Divide Chocolate

Hard

Google

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Topics:

ArraysBinary Search

Imagine you have a chocolate bar represented as an array of sweetness values, sweetness = [1, 2, 3, 4, 5, 6, 7, 8, 9]. You want to divide this chocolate bar into k + 1 pieces and give one piece to each of your k friends, keeping one piece for yourself. The sweetness of each piece is the sum of the sweetness values of the segments in that piece. The goal is to maximize the minimum sweetness of any piece. In other words, you want to find the largest possible value x such that you can divide the chocolate bar into k + 1 pieces, and each piece has a sweetness of at least x.

For example:

If sweetness = [1, 2, 3, 4, 5, 6, 7, 8, 9] and k = 5, what is the maximum minimum sweetness you can achieve?
If sweetness = [5, 6, 7, 8, 9, 1, 2, 3, 4] and k = 8, what is the maximum minimum sweetness you can achieve?
If sweetness = [1, 2, 2, 1, 2, 2, 1, 2, 2] and k = 2, what is the maximum minimum sweetness you can achieve?

Write a function that takes the sweetness array and the number of friends k as input and returns the maximum minimum sweetness you can achieve. Provide an efficient algorithm to solve this problem.

Divide Chocolate

Hard

Google

2 views

Topics:

ArraysBinary Search

Imagine you have a chocolate bar represented as an array of sweetness values, sweetness = [1, 2, 3, 4, 5, 6, 7, 8, 9]. You want to divide this chocolate bar into k + 1 pieces and give one piece to each of your k friends, keeping one piece for yourself. The sweetness of each piece is the sum of the sweetness values of the segments in that piece. The goal is to maximize the minimum sweetness of any piece. In other words, you want to find the largest possible value x such that you can divide the chocolate bar into k + 1 pieces, and each piece has a sweetness of at least x.

For example:

If sweetness = [1, 2, 3, 4, 5, 6, 7, 8, 9] and k = 5, what is the maximum minimum sweetness you can achieve?
If sweetness = [5, 6, 7, 8, 9, 1, 2, 3, 4] and k = 8, what is the maximum minimum sweetness you can achieve?
If sweetness = [1, 2, 2, 1, 2, 2, 1, 2, 2] and k = 2, what is the maximum minimum sweetness you can achieve?

Write a function that takes the sweetness array and the number of friends k as input and returns the maximum minimum sweetness you can achieve. Provide an efficient algorithm to solve this problem.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the values in the chocolate array, and can they be negative or zero?
What is the expected return value if it's not possible to divide the chocolate as requested (e.g., is there a valid division)?
Can I assume that `K` (number of friends) is always a positive integer greater than zero, and what is the upper bound for K?
What does 'fairly' mean in this context? Are we looking for the maximum possible value for the smallest piece after dividing, and should all friends get at least one piece?
Is the input chocolate array guaranteed to be non-empty?

Brute Force Solution

Approach

The brute force approach to dividing chocolate involves trying every possible way to cut the chocolate bar into pieces. We will explore all the combinations to find a solution that satisfies the given criteria. We will check each possible division to determine its validity.

Here's how the algorithm would work step-by-step:

Imagine all the possible places you could cut the chocolate bar.
Start by considering cutting after the first piece, and see if that creates a valid division.
Then consider cutting after the second piece, the third piece, and so on, always checking if each division is valid according to the rules.
If you find a valid division after some number of pieces, try all the possible ways to divide the remaining chocolate bar.
Keep doing this until you've considered every single possible way to cut the chocolate bar into the required number of pieces.
After exploring every option, pick the 'best' division based on the specific conditions defined in the problem such as maximizing the smallest piece or something similar.

Code Implementation

def divide_chocolate_brute_force(chocolate_bar, number_of_pieces):
    number_of_pieces_in_chocolate_bar = len(chocolate_bar)

    if number_of_pieces > number_of_pieces_in_chocolate_bar:
        return -1  # Impossible to divide

    best_division_quality = -1
    best_division = []

    def explore_divisions(current_division, remaining_chocolate, pieces_left):
        nonlocal best_division_quality, best_division

        if pieces_left == 1:
            # Base case: assign rest to final piece
            current_division.append(sum(remaining_chocolate))
            
            division_quality = min(current_division)

            if division_quality > best_division_quality:
                best_division_quality = division_quality
                best_division = current_division.copy()

            current_division.pop()
            return

        # Iterate over possible cut locations
        for cut_location in range(1, len(remaining_chocolate) - pieces_left + 2):
            first_piece = sum(remaining_chocolate[:cut_location])
            
            # Explore subsequent cuts
            explore_divisions(current_division + [first_piece],
                                remaining_chocolate[cut_location:],
                                pieces_left - 1)

    explore_divisions([], chocolate_bar, number_of_pieces)

    #Find smallest piece as the quality of the cut.
    return best_division_quality

Big(O) Analysis

Time Complexity

O(2^n) – The described brute-force approach explores all possible ways to cut the chocolate bar. For a bar of length n, there are n-1 possible cut locations. At each of these locations, we either make a cut or we don't. This results in 2^(n-1) possible combinations of cuts. Therefore, the algorithm essentially iterates through all these combinations to check if they are valid. Since the number of combinations grows exponentially with the size of the chocolate bar, the time complexity is O(2^n).

Space Complexity

O(N) – The brute force approach explores all possible ways to cut the chocolate, implying a recursive algorithm. In the worst-case scenario, the recursion depth could reach N, where N is the number of pieces in the chocolate bar, as we consider cutting after each piece. Each recursive call creates a new stack frame, storing function arguments and local variables. Therefore, the auxiliary space used by the recursion stack can grow linearly with the input size N, resulting in O(N) space complexity.

Optimal Solution

Approach

The most efficient approach to dividing chocolate fairly involves using a method to quickly find the best piece size. Instead of checking every possible piece size, we'll use a smart guessing technique to narrow down the options until we find the perfect one.

Here's how the algorithm would work step-by-step:

First, think about the smallest and largest possible sizes a piece of chocolate could be. The smallest is 1 (taking only the smallest value in the array), and the largest is the sum of all the chocolate sizes.
Pick a piece size somewhere in the middle of these two values. This is your first guess.
Go through the chocolate sizes and see how many pieces of your guessed size you can make, given the number of people.
If you can make enough pieces for everyone, then your guessed piece size might be too small. Try guessing a bigger size in the upper half of the current possible sizes. This means you throw away all the smaller values since you are trying to maximize the size.
If you can't make enough pieces for everyone, your guessed piece size is too big. Try guessing a smaller size in the lower half of the current possible sizes. This means you can throw away all the bigger values since you are trying to maximize the size.
Keep guessing, each time halving the range of possible piece sizes. Eventually, the smallest and largest possible sizes will meet. That final size is the largest possible size that allows you to give everyone a piece of chocolate.

Code Implementation

def divide_chocolate(chocolate_sizes, number_of_people):
    smallest_possible_size = min(chocolate_sizes)
    largest_possible_size = sum(chocolate_sizes)

    while smallest_possible_size <= largest_possible_size:
        mid_size = (smallest_possible_size + largest_possible_size) // 2
        
        # Check if this piece size allows us to divide the chocolate for everyone
        number_of_pieces = 0
        current_chocolate_size = 0
        for chocolate_size in chocolate_sizes:
            current_chocolate_size += chocolate_size
            if current_chocolate_size >= mid_size:
                number_of_pieces += 1
                current_chocolate_size = 0

        # Need a larger size, so update smallest possible size
        if number_of_pieces >= number_of_people + 1:
            smallest_possible_size = mid_size + 1
        # Current size yields too few pieces, need a smaller size
        else:
            largest_possible_size = mid_size - 1

    # Return the maximum possible size.
    return largest_possible_size

Big(O) Analysis

Time Complexity

O(n log m) – The algorithm uses binary search to find the optimal size of the chocolate pieces. The search space for the piece size is from 1 to the sum of all chocolate sizes, which we denote as 'm'. Within the binary search loop, we iterate through the chocolate pieces array of size 'n' to check how many pieces of the guessed size can be made. Therefore, for each binary search iteration, we perform O(n) operations. Since binary search takes O(log m) iterations, the overall time complexity is O(n log m), where 'n' is the number of chocolate pieces and 'm' is the sum of all chocolate sizes.

Space Complexity

O(1) – The algorithm uses a binary search approach with variables to track the smallest and largest possible piece sizes. It also uses a variable to store the guessed piece size. Since no auxiliary data structures that scale with the input size (array of chocolate sizes) are created, the space complexity is constant, O(1).

Edge Cases

Case	How to Handle
Empty chocolate bar (empty array)	Return 0 as no cuts can be made.
All pieces have zero sweetness	The minimum sweetness will always be zero, which is the minimum possible outcome from a greedy approach.
k (number of friends) is zero	Return the total sweetness of the entire chocolate bar, as no cuts are needed.
Pieces have negative sweetness values	Handle negative values appropriately during the binary search process, potentially skewing sweetness distributions.
Large array size exceeding memory limits	Consider a streaming approach if the entire array does not fit in memory.
k (number of friends) is greater than or equal to length of sweetness array	Return the smallest sweetness value since each friend can only get at most one piece.
The array contains extremely large sweetness values leading to integer overflow when summed	Use a larger data type like long to store the sum if the language allows, or use a modular arithmetic approach to prevent overflow during the calculation if appropriate.
No valid solution exists because the minimum sweetness requirement cannot be met even if distributed optimally	Return 0, indicating it is impossible to divide the chocolate with the required minimum sweetness for each friend.