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Rearrange Array to Maximize Prefix Score

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ArraysGreedy Algorithms

You are given a 0-indexed integer array nums. You can rearrange the elements of nums to any order (including the given order).

Let prefix be the array containing the prefix sums of nums after rearranging it. In other words, prefix[i] is the sum of the elements from 0 to i in nums after rearranging it. The score of nums is the number of positive integers in the array prefix.

Return the maximum score you can achieve.

Example 1:

Input: nums = [2,-1,0,1,-3,3,-3]
Output: 6
Explanation: We can rearrange the array into nums = [2,3,1,-1,-3,0,-3].
prefix = [2,5,6,5,2,2,-1], so the score is 6.
It can be shown that 6 is the maximum score we can obtain.

Example 2:

Input: nums = [-2,-3,0]
Output: 0
Explanation: Any rearrangement of the array will result in a score of 0.

Constraints:

1 <= nums.length <= 10⁵
-10⁶ <= nums[i] <= 10⁶

Try coding it in LeetCode

Rearrange Array to Maximize Prefix Score

Medium

Asked by:

38 views

Topics:

ArraysGreedy Algorithms

You are given a 0-indexed integer array nums. You can rearrange the elements of nums to any order (including the given order).

Return the maximum score you can achieve.

Example 1:

Input: nums = [2,-1,0,1,-3,3,-3]
Output: 6
Explanation: We can rearrange the array into nums = [2,3,1,-1,-3,0,-3].
prefix = [2,5,6,5,2,2,-1], so the score is 6.
It can be shown that 6 is the maximum score we can obtain.

Example 2:

Input: nums = [-2,-3,0]
Output: 0
Explanation: Any rearrangement of the array will result in a score of 0.

Constraints:

1 <= nums.length <= 10⁵
-10⁶ <= nums[i] <= 10⁶

Try coding it in LeetCode

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 is the range of values for the integers in the input array? Can they be negative, zero, or only positive?
What is the maximum size of the input array?
Are duplicate numbers allowed in the input array, and if so, how should they be handled when trying to maximize the prefix score?
Are there multiple possible rearrangements that result in the same maximum prefix score? If so, is any valid arrangement acceptable, or is there a specific requirement (e.g., lexicographically smallest)?
Could you define more formally what is considered the 'prefix score' that we are trying to maximize? Is it simply the sum of positive prefix sums?

Brute Force Solution

Approach

The brute force approach tackles this problem by trying absolutely every single possible arrangement of the numbers. For each arrangement, we calculate a 'score' to see how good it is, and at the end, we pick the arrangement with the best score. It's like trying every combination lock setting until you find the right one.

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

Consider the first number in the set.
Now consider the first two numbers, and all the different ways they can be ordered.
Next, add the third number. Explore every possible location for that third number amongst the first two (first, second, third position).
Continue this process, each time adding one more number, and figuring out every single spot it could possibly go among the previously arranged numbers.
For each complete arrangement you create, calculate its 'score' according to the specific scoring rule.
Keep track of the arrangement that gives you the highest score.
Once you've tried all possible arrangements, the arrangement that gave you the highest score is the answer.

Code Implementation

def rearrange_array_brute_force(numbers):
    best_score = float('-inf')
    best_arrangement = []

    def calculate_prefix_score(arrangement):
        current_score = 0
        prefix_sum = 0
        for number in arrangement:
            prefix_sum += number
            if prefix_sum > 0:
                current_score += 1
        return current_score

    def generate_permutations(index, current_arrangement):
        nonlocal best_score
        nonlocal best_arrangement

        if index == len(numbers):
            # We have a complete arrangement; calculate score and update if necessary
            current_score = calculate_prefix_score(current_arrangement)

            if current_score > best_score:
                best_score = current_score
                best_arrangement = current_arrangement[:]
            return

        for i in range(index + 1):
            # Try inserting the current number at every possible position
            new_arrangement = current_arrangement[:i] + [numbers[index]] + current_arrangement[i:]

            # Recursively generate permutations with the new arrangement
            generate_permutations(index + 1, new_arrangement)

    # Start generating permutations from the beginning
    generate_permutations(0, [])

    return best_arrangement

Big(O) Analysis

Time Complexity

O(n!) – The brute force approach explores all possible permutations of the input array of size n. Generating all permutations takes n! (n factorial) time. For each permutation, we calculate the prefix score, which takes O(n) time. Therefore, the overall time complexity is O(n * n!), which is dominated by the factorial component, and we simplify to O(n!).

Space Complexity

O(N) – The brute force approach generates all permutations of the input array of size N. Although the permutations are generated and evaluated iteratively, keeping track of the current permutation requires auxiliary space. The algorithm maintains a copy of the input array to represent a possible arrangement. Therefore, auxiliary space of size N is needed to store each arrangement, leading to a space complexity of O(N).

Optimal Solution

Approach

To maximize the prefix score, we need to arrange the input numbers in a special order. The key idea is to prioritize putting larger numbers earlier in the sequence, because they contribute more to the sums that make up the score.

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

First, sort all the given numbers from largest to smallest.
Next, use the sorted numbers in their new order to construct the rearranged sequence.
The score is calculated by taking the cumulative sums (prefix sums) of this new sequence and then adding up all those sums.

Code Implementation

def rearrange_array_to_maximize_prefix_score(numbers):
    # Sort the numbers in descending order
    numbers.sort(reverse=True)

    prefix_sums = []
    current_prefix_sum = 0
    for number in numbers:
        current_prefix_sum += number
        prefix_sums.append(current_prefix_sum)

    # Calculate the score by summing up all prefix sums
    total_score = sum(prefix_sums)

    return total_score

Big(O) Analysis

Time Complexity

O(n log n) – The dominant operation is sorting the input array of size n in descending order. Common sorting algorithms like merge sort or quicksort have a time complexity of O(n log n). Constructing the rearranged sequence and calculating the prefix sums involves iterating through the sorted array once, which takes O(n) time. Since O(n log n) grows faster than O(n), the overall time complexity is determined by the sorting step, making it O(n log n).

Space Complexity

O(N) – The algorithm sorts the input array of size N in place or creates a new sorted array of size N depending on the sorting algorithm used. Even if sorted in place, many sorting algorithms implicitly use space proportional to the depth of recursion, which in the worst case for some algorithms like quicksort is O(N). Therefore, the auxiliary space complexity is O(N) due to the sorting operation.

Edge Cases

Case	How to Handle
Null or empty input array	Return an empty array or throw an IllegalArgumentException, depending on the specified behavior.
Array with one element	Return the array as is, as no rearrangement is possible.
Array with all identical values	Sorting the array will not change its order, but it should still produce a valid, although not necessarily 'optimized', score; the solution should execute without errors.
Array with extremely large numbers that could lead to integer overflow when summing the prefix	Use a data type with a larger range, like long, to store the prefix sums and handle potential overflows.
Array with very large input size (e.g., 10^5 or more elements)	Ensure that the sorting algorithm used is efficient (e.g., O(n log n)) to avoid exceeding time limits.
Array with negative numbers, potentially causing negative prefix scores.	The algorithm should correctly handle negative prefix scores without errors and optimize to maximize the sum even with negative values.
Array contains a mix of large positive and large negative numbers that could result in an intermediate overflow, even with long.	Consider using a datatype that can handle larger ranges than long or using an alternative algorithm that avoids intermediate sums, if feasible.
Array with input values equal to max or min integer values	Check for integer overflow/underflow issues and handle them by casting to long before operations and casting back, handling overflow by using max or min long values

Null or empty input array

How to Handle:

Return an empty array or throw an IllegalArgumentException, depending on the specified behavior.

Array with one element

How to Handle:

Return the array as is, as no rearrangement is possible.

Array with all identical values

How to Handle:

Sorting the array will not change its order, but it should still produce a valid, although not necessarily 'optimized', score; the solution should execute without errors.

Array with extremely large numbers that could lead to integer overflow when summing the prefix

How to Handle:

Use a data type with a larger range, like long, to store the prefix sums and handle potential overflows.

Array with very large input size (e.g., 10^5 or more elements)

How to Handle:

Ensure that the sorting algorithm used is efficient (e.g., O(n log n)) to avoid exceeding time limits.

Array with negative numbers, potentially causing negative prefix scores.

How to Handle:

The algorithm should correctly handle negative prefix scores without errors and optimize to maximize the sum even with negative values.

Array contains a mix of large positive and large negative numbers that could result in an intermediate overflow, even with long.

How to Handle:

Consider using a datatype that can handle larger ranges than long or using an alternative algorithm that avoids intermediate sums, if feasible.

Array with input values equal to max or min integer values

How to Handle:

Check for integer overflow/underflow issues and handle them by casting to long before operations and casting back, handling overflow by using max or min long values

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 is the range of values for the integers in the input array? Can they be negative, zero, or only positive?
What is the maximum size of the input array?
Are duplicate numbers allowed in the input array, and if so, how should they be handled when trying to maximize the prefix score?
Are there multiple possible rearrangements that result in the same maximum prefix score? If so, is any valid arrangement acceptable, or is there a specific requirement (e.g., lexicographically smallest)?
Could you define more formally what is considered the 'prefix score' that we are trying to maximize? Is it simply the sum of positive prefix sums?

Brute Force Solution

Approach

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

Consider the first number in the set.
Now consider the first two numbers, and all the different ways they can be ordered.
Next, add the third number. Explore every possible location for that third number amongst the first two (first, second, third position).
Continue this process, each time adding one more number, and figuring out every single spot it could possibly go among the previously arranged numbers.
For each complete arrangement you create, calculate its 'score' according to the specific scoring rule.
Keep track of the arrangement that gives you the highest score.
Once you've tried all possible arrangements, the arrangement that gave you the highest score is the answer.

Code Implementation

def rearrange_array_brute_force(numbers):
    best_score = float('-inf')
    best_arrangement = []

    def calculate_prefix_score(arrangement):
        current_score = 0
        prefix_sum = 0
        for number in arrangement:
            prefix_sum += number
            if prefix_sum > 0:
                current_score += 1
        return current_score

    def generate_permutations(index, current_arrangement):
        nonlocal best_score
        nonlocal best_arrangement

        if index == len(numbers):
            # We have a complete arrangement; calculate score and update if necessary
            current_score = calculate_prefix_score(current_arrangement)

            if current_score > best_score:
                best_score = current_score
                best_arrangement = current_arrangement[:]
            return

        for i in range(index + 1):
            # Try inserting the current number at every possible position
            new_arrangement = current_arrangement[:i] + [numbers[index]] + current_arrangement[i:]

            # Recursively generate permutations with the new arrangement
            generate_permutations(index + 1, new_arrangement)

    # Start generating permutations from the beginning
    generate_permutations(0, [])

    return best_arrangement

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

First, sort all the given numbers from largest to smallest.
Next, use the sorted numbers in their new order to construct the rearranged sequence.
The score is calculated by taking the cumulative sums (prefix sums) of this new sequence and then adding up all those sums.

Code Implementation

def rearrange_array_to_maximize_prefix_score(numbers):
    # Sort the numbers in descending order
    numbers.sort(reverse=True)

    prefix_sums = []
    current_prefix_sum = 0
    for number in numbers:
        current_prefix_sum += number
        prefix_sums.append(current_prefix_sum)

    # Calculate the score by summing up all prefix sums
    total_score = sum(prefix_sums)

    return total_score

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input array	Return an empty array or throw an IllegalArgumentException, depending on the specified behavior.
Array with one element	Return the array as is, as no rearrangement is possible.
Array with all identical values	Sorting the array will not change its order, but it should still produce a valid, although not necessarily 'optimized', score; the solution should execute without errors.
Array with extremely large numbers that could lead to integer overflow when summing the prefix	Use a data type with a larger range, like long, to store the prefix sums and handle potential overflows.
Array with very large input size (e.g., 10^5 or more elements)	Ensure that the sorting algorithm used is efficient (e.g., O(n log n)) to avoid exceeding time limits.
Array with negative numbers, potentially causing negative prefix scores.	The algorithm should correctly handle negative prefix scores without errors and optimize to maximize the sum even with negative values.
Array contains a mix of large positive and large negative numbers that could result in an intermediate overflow, even with long.	Consider using a datatype that can handle larger ranges than long or using an alternative algorithm that avoids intermediate sums, if feasible.
Array with input values equal to max or min integer values	Check for integer overflow/underflow issues and handle them by casting to long before operations and casting back, handling overflow by using max or min long values

Null or empty input array

How to Handle:

Return an empty array or throw an IllegalArgumentException, depending on the specified behavior.

Array with one element

How to Handle:

Return the array as is, as no rearrangement is possible.

Array with all identical values

How to Handle:

Sorting the array will not change its order, but it should still produce a valid, although not necessarily 'optimized', score; the solution should execute without errors.

Array with extremely large numbers that could lead to integer overflow when summing the prefix

How to Handle:

Use a data type with a larger range, like long, to store the prefix sums and handle potential overflows.

Array with very large input size (e.g., 10^5 or more elements)

How to Handle:

Ensure that the sorting algorithm used is efficient (e.g., O(n log n)) to avoid exceeding time limits.

Array with negative numbers, potentially causing negative prefix scores.

How to Handle:

The algorithm should correctly handle negative prefix scores without errors and optimize to maximize the sum even with negative values.

Array contains a mix of large positive and large negative numbers that could result in an intermediate overflow, even with long.

How to Handle:

Consider using a datatype that can handle larger ranges than long or using an alternative algorithm that avoids intermediate sums, if feasible.

Array with input values equal to max or min integer values

How to Handle:

Check for integer overflow/underflow issues and handle them by casting to long before operations and casting back, handling overflow by using max or min long values