantities: List[int]) -> int:
    def canDistribute(maximum: int) -> bool:
        stores_needed = 0
        for quantity in quantities:
            stores_needed += (quantity + maximum - 1) // maximum  # Calculate needed stores for this product type
        return stores_needed <= n  # Check if we can distribute within the limit of n stores

    left, right = 1, max(quantities)  # Binary search range
    while left < right:
        mid = (left + right) // 2
        if canDistribute(mid):
            right = mid  # Try for a smaller maximum
        else:
            left = mid + 1  # Increase the minimum possible maximum

    return left  # This will be the minimized maximum
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