Sharing the Coffee - codepath/compsci_guides GitHub Wiki

TIP102 Unit 7 Session 1 Advanced (Click for link to problem statements)

Problem 3: Sharing the Coffee

The deli staff is in desperate need of caffeine to keep them going through their shift and has decided to divide the coffee supply equally among themselves. Each batch of coffee is stored in containers of different sizes. Write a recursive function can_split_coffee() that accepts a list of integers coffee representing the volume of each batch of coffee and returns True if the coffee can be split evenly by volume among n staff and False otherwise.

Problem Highlights

  • 💡 Difficulty: Medium
  • ⏰ Time to complete: 25-30 mins
  • 🛠️ Topics: Recursion, Subset Sum Problem, Partition Problem

1: U-nderstand

Understand what the interviewer is asking for by using test cases and questions about the problem.

  • Established a set (2-3) of test cases to verify their own solution later.
  • Established a set (1-2) of edge cases to verify their solution handles complexities.
  • Have fully understood the problem and have no clarifying questions.
  • Have you verified any Time/Space Constraints for this problem?
  • Q: What is the main task in this problem?
    • A: The task is to determine if the list of coffee volumes can be evenly split among n staff members using recursion.
  • Q: What should the function return if the total volume of coffee cannot be evenly divided by n?
    • A: The function should return False.
HAPPY CASE
Input: [4, 4, 8], 2
Output: True
Explanation: The total volume is 16, which can be split into two equal parts of 8.

Input: [5, 10, 15], 4
Output: False
Explanation: The total volume is 30, which cannot be split into four equal parts.

EDGE CASE
Input: [7, 3, 2], 2
Output: True
Explanation: The total volume is 12, which can be split into two equal parts of 6.

2: M-atch

Match what this problem looks like to known categories of problems, e.g. Linked List or Dynamic Programming, and strategies or patterns in those categories.

For Partitioning a Set, we want to consider the following approaches:

  • Subset Sum Problem: This problem is closely related to the subset sum problem, where we check if we can partition the array into n subsets each with a sum equal to the total sum divided by n.

3: P-lan

Plan the solution with appropriate visualizations and pseudocode.

General Idea:

  • First, check if the total volume of coffee can be evenly divided by n. If not, return False.
  • Then, attempt to partition the coffee into n subsets, each with a sum equal to total_volume // n.
  • Use recursion to explore different combinations of coffee batches to achieve the target volume for each subset.

Recursive Approach:

1) Calculate the `total_volume` of the coffee list.
2) If `total_volume` is not divisible by `n`, return `False`.
3) Define a helper function `can_divide(coffee, n, target, current_sum)`:
    a) Base case: If `n` is 0, return `True` (all staff members have received their share).
    b) If `current_sum` equals `target`, start partitioning for the next staff member.
    c) If the coffee list is empty, return `False`.
    d) Recursively try to include or exclude the first coffee batch in the current partition.
4) Return the result of `can_divide(coffee, n, target, 0)`.

⚠️ Common Mistakes

  • Forgetting to handle the base cases correctly, leading to incorrect results or infinite recursion.
  • Not correctly handling the case where a partition is successfully completed, resulting in incorrect further partitions.

4: I-mplement

Implement the code to solve the algorithm.

def can_split_coffee(coffee, n):
    total_volume = sum(coffee)
    
    # If the total volume isn't divisible by n, return False
    if total_volume % n != 0:
        return False
    
    target = total_volume // n
    return can_divide(coffee, n, target, 0)

def can_divide(coffee, n, target, current_sum):
    if n == 0:
        return True  # If we've successfully partitioned for all staff
    if current_sum == target:  # Current staff member has a full share
        return can_divide(coffee, n - 1, target, 0)  # Move to the next staff member
    if not coffee:
        return False  # No more coffee batches to partition

    # Try including the first batch of coffee in the current partition
    include = can_divide(coffee[1:], n, target, current_sum + coffee[0])
    
    # Try excluding the first batch of coffee from the current partition
    exclude = can_divide(coffee[1:], n, target, current_sum)
    
    return include or exclude

5: R-eview

Review the code by running specific example(s) and recording values (watchlist) of your code's variables along the way.

  • Trace through the can_split_coffee function with the input [4, 4, 8] and 2. The function should return True after partitioning the coffee into two equal parts.
  • Test the function with edge cases like a list that cannot be evenly divided.

6: E-valuate

Evaluate the performance of your algorithm and state any strong/weak or future potential work.

Time Complexity:

  • Time Complexity: The time complexity is O(2^N) in the worst case due to the recursive exploration of all subsets, where N is the number of coffee batches. The problem is similar to the subset sum problem, which is NP-complete.
  • Space Complexity: The space complexity is O(N) due to the recursion stack, where N is the depth of recursion proportional to the number of coffee batches.

Discussion:

  • This recursive solution is straightforward and leverages the recursive nature of partitioning problems. However, the exponential time complexity makes it inefficient for large inputs.
  • Dynamic programming or memoization could be explored to optimize the solution and reduce redundant calculations.