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Maximum of Absolute Value Expression

Medium

Asked by:

60 views

Topics:

Arrays

Given two arrays of integers with equal lengths, return the maximum value of:

|arr1[i] - arr1[j]| + |arr2[i] - arr2[j]| + |i - j|

where the maximum is taken over all 0 <= i, j < arr1.length.

Example 1:

Input: arr1 = [1,2,3,4], arr2 = [-1,4,5,6]
Output: 13

Example 2:

Input: arr1 = [1,-2,-5,0,10], arr2 = [0,-2,-1,-7,-4]
Output: 20

Constraints:

2 <= arr1.length == arr2.length <= 40000
-10^6 <= arr1[i], arr2[i] <= 10^6

Try coding it in LeetCode

Maximum of Absolute Value Expression

Medium

Asked by:

60 views

Topics:

Arrays

Given two arrays of integers with equal lengths, return the maximum value of:

|arr1[i] - arr1[j]| + |arr2[i] - arr2[j]| + |i - j|

where the maximum is taken over all 0 <= i, j < arr1.length.

Example 1:

Input: arr1 = [1,2,3,4], arr2 = [-1,4,5,6]
Output: 13

Example 2:

Input: arr1 = [1,-2,-5,0,10], arr2 = [0,-2,-1,-7,-4]
Output: 20

Constraints:

2 <= arr1.length == arr2.length <= 40000
-10^6 <= arr1[i], arr2[i] <= 10^6

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 are the possible ranges for the integer values within the input arrays `arr1` and `arr2`? Can they be negative, zero, or very large?
What is the expected behavior if either `arr1` or `arr2` is null or empty?
Are `arr1` and `arr2` guaranteed to have the same length?
Could you clarify the return value if the input arrays are of length 1 (i.e., only one data point is available)?
Are we optimizing for time complexity (assuming the most optimal solution is expected, but confirming this explicitly to show awareness)?

Brute Force Solution

Approach

The brute force method here means we will check all possible combinations. We will compute the result of the expression for every possible pair of values from our input and select the largest one.

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

Take the first number from the first list and the first number from the second list.
Plug these two numbers into the given formula to calculate the result.
Store this result.
Now, take the first number from the first list again, but this time pair it with the second number from the second list.
Calculate the result using the formula and compare it with the stored result.
If the new result is larger, replace the stored result with this larger value.
Keep pairing the first number from the first list with every other number from the second list, updating the stored result whenever you find a larger value.
Once you've exhausted all pairings with the first number, move on to the second number from the first list.
Repeat the process: pair it with every number from the second list, calculating the formula each time and updating the stored result if a larger value is found.
Continue this process until you have paired every number from the first list with every number from the second list.
The stored result at the end of this process will be the maximum value you can get from the formula.

Code Implementation

def max_absolute_expression_brute_force(arr1, arr2):
    max_value = 0

    # Iterate through all possible pairs of elements from the two arrays
    for first_index in range(len(arr1)):
        for second_index in range(len(arr2)):

            # Calculate the value of the expression for the current pair
            current_value = abs(arr1[first_index] - arr1[second_index]) + abs(arr2[first_index] - arr2[second_index]) + abs(first_index - second_index)

            # Update the maximum value if the current value is greater
            if current_value > max_value:
                max_value = current_value

    return max_value

Big(O) Analysis

Time Complexity

O(n²) – The given solution involves iterating through two input lists, `arr1` and `arr2`, to check every possible pair of elements. For each element in `arr1` (of size n, assuming both lists have the same size for simplicity), the solution iterates through all elements in `arr2`, also of size n. The expression is evaluated for each pair. Thus, the number of operations grows proportionally to n * n, which simplifies to O(n²).

Space Complexity

O(1) – The brute force algorithm described only uses a single variable to store the maximum result found so far. This variable requires a constant amount of memory, independent of the input array sizes. The algorithm does not create any auxiliary data structures like lists, maps, or recursion stacks whose size depends on the input size N (where N is the size of the input lists). Therefore, the space complexity is O(1).

Optimal Solution

Approach

The key is to realize that we can simplify the absolute value expression by considering all possible sign combinations. This breaks the problem down into finding the maximum value across a few simple linear equations, avoiding the need to check every possible pair of numbers.

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

Notice that the absolute value signs create a limited number of possible expressions based on the signs inside them.
Figure out all of the possible combinations of plus and minus signs in front of each term in the expression. Each combination represents a separate case.
For each of these sign combinations, rewrite the expression as a simple linear equation. For example, you might end up with an equation like: (one number) + (another number) + (index of the first number) - (index of the second number).
For each of those simple equations, go through all the numbers to keep track of and remember the biggest value you find.
Once you've checked all possible sign combinations, compare the biggest values you found in each of those cases, and choose the very biggest one. This final value is the answer.

Code Implementation

def find_max_absolute_value_expression(
    array_one, array_two
):
    max_value = 0
    number_of_elements = len(array_one)

    # Iterate through all 4 possible sign combinations
    possible_sign_combinations = [
        (1, 1),
        (1, -1),
        (-1, 1),
        (-1, -1),
    ]

    for sign_one, sign_two in possible_sign_combinations:
        max_sum = float('-inf')
        min_sum = float('inf')

        # Find the max and min sum for this combination
        for index in range(number_of_elements):
            current_sum = (
                sign_one * array_one[index]
                + sign_two * array_two[index]
                + index
            )
            max_sum = max(max_sum, current_sum)
            min_sum = min(min_sum, current_sum)

        # Update max_value with the largest difference
        max_value = max(max_value, max_sum - min_sum)

    return max_value

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through all possible sign combinations of the absolute value expression. Since there are a fixed number of combinations (based on the absolute value signs), the number of cases is constant. For each of these constant number of cases, the algorithm iterates through the input arrays nums1 and nums2 once. Therefore, the time complexity is driven by a constant number of iterations over the n elements of the input arrays, resulting in a linear time complexity of O(n).

Space Complexity

O(1) – The solution iterates through a fixed number of sign combinations and for each, it only needs to store the maximum value found so far. The number of sign combinations is constant (derived from the number of absolute value expressions), and only a few scalar variables are used to keep track of maximum values across these combinations. These variables occupy constant space regardless of the size of the input arrays. Therefore, the auxiliary space complexity is O(1).

Edge Cases

Case	How to Handle
Empty arrays for arr1 or arr2	Return 0 immediately as there are no elements to compute the expression.
Arrays arr1 and arr2 have different lengths	Return 0 or throw an exception, as the expression is not defined for unequal length arrays.
Arrays with only two elements	The solution should correctly calculate the absolute value expression for the minimal possible input size.
Arrays with all identical values	The absolute differences will all be zero, resulting in a maximum value of 0 which the solution should handle correctly.
Arrays with a large number of elements (scalability)	Ensure the solution uses an algorithm with optimal time complexity to avoid timeouts for large inputs.
Integer overflow during addition/subtraction	Use long integers or appropriate data types to prevent overflow when calculating the expression components.
Arrays containing negative numbers	The absolute value function handles negative numbers correctly, but verify the overall algorithm works.
Arrays containing very large positive numbers near the maximum integer limit	Ensure that intermediate calculations don't exceed the maximum integer value to avoid incorrect results due to overflow.

Empty arrays for arr1 or arr2

How to Handle:

Return 0 immediately as there are no elements to compute the expression.

Arrays arr1 and arr2 have different lengths

How to Handle:

Return 0 or throw an exception, as the expression is not defined for unequal length arrays.

Arrays with only two elements

How to Handle:

The solution should correctly calculate the absolute value expression for the minimal possible input size.

Arrays with all identical values

How to Handle:

The absolute differences will all be zero, resulting in a maximum value of 0 which the solution should handle correctly.

Arrays with a large number of elements (scalability)

How to Handle:

Ensure the solution uses an algorithm with optimal time complexity to avoid timeouts for large inputs.

Integer overflow during addition/subtraction

How to Handle:

Use long integers or appropriate data types to prevent overflow when calculating the expression components.

Arrays containing negative numbers

How to Handle:

The absolute value function handles negative numbers correctly, but verify the overall algorithm works.

Arrays containing very large positive numbers near the maximum integer limit

How to Handle:

Ensure that intermediate calculations don't exceed the maximum integer value to avoid incorrect results due to overflow.

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 integer values within the input arrays `arr1` and `arr2`? Can they be negative, zero, or very large?
What is the expected behavior if either `arr1` or `arr2` is null or empty?
Are `arr1` and `arr2` guaranteed to have the same length?
Could you clarify the return value if the input arrays are of length 1 (i.e., only one data point is available)?
Are we optimizing for time complexity (assuming the most optimal solution is expected, but confirming this explicitly to show awareness)?

Brute Force Solution

Approach

The brute force method here means we will check all possible combinations. We will compute the result of the expression for every possible pair of values from our input and select the largest one.

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

Take the first number from the first list and the first number from the second list.
Plug these two numbers into the given formula to calculate the result.
Store this result.
Now, take the first number from the first list again, but this time pair it with the second number from the second list.
Calculate the result using the formula and compare it with the stored result.
If the new result is larger, replace the stored result with this larger value.
Keep pairing the first number from the first list with every other number from the second list, updating the stored result whenever you find a larger value.
Once you've exhausted all pairings with the first number, move on to the second number from the first list.
Repeat the process: pair it with every number from the second list, calculating the formula each time and updating the stored result if a larger value is found.
Continue this process until you have paired every number from the first list with every number from the second list.
The stored result at the end of this process will be the maximum value you can get from the formula.

Code Implementation

def max_absolute_expression_brute_force(arr1, arr2):
    max_value = 0

    # Iterate through all possible pairs of elements from the two arrays
    for first_index in range(len(arr1)):
        for second_index in range(len(arr2)):

            # Calculate the value of the expression for the current pair
            current_value = abs(arr1[first_index] - arr1[second_index]) + abs(arr2[first_index] - arr2[second_index]) + abs(first_index - second_index)

            # Update the maximum value if the current value is greater
            if current_value > max_value:
                max_value = current_value

    return max_value

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Notice that the absolute value signs create a limited number of possible expressions based on the signs inside them.
Figure out all of the possible combinations of plus and minus signs in front of each term in the expression. Each combination represents a separate case.
For each of these sign combinations, rewrite the expression as a simple linear equation. For example, you might end up with an equation like: (one number) + (another number) + (index of the first number) - (index of the second number).
For each of those simple equations, go through all the numbers to keep track of and remember the biggest value you find.
Once you've checked all possible sign combinations, compare the biggest values you found in each of those cases, and choose the very biggest one. This final value is the answer.

Code Implementation

def find_max_absolute_value_expression(
    array_one, array_two
):
    max_value = 0
    number_of_elements = len(array_one)

    # Iterate through all 4 possible sign combinations
    possible_sign_combinations = [
        (1, 1),
        (1, -1),
        (-1, 1),
        (-1, -1),
    ]

    for sign_one, sign_two in possible_sign_combinations:
        max_sum = float('-inf')
        min_sum = float('inf')

        # Find the max and min sum for this combination
        for index in range(number_of_elements):
            current_sum = (
                sign_one * array_one[index]
                + sign_two * array_two[index]
                + index
            )
            max_sum = max(max_sum, current_sum)
            min_sum = min(min_sum, current_sum)

        # Update max_value with the largest difference
        max_value = max(max_value, max_sum - min_sum)

    return max_value

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty arrays for arr1 or arr2	Return 0 immediately as there are no elements to compute the expression.
Arrays arr1 and arr2 have different lengths	Return 0 or throw an exception, as the expression is not defined for unequal length arrays.
Arrays with only two elements	The solution should correctly calculate the absolute value expression for the minimal possible input size.
Arrays with all identical values	The absolute differences will all be zero, resulting in a maximum value of 0 which the solution should handle correctly.
Arrays with a large number of elements (scalability)	Ensure the solution uses an algorithm with optimal time complexity to avoid timeouts for large inputs.
Integer overflow during addition/subtraction	Use long integers or appropriate data types to prevent overflow when calculating the expression components.
Arrays containing negative numbers	The absolute value function handles negative numbers correctly, but verify the overall algorithm works.
Arrays containing very large positive numbers near the maximum integer limit	Ensure that intermediate calculations don't exceed the maximum integer value to avoid incorrect results due to overflow.

Empty arrays for arr1 or arr2

How to Handle:

Return 0 immediately as there are no elements to compute the expression.

Arrays arr1 and arr2 have different lengths

How to Handle:

Return 0 or throw an exception, as the expression is not defined for unequal length arrays.

Arrays with only two elements

How to Handle:

The solution should correctly calculate the absolute value expression for the minimal possible input size.

Arrays with all identical values

How to Handle:

The absolute differences will all be zero, resulting in a maximum value of 0 which the solution should handle correctly.

Arrays with a large number of elements (scalability)

How to Handle:

Ensure the solution uses an algorithm with optimal time complexity to avoid timeouts for large inputs.

Integer overflow during addition/subtraction

How to Handle:

Use long integers or appropriate data types to prevent overflow when calculating the expression components.

Arrays containing negative numbers

How to Handle:

The absolute value function handles negative numbers correctly, but verify the overall algorithm works.

Arrays containing very large positive numbers near the maximum integer limit

How to Handle:

Ensure that intermediate calculations don't exceed the maximum integer value to avoid incorrect results due to overflow.