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Maximum Number of Alloys

Difficulty: Medium

Problem Description

You are the owner of a company that creates alloys using various types of metals. There are n different types of metals available, and you have access to k machines that can be used to create alloys. Each machine requires a specific amount of each metal type to create an alloy. Initially, you have stock[i] units of metal type i, and purchasing one unit of metal type i costs cost[i] coins. Given integers n, k, budget, a 1-indexed 2D array composition, and 1-indexed arrays stock and cost, your goal is to maximize the number of alloys the company can create while staying within the budget of budget coins. All alloys must be created with the same machine. Return the maximum number of alloys that the company can create.

Key Insights

  • Each machine has different requirements for the types of metals needed to produce an alloy.
  • The total cost to produce a certain number of alloys is based on the composition requirements and the available stock of metals.
  • Binary search can be employed to efficiently determine the maximum number of alloys that can be produced within the given budget.

Space and Time Complexity

Time Complexity: O(k * log(budget)), where k is the number of machines and budget is the maximum budget. Space Complexity: O(n), for storing the required metal amounts.

Solution

To solve this problem, we will use a binary search approach. The idea is to determine the maximum number of alloys that can be produced by iterating through each machine's composition and calculating the cost based on the required metals and their respective stocks. The binary search will help us efficiently find the highest number of alloys that can be produced without exceeding the budget. The key steps are:

  1. For each machine, calculate the total amount of each metal required to produce a certain number of alloys.
  2. Compute the additional metal needed based on available stock.
  3. Calculate the total cost to purchase the additional metals needed.
  4. Use binary search to maximize the number of alloys produced while keeping the total cost within the budget.

Code Solutions

def maxAlloys(n, k, budget, composition, stock, cost):
    def canProduceAlloys(machine_idx, alloys):
        total_cost = 0
        for j in range(n):
            required = max(0, composition[machine_idx][j] * alloys - stock[j])
            total_cost += required * cost[j]
            if total_cost > budget:  # Early exit if cost exceeds budget
                return False
        return total_cost <= budget

    left, right = 0, budget + 1  # max alloys cannot exceed budget
    result = 0

    for machine_idx in range(k):
        while left < right:
            mid = (left + right) // 2
            if canProduceAlloys(machine_idx, mid):
                result = max(result, mid)
                left = mid + 1
            else:
                right = mid
        left, right = 0, budget + 1  # Reset for the next machine

    return result
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