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Collecting Chocolates

Difficulty: Medium

Problem Description

You are given a 0-indexed integer array nums of size n representing the cost of collecting different chocolates. The cost of collecting the chocolate at the index i is nums[i]. Each chocolate is of a different type, and initially, the chocolate at the index i is of i^th type. In one operation, you can do the following with an incurred cost of x: Simultaneously change the chocolate of i^th type to ((i + 1) mod n)^th type for all chocolates. Return the minimum cost to collect chocolates of all types, given that you can perform as many operations as you would like.

Key Insights

  • Each chocolate type can be collected at its own cost.
  • Performing operations incurs a fixed cost and shifts all chocolate types.
  • The cost of collecting chocolates can either be done directly or by performing operations first.
  • The optimal strategy involves comparing the total cost of direct collection vs. collection after a series of operations.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we can utilize a greedy approach. We will calculate the total cost of directly collecting each chocolate type without performing any operations. Then, for each possible starting position (representing the initial type of chocolate), we will calculate the cost of collecting all the chocolates after performing a series of operations. The key is to consider the cost of performing operations and the cost of collecting chocolates after the operations.

  1. Calculate the total cost of collecting all chocolates without any operations.
  2. For each starting type, calculate the cost of performing operations:
    • For each chocolate type, compute the cost of collecting that type after the necessary shifts.
    • Add the cost of each operation performed.

Finally, return the minimum cost found.

Code Solutions

def minCost(nums, x):
    n = len(nums)
    
    # Calculate the cost of collecting all types without any operations
    direct_cost = sum(nums)
    
    # Initialize the minimum cost with the direct cost
    min_cost = direct_cost
    
    # Iterate through each type as the starting point
    for start in range(n):
        operation_cost = 0
        # Calculate the cost if starting from this type
        for i in range(n):
            # The type after shifts
            type_index = (start + i) % n
            operation_cost += nums[type_index] + (i * x)
        
        # Update the minimum cost
        min_cost = min(min_cost, operation_cost)
    
    return min_cost
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