Jobs

Interviews

Discussions

Sum of Mutated Array Closest to Target

Medium

Asked by:

9 views

Topics:

ArraysBinary Search

Given an integer array arr and a target value target, return the integer value such that when we change all the integers larger than value in the given array to be equal to value, the sum of the array gets as close as possible (in absolute difference) to target.

In case of a tie, return the minimum such integer.

Notice that the answer is not neccesarilly a number from arr.

Example 1:

Input: arr = [4,9,3], target = 10
Output: 3
Explanation: When using 3 arr converts to [3, 3, 3] which sums 9 and that's the optimal answer.

Example 2:

Input: arr = [2,3,5], target = 10
Output: 5

Example 3:

Input: arr = [60864,25176,27249,21296,20204], target = 56803
Output: 11361

Constraints:

1 <= arr.length <= 10⁴
1 <= arr[i], target <= 10⁵

Try coding it in LeetCode

Sum of Mutated Array Closest to Target

Medium

Asked by:

9 views

Topics:

ArraysBinary Search

In case of a tie, return the minimum such integer.

Notice that the answer is not neccesarilly a number from arr.

Example 1:

Input: arr = [4,9,3], target = 10
Output: 3
Explanation: When using 3 arr converts to [3, 3, 3] which sums 9 and that's the optimal answer.

Example 2:

Input: arr = [2,3,5], target = 10
Output: 5

Example 3:

Input: arr = [60864,25176,27249,21296,20204], target = 56803
Output: 11361

Constraints:

1 <= arr.length <= 10⁴
1 <= arr[i], target <= 10⁵

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 values within the input array and the target value? Can they be negative?
What should I return if the input array is null or empty?
If multiple mutation values result in the same minimum difference from the target, which one should I return? The smallest, the largest, or any of them?
Are the numbers in the input array integers or floating-point numbers? If they are floating-point numbers, what level of precision is expected in the result?
Is the target value guaranteed to be achievable by some mutation value, or is it possible that no mutation value produces a sum exactly equal to the target?

Brute Force Solution

Approach

The brute force method aims to find a special value by trying every possible number within a specific range. For each of these numbers, we modify the original set of numbers and calculate the sum of the modified set. Finally, we determine which of these modified sums is the closest to a target value.

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

Start by considering the smallest possible value for modification.
Go through every possible number from that smallest value up to the largest number in the original set, one by one.
For each of these numbers, go through each number in the original set.
If a number in the original set is bigger than the current possible modification number, replace it with that modification number.
Add up all the numbers in this newly modified set.
Figure out how far away that sum is from the target value.
Keep track of which modification number gave a sum that was closest to the target value.
Once you've tried all possible modification numbers, pick the one that resulted in the closest sum to the target value.

Code Implementation

def find_best_value_brute_force(numbers, target):
    best_value = 0
    minimum_difference = float('inf')
    
    # Iterate through all possible values
    for value_to_try in range(1, max(numbers) + 1):
        modified_sum = 0
        
        # Calculate the sum of the modified array
        for number in numbers:
            modified_sum += min(number, value_to_try)
        
        # Check if current value is closer to the target
        difference = abs(modified_sum - target)
        
        # Update result if difference is smaller
        if difference < minimum_difference:
            minimum_difference = difference
            best_value = value_to_try
        elif difference == minimum_difference and value_to_try < best_value:
            best_value = value_to_try
            
    return best_value

Big(O) Analysis

Time Complexity

O(n²) – The algorithm iterates through a range of possible modification values. In the worst-case scenario, this range is from 0 to the largest element in the input array, which we can consider proportional to n, where n is the size of the input array (nums). For each of these potential modification values, the algorithm iterates through the entire input array (nums) of size n to modify the elements and calculate the sum. Thus, the outer loop runs on the order of n times, and the inner loop also runs on the order of n times, leading to approximately n*n operations. Therefore, the time complexity is O(n²).

Space Complexity

O(1) – The algorithm's space complexity is O(1) because it only uses a few variables to store the current modification number, the closest sum, and the corresponding special value. It does not create any additional data structures such as arrays or hash maps whose size depends on the input array's size, denoted by N. The modifications are implicitly done in place through direct calculations, and no intermediate data structures are used to store the modified array. Therefore, the space usage remains constant regardless of the input size N.

Optimal Solution

Approach

The goal is to find a single value that, when used to modify an array, results in a sum as close as possible to a target. Instead of exhaustively trying every possible value, we use a clever strategy that focuses on narrowing down the potential range and making educated guesses to efficiently find the optimal one.

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

First, recognize that setting the modification value too low will result in a sum smaller than the target, and setting it too high will result in a sum larger than the target.
Figure out the largest possible value in the array. There's no need to test a value larger than this, because any array element already smaller than this can't get bigger than this value, and any number in the array bigger than this number will just be the number itself after mutation.
Use a method called binary search. This method works like searching for a word in a dictionary: start in the middle. Check if using the middle value as the modification number results in a sum close to the target.
If the sum is too big, then we know the correct modification value is smaller, so eliminate the higher half of values. Then try the middle of the remaining lower half.
If the sum is too small, then we know the correct modification value is bigger, so eliminate the lower half of values. Then try the middle of the remaining upper half.
Repeat this process, continually halving the possible values until the closest possible sum to the target is found.

Code Implementation

def sum_of_mutated_array_closest_to_target(array, target):
    left = 0
    right = max(array)

    while left <= right:
        mutation_value = (left + right) // 2
        
        mutated_sum = 0
        for element in array:
            mutated_sum += min(element, mutation_value)
        
        # Adjust search range based on mutated sum vs target.
        if mutated_sum < target:
            left = mutation_value + 1
        else:
            right = mutation_value - 1

    # At this point, left is the potential closest value. Check left and right.
    best_value = left
    
    sum_for_left = 0
    for element in array:
        sum_for_left += min(element, left)

    #Consider the value to the left of 'left' as potentially closer.
    if left > 0:
        sum_for_right = 0
        for element in array:
            sum_for_right += min(element, left - 1)

        # Choose the value (left or left - 1) that gives a sum closest to target
        if abs(sum_for_right - target) < abs(sum_for_left - target):
            best_value = left - 1

    return best_value

Big(O) Analysis

Time Complexity

O(n log n) – The algorithm performs a binary search to find the optimal value. The binary search iterates a logarithmic number of times, specifically log(m), where m is the maximum element in the input array (as we are searching a space from 0 to the max element). In each iteration of the binary search, we iterate through all n elements of the array to calculate the sum after the mutation. Therefore, the overall time complexity is O(n log m), which is often written as O(n log n) since m is related to n in many problem constraints and the logarithm grows slowly.

Space Complexity

O(1) – The algorithm utilizes binary search, which involves calculating sums based on a modification value. It iteratively narrows down the range but doesn't create any auxiliary data structures dependent on the input array's size (N). The space used is primarily for a few constant variables used in the binary search process, such as `low`, `high`, and `mid` for the binary search range. Therefore, the space complexity remains constant, regardless of the input array's size.

Edge Cases

Case	How to Handle
Empty input array	Return 0, or throw an exception indicating invalid input as the problem statement does not define how to handle it.
Array with a single element	Return that single element if target is close, or apply a predefined mutation rule as needed, then return the mutated value.
Target value is extremely small (negative large)	The solution should correctly handle negative target values, potentially requiring absolute value calculations when determining the closest sum.
Target value is extremely large	The solution should avoid integer overflow issues by using appropriate data types or scaling calculations.
Array contains very large numbers that could cause overflow during summation	Use long or double data types to prevent integer overflow during intermediate calculations and when summing.
All elements in the array are identical	The algorithm should not get stuck in a loop and will correctly converge to the solution.
No mutation value leads to a sum closer to the target than the sum of the original array	Return the sum of the original array as the closest value, as the problem seeks to minimize the difference with the target.
The target is close to the sum of a sub-array, but mutating a different value gives an equal result	The problem states it can return any such integer in such a case, and the algorithm needs to be consistent.

Empty input array

How to Handle:

Return 0, or throw an exception indicating invalid input as the problem statement does not define how to handle it.

Array with a single element

How to Handle:

Return that single element if target is close, or apply a predefined mutation rule as needed, then return the mutated value.

Target value is extremely small (negative large)

How to Handle:

The solution should correctly handle negative target values, potentially requiring absolute value calculations when determining the closest sum.

Target value is extremely large

How to Handle:

The solution should avoid integer overflow issues by using appropriate data types or scaling calculations.

Array contains very large numbers that could cause overflow during summation

How to Handle:

Use long or double data types to prevent integer overflow during intermediate calculations and when summing.

All elements in the array are identical

How to Handle:

The algorithm should not get stuck in a loop and will correctly converge to the solution.

No mutation value leads to a sum closer to the target than the sum of the original array

How to Handle:

Return the sum of the original array as the closest value, as the problem seeks to minimize the difference with the target.

The target is close to the sum of a sub-array, but mutating a different value gives an equal result

How to Handle:

The problem states it can return any such integer in such a case, and the algorithm needs to be consistent.

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 within the input array and the target value? Can they be negative?
What should I return if the input array is null or empty?
If multiple mutation values result in the same minimum difference from the target, which one should I return? The smallest, the largest, or any of them?
Are the numbers in the input array integers or floating-point numbers? If they are floating-point numbers, what level of precision is expected in the result?
Is the target value guaranteed to be achievable by some mutation value, or is it possible that no mutation value produces a sum exactly equal to the target?

Brute Force Solution

Approach

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

Start by considering the smallest possible value for modification.
Go through every possible number from that smallest value up to the largest number in the original set, one by one.
For each of these numbers, go through each number in the original set.
If a number in the original set is bigger than the current possible modification number, replace it with that modification number.
Add up all the numbers in this newly modified set.
Figure out how far away that sum is from the target value.
Keep track of which modification number gave a sum that was closest to the target value.
Once you've tried all possible modification numbers, pick the one that resulted in the closest sum to the target value.

Code Implementation

def find_best_value_brute_force(numbers, target):
    best_value = 0
    minimum_difference = float('inf')
    
    # Iterate through all possible values
    for value_to_try in range(1, max(numbers) + 1):
        modified_sum = 0
        
        # Calculate the sum of the modified array
        for number in numbers:
            modified_sum += min(number, value_to_try)
        
        # Check if current value is closer to the target
        difference = abs(modified_sum - target)
        
        # Update result if difference is smaller
        if difference < minimum_difference:
            minimum_difference = difference
            best_value = value_to_try
        elif difference == minimum_difference and value_to_try < best_value:
            best_value = value_to_try
            
    return best_value

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

First, recognize that setting the modification value too low will result in a sum smaller than the target, and setting it too high will result in a sum larger than the target.
Figure out the largest possible value in the array. There's no need to test a value larger than this, because any array element already smaller than this can't get bigger than this value, and any number in the array bigger than this number will just be the number itself after mutation.
Use a method called binary search. This method works like searching for a word in a dictionary: start in the middle. Check if using the middle value as the modification number results in a sum close to the target.
If the sum is too big, then we know the correct modification value is smaller, so eliminate the higher half of values. Then try the middle of the remaining lower half.
If the sum is too small, then we know the correct modification value is bigger, so eliminate the lower half of values. Then try the middle of the remaining upper half.
Repeat this process, continually halving the possible values until the closest possible sum to the target is found.

Code Implementation

def sum_of_mutated_array_closest_to_target(array, target):
    left = 0
    right = max(array)

    while left <= right:
        mutation_value = (left + right) // 2
        
        mutated_sum = 0
        for element in array:
            mutated_sum += min(element, mutation_value)
        
        # Adjust search range based on mutated sum vs target.
        if mutated_sum < target:
            left = mutation_value + 1
        else:
            right = mutation_value - 1

    # At this point, left is the potential closest value. Check left and right.
    best_value = left
    
    sum_for_left = 0
    for element in array:
        sum_for_left += min(element, left)

    #Consider the value to the left of 'left' as potentially closer.
    if left > 0:
        sum_for_right = 0
        for element in array:
            sum_for_right += min(element, left - 1)

        # Choose the value (left or left - 1) that gives a sum closest to target
        if abs(sum_for_right - target) < abs(sum_for_left - target):
            best_value = left - 1

    return best_value

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty input array	Return 0, or throw an exception indicating invalid input as the problem statement does not define how to handle it.
Array with a single element	Return that single element if target is close, or apply a predefined mutation rule as needed, then return the mutated value.
Target value is extremely small (negative large)	The solution should correctly handle negative target values, potentially requiring absolute value calculations when determining the closest sum.
Target value is extremely large	The solution should avoid integer overflow issues by using appropriate data types or scaling calculations.
Array contains very large numbers that could cause overflow during summation	Use long or double data types to prevent integer overflow during intermediate calculations and when summing.
All elements in the array are identical	The algorithm should not get stuck in a loop and will correctly converge to the solution.
No mutation value leads to a sum closer to the target than the sum of the original array	Return the sum of the original array as the closest value, as the problem seeks to minimize the difference with the target.
The target is close to the sum of a sub-array, but mutating a different value gives an equal result	The problem states it can return any such integer in such a case, and the algorithm needs to be consistent.

Empty input array

How to Handle:

Return 0, or throw an exception indicating invalid input as the problem statement does not define how to handle it.

Array with a single element

How to Handle:

Return that single element if target is close, or apply a predefined mutation rule as needed, then return the mutated value.

Target value is extremely small (negative large)

How to Handle:

The solution should correctly handle negative target values, potentially requiring absolute value calculations when determining the closest sum.

Target value is extremely large

How to Handle:

The solution should avoid integer overflow issues by using appropriate data types or scaling calculations.

Array contains very large numbers that could cause overflow during summation

How to Handle:

Use long or double data types to prevent integer overflow during intermediate calculations and when summing.

All elements in the array are identical

How to Handle:

The algorithm should not get stuck in a loop and will correctly converge to the solution.

No mutation value leads to a sum closer to the target than the sum of the original array

How to Handle:

Return the sum of the original array as the closest value, as the problem seeks to minimize the difference with the target.

The target is close to the sum of a sub-array, but mutating a different value gives an equal result

How to Handle:

The problem states it can return any such integer in such a case, and the algorithm needs to be consistent.