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Maximum Absolute Sum of Any Subarray

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ArraysDynamic Programming

You are given an integer array nums. The absolute sum of a subarray [nums_l, nums_l+1, ..., nums_r-1, nums_r] is abs(nums_l + nums_l+1 + ... + nums_r-1 + nums_r).

Return the maximum absolute sum of any (possibly empty) subarray of nums.

Note that abs(x) is defined as follows:

If x is a negative integer, then abs(x) = -x.
If x is a non-negative integer, then abs(x) = x.

Example 1:

Input: nums = [1,-3,2,3,-4]
Output: 5
Explanation: The subarray [2,3] has absolute sum = abs(2+3) = abs(5) = 5.

Example 2:

Input: nums = [2,-5,1,-4,3,-2]
Output: 8
Explanation: The subarray [-5,1,-4] has absolute sum = abs(-5+1-4) = abs(-8) = 8.

Constraints:

1 <= nums.length <= 10⁵
-10⁴ <= nums[i] <= 10⁴

Try coding it in LeetCode

Maximum Absolute Sum of Any Subarray

Medium

Asked by:

26 views

Topics:

ArraysDynamic Programming

You are given an integer array nums. The absolute sum of a subarray [nums_l, nums_l+1, ..., nums_r-1, nums_r] is abs(nums_l + nums_l+1 + ... + nums_r-1 + nums_r).

Return the maximum absolute sum of any (possibly empty) subarray of nums.

Note that abs(x) is defined as follows:

If x is a negative integer, then abs(x) = -x.
If x is a non-negative integer, then abs(x) = x.

Example 1:

Input: nums = [1,-3,2,3,-4]
Output: 5
Explanation: The subarray [2,3] has absolute sum = abs(2+3) = abs(5) = 5.

Example 2:

Input: nums = [2,-5,1,-4,3,-2]
Output: 8
Explanation: The subarray [-5,1,-4] has absolute sum = abs(-5+1-4) = abs(-8) = 8.

Constraints:

1 <= nums.length <= 10⁵
-10⁴ <= nums[i] <= 10⁴

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 expected data type of the array elements (e.g., integers, floats)? What is the range of possible values for the elements, including the possibility of negative numbers or zero?
What should I return if the input array is null or empty?
Is there a constraint on the size of the input array (e.g., maximum number of elements)?
By 'absolute sum', do you mean the absolute value of the sum of the subarray (i.e., |sum(subarray)|)?
If there are multiple subarrays with the same maximum absolute sum, am I allowed to return the value of the absolute sum of any one of them?

Brute Force Solution

Approach

We want to find the largest possible total value (positive or negative) that can be achieved from a continuous section of numbers. The brute force method checks every single possible section and calculates its total value.

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

First, consider all sections that start with the very first number.
Calculate the total value of just the first number alone, then the total value of the first two numbers combined, then the first three, and so on, until you've included all the numbers.
Next, consider all sections that start with the second number. Again, calculate the total value of just the second number, then the second and third, and so on, until you've included all the remaining numbers.
Continue this process, each time starting at the next number and calculating the total value of all possible sections starting from that number.
As you calculate the total values for each section, keep track of both the biggest positive value you find and the biggest negative value you find.
Finally, compare the biggest positive value and the biggest negative value (converted to positive by taking its absolute value). The larger of these two is your answer.

Code Implementation

def max_absolute_sum_subarray_brute_force(numbers):
    maximum_positive_sum = 0
    maximum_negative_sum = 0

    # Iterate through all possible starting positions
    for start_index in range(len(numbers)):
        current_subarray_sum = 0

        # Iterate through all possible ending positions
        for end_index in range(start_index, len(numbers)):
            current_subarray_sum += numbers[end_index]

            # Check if the current subarray sum is the largest positive sum
            if current_subarray_sum > maximum_positive_sum:
                maximum_positive_sum = current_subarray_sum

            # Check if the current subarray sum is the smallest negative sum
            if current_subarray_sum < maximum_negative_sum:
                maximum_negative_sum = current_subarray_sum

    # Need to take the absolute value to compare
    absolute_maximum_negative_sum = abs(maximum_negative_sum)

    # Return the larger of the max positive sum and absolute max negative sum
    if maximum_positive_sum > absolute_maximum_negative_sum:
        return maximum_positive_sum
    else:
        return absolute_maximum_negative_sum

Big(O) Analysis

Time Complexity

O(n²) – The algorithm iterates through the array to consider all possible starting positions of subarrays. For each starting position, it calculates the sum of all possible subarrays starting from that position. This means that for each of the n elements, we perform up to n sum calculations. Thus the total number of operations is proportional to n * n which results in O(n²).

Space Complexity

O(1) – The described brute force algorithm only uses a few variables to keep track of the current subarray sum, the maximum positive sum, and the minimum negative sum. These variables consume a constant amount of space, irrespective of the input array's size (N). No auxiliary data structures like arrays or hash maps are created. Therefore, the space complexity is O(1).

Optimal Solution

Approach

The goal is to find the largest possible sum (either positive or negative) you can get from a continuous chunk of numbers. Instead of checking every possible chunk, we focus on tracking the largest positive sum and the smallest negative sum we've seen so far.

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

Imagine walking through the numbers one by one, keeping track of the biggest sum you've encountered that ends at your current number.
Also, keep track of the smallest sum (most negative) you've encountered that ends at your current number.
At each number, decide whether adding it to the current running sum makes the sum bigger (for the maximum case) or smaller (for the minimum case). If adding the current number makes the sum worse, just start a new sum from the current number.
The biggest positive sum found so far and the absolute value of the smallest negative sum found so far are our candidates for the final answer.
The largest of these two values is the maximum absolute sum of any subarray.

Code Implementation

def max_absolute_sum(nums):
    max_so_far = 0
    min_so_far = 0
    current_max = 0
    current_min = 0

    for num in nums:
        # Keep track of the largest positive sum.
        current_max = max(num, current_max + num)

        max_so_far = max(max_so_far, current_max)

        # Keep track of the smallest negative sum.
        current_min = min(num, current_min + num)

        min_so_far = min(min_so_far, current_min)

    #The largest of these two values is our answer
    return max(max_so_far, abs(min_so_far))

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the input array of size n exactly once to calculate the maximum positive sum and minimum negative sum ending at each element. Within the loop, only constant time operations like addition, comparison, and assignment are performed. Therefore, the time complexity is directly proportional to the size of the input array, resulting in O(n).

Space Complexity

O(1) – The algorithm described only uses a few variables to store the current maximum positive sum, the current minimum negative sum, the overall maximum absolute sum, and potentially an index or two for iteration. No additional data structures, like arrays or hash maps, are created that scale with the input size N (the number of elements in the input array). Therefore, the space used is constant and does not depend on the input size.

Edge Cases

Case	How to Handle
Empty input array	Return 0, as the sum of an empty subarray is 0, and the absolute value is also 0.
Array with a single element	Return the absolute value of the single element.
Array with all positive numbers	Return the sum of all elements in the array.
Array with all negative numbers	Return the absolute value of the sum of all elements in the array.
Array with all zeros	Return 0, as the sum of any subarray will be 0.
Array with very large positive and negative numbers (potential overflow)	Use a data type that can accommodate large sums, like long, to prevent integer overflow during calculation.
Array with alternating positive and negative numbers	Kadane's Algorithm adapted for maximum and minimum subarray sums should handle this correctly by considering both positive and negative contributions.
Maximum-sized input array (memory constraints)	Kadane's algorithm has O(n) time complexity and O(1) space complexity, so it should scale efficiently without exceeding memory limits for large arrays.

Empty input array

How to Handle:

Return 0, as the sum of an empty subarray is 0, and the absolute value is also 0.

Array with a single element

How to Handle:

Return the absolute value of the single element.

Array with all positive numbers

How to Handle:

Return the sum of all elements in the array.

Array with all negative numbers

How to Handle:

Return the absolute value of the sum of all elements in the array.

Array with all zeros

How to Handle:

Return 0, as the sum of any subarray will be 0.

Array with very large positive and negative numbers (potential overflow)

How to Handle:

Use a data type that can accommodate large sums, like long, to prevent integer overflow during calculation.

Array with alternating positive and negative numbers

How to Handle:

Kadane's Algorithm adapted for maximum and minimum subarray sums should handle this correctly by considering both positive and negative contributions.

Maximum-sized input array (memory constraints)

How to Handle:

Kadane's algorithm has O(n) time complexity and O(1) space complexity, so it should scale efficiently without exceeding memory limits for large arrays.

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 expected data type of the array elements (e.g., integers, floats)? What is the range of possible values for the elements, including the possibility of negative numbers or zero?
What should I return if the input array is null or empty?
Is there a constraint on the size of the input array (e.g., maximum number of elements)?
By 'absolute sum', do you mean the absolute value of the sum of the subarray (i.e., |sum(subarray)|)?
If there are multiple subarrays with the same maximum absolute sum, am I allowed to return the value of the absolute sum of any one of them?

Brute Force Solution

Approach

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

First, consider all sections that start with the very first number.
Calculate the total value of just the first number alone, then the total value of the first two numbers combined, then the first three, and so on, until you've included all the numbers.
Next, consider all sections that start with the second number. Again, calculate the total value of just the second number, then the second and third, and so on, until you've included all the remaining numbers.
Continue this process, each time starting at the next number and calculating the total value of all possible sections starting from that number.
As you calculate the total values for each section, keep track of both the biggest positive value you find and the biggest negative value you find.
Finally, compare the biggest positive value and the biggest negative value (converted to positive by taking its absolute value). The larger of these two is your answer.

Code Implementation

def max_absolute_sum_subarray_brute_force(numbers):
    maximum_positive_sum = 0
    maximum_negative_sum = 0

    # Iterate through all possible starting positions
    for start_index in range(len(numbers)):
        current_subarray_sum = 0

        # Iterate through all possible ending positions
        for end_index in range(start_index, len(numbers)):
            current_subarray_sum += numbers[end_index]

            # Check if the current subarray sum is the largest positive sum
            if current_subarray_sum > maximum_positive_sum:
                maximum_positive_sum = current_subarray_sum

            # Check if the current subarray sum is the smallest negative sum
            if current_subarray_sum < maximum_negative_sum:
                maximum_negative_sum = current_subarray_sum

    # Need to take the absolute value to compare
    absolute_maximum_negative_sum = abs(maximum_negative_sum)

    # Return the larger of the max positive sum and absolute max negative sum
    if maximum_positive_sum > absolute_maximum_negative_sum:
        return maximum_positive_sum
    else:
        return absolute_maximum_negative_sum

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Imagine walking through the numbers one by one, keeping track of the biggest sum you've encountered that ends at your current number.
Also, keep track of the smallest sum (most negative) you've encountered that ends at your current number.
At each number, decide whether adding it to the current running sum makes the sum bigger (for the maximum case) or smaller (for the minimum case). If adding the current number makes the sum worse, just start a new sum from the current number.
The biggest positive sum found so far and the absolute value of the smallest negative sum found so far are our candidates for the final answer.
The largest of these two values is the maximum absolute sum of any subarray.

Code Implementation

def max_absolute_sum(nums):
    max_so_far = 0
    min_so_far = 0
    current_max = 0
    current_min = 0

    for num in nums:
        # Keep track of the largest positive sum.
        current_max = max(num, current_max + num)

        max_so_far = max(max_so_far, current_max)

        # Keep track of the smallest negative sum.
        current_min = min(num, current_min + num)

        min_so_far = min(min_so_far, current_min)

    #The largest of these two values is our answer
    return max(max_so_far, abs(min_so_far))

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty input array	Return 0, as the sum of an empty subarray is 0, and the absolute value is also 0.
Array with a single element	Return the absolute value of the single element.
Array with all positive numbers	Return the sum of all elements in the array.
Array with all negative numbers	Return the absolute value of the sum of all elements in the array.
Array with all zeros	Return 0, as the sum of any subarray will be 0.
Array with very large positive and negative numbers (potential overflow)	Use a data type that can accommodate large sums, like long, to prevent integer overflow during calculation.
Array with alternating positive and negative numbers	Kadane's Algorithm adapted for maximum and minimum subarray sums should handle this correctly by considering both positive and negative contributions.
Maximum-sized input array (memory constraints)	Kadane's algorithm has O(n) time complexity and O(1) space complexity, so it should scale efficiently without exceeding memory limits for large arrays.

Empty input array

How to Handle:

Return 0, as the sum of an empty subarray is 0, and the absolute value is also 0.

Array with a single element

How to Handle:

Return the absolute value of the single element.

Array with all positive numbers

How to Handle:

Return the sum of all elements in the array.

Array with all negative numbers

How to Handle:

Return the absolute value of the sum of all elements in the array.

Array with all zeros

How to Handle:

Return 0, as the sum of any subarray will be 0.

Array with very large positive and negative numbers (potential overflow)

How to Handle:

Use a data type that can accommodate large sums, like long, to prevent integer overflow during calculation.

Array with alternating positive and negative numbers

How to Handle:

Kadane's Algorithm adapted for maximum and minimum subarray sums should handle this correctly by considering both positive and negative contributions.

Maximum-sized input array (memory constraints)

How to Handle:

Kadane's algorithm has O(n) time complexity and O(1) space complexity, so it should scale efficiently without exceeding memory limits for large arrays.