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Distribute Repeating Integers

Difficulty: Hard

Problem Description

You are given an array of n integers, nums, where there are at most 50 unique values in the array. You are also given an array of m customer order quantities, quantity, where quantity[i] is the amount of integers the i-th customer ordered. Determine if it is possible to distribute nums such that:

  • The i-th customer gets exactly quantity[i] integers,
  • The integers the i-th customer gets are all equal, and
  • Every customer is satisfied.

Return true if it is possible to distribute nums according to the above conditions.

Key Insights

  • Each customer requires a specific quantity of identical integers.
  • The maximum number of unique integers in nums is limited to 50, allowing for combinatorial approaches.
  • The sum of quantities must not exceed the total available integers in nums.
  • Backtracking can be employed to explore different combinations of integer distributions to customers.

Space and Time Complexity

Time Complexity: O(2^m * n), where m is the number of customers and n is the number of integers in nums, due to the combinatorial nature of the backtracking approach. Space Complexity: O(m), for the recursion stack in the backtracking algorithm.

Solution

The algorithm uses a backtracking approach to attempt to allocate integers from the nums array to satisfy each customer's quantity requirement. It keeps track of how many of each unique integer is available and checks recursively if it can fulfill the orders.

  1. Count the occurrences of each unique integer in nums.
  2. Sort the quantity array in descending order to try to satisfy larger orders first.
  3. Use a backtracking function that attempts to allocate integers to each customer based on their required quantity.
  4. If all customers are satisfied, return true; otherwise, backtrack and try different allocations.

Code Solutions

from collections import Counter

def canDistribute(nums, quantity):
    count = Counter(nums)  # Count occurrences of each integer
    unique_nums = list(count.values())  # Get the counts of unique integers
    quantity.sort(reverse=True)  # Sort quantities in descending order
    
    def backtrack(i):
        if i == len(quantity):  # All customers satisfied
            return True
        for j in range(len(unique_nums)):
            if unique_nums[j] >= quantity[i]:  # Check if we can satisfy the i-th customer
                unique_nums[j] -= quantity[i]  # Allocate the integers
                if backtrack(i + 1):  # Move to the next customer
                    return True
                unique_nums[j] += quantity[i]  # Backtrack
        return False
    
    return backtrack(0)
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