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Make Array Zero by Subtracting Equal Amounts

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ArraysGreedy Algorithms

You are given a non-negative integer array nums. In one operation, you must:

Choose a positive integer x such that x is less than or equal to the smallest non-zero element in nums.
Subtract x from every positive element in nums.

Return the minimum number of operations to make every element in nums equal to 0.

Example 1:

Input: nums = [1,5,0,3,5]
Output: 3
Explanation:
In the first operation, choose x = 1. Now, nums = [0,4,0,2,4].
In the second operation, choose x = 2. Now, nums = [0,2,0,0,2].
In the third operation, choose x = 2. Now, nums = [0,0,0,0,0].

Example 2:

Input: nums = [0]
Output: 0
Explanation: Each element in nums is already 0 so no operations are needed.

Constraints:

1 <= nums.length <= 100
0 <= nums[i] <= 100

Try coding it in LeetCode

Make Array Zero by Subtracting Equal Amounts

Easy

Asked by:

10 views

Topics:

ArraysGreedy Algorithms

You are given a non-negative integer array nums. In one operation, you must:

Choose a positive integer x such that x is less than or equal to the smallest non-zero element in nums.
Subtract x from every positive element in nums.

Return the minimum number of operations to make every element in nums equal to 0.

Example 1:

Input: nums = [1,5,0,3,5]
Output: 3
Explanation:
In the first operation, choose x = 1. Now, nums = [0,4,0,2,4].
In the second operation, choose x = 2. Now, nums = [0,2,0,0,2].
In the third operation, choose x = 2. Now, nums = [0,0,0,0,0].

Example 2:

Input: nums = [0]
Output: 0
Explanation: Each element in nums is already 0 so no operations are needed.

Constraints:

1 <= nums.length <= 100
0 <= nums[i] <= 100

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 maximum size of the input array `nums`? Are there any memory constraints I should be aware of?
The problem states non-negative integers, but is it possible for the input array `nums` to be null or empty?
If all the elements in `nums` are initially zero, what should the return value be?
Are there any specific data type constraints on the integers in `nums` beyond being non-negative (e.g., can they be very large, potentially causing integer overflow issues if not handled carefully)?
If there are multiple identical non-zero minimum values in `nums`, does the order in which I conceptually subtract `x` from the positive elements matter, or can I assume any of them can be chosen as `x`?

Brute Force Solution

Approach

To make every number in a group zero, we can repeatedly subtract the smallest positive number until everything is zero. This brute force approach directly simulates the process of repeatedly finding the minimum and subtracting.

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

Look at all the positive numbers in the group.
Find the smallest positive number.
Subtract that smallest number from every other positive number in the group.
Count this as one operation.
Repeat this process of finding the smallest positive number and subtracting until there are no more positive numbers left.
The number of times you subtract is the answer.

Code Implementation

def make_array_zero(numbers):
    operation_count = 0

    while True:
        positive_numbers = [number for number in numbers if number > 0]

        # If there are no positive numbers, we're done.
        if not positive_numbers:
            break

        smallest_positive = min(positive_numbers)

        # Subtract the smallest positive number from all positive numbers.
        for index in range(len(numbers)):
            if numbers[index] > 0:
                numbers[index] -= smallest_positive

        operation_count += 1

    return operation_count

Big(O) Analysis

Time Complexity

O(n²) – The algorithm iterates until all numbers in the array of size n are zero. In each iteration, it finds the smallest positive number, which requires examining up to n elements in the worst case. Then, it subtracts this smallest number from all other positive numbers, requiring another scan of up to n elements. Since this process repeats until all elements are zero, and in the worst-case scenario where numbers are close to each other, this can take up to n iterations. Therefore, the overall time complexity is approximately n * n, which simplifies to O(n²).

Space Complexity

O(1) – The algorithm iteratively finds the smallest positive number and subtracts it. It doesn't use any auxiliary data structures like arrays, lists, or hash maps to store intermediate results or track visited numbers. The operations are performed directly on the input array without creating any extra memory dependent on the input size N (where N is the number of elements in the input array). Therefore, the space complexity is constant.

Optimal Solution

Approach

The key idea is to repeatedly subtract the smallest positive number in the list from all the positive numbers. This efficiently brings the array closer to all zeros without unnecessary calculations. By focusing on the smallest positive number each time, we minimize the number of operations.

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

Find the smallest positive number in the list.
If there are no positive numbers, you're done.
Subtract this smallest positive number from every positive number in the list.
Count this subtraction as one operation.
Repeat this process until all numbers in the list are zero.

Code Implementation

def make_array_zero(numbers):
    operations_count = 0

    while True:
        positive_numbers = [number for number in numbers if number > 0]

        if not positive_numbers:
            break

        smallest_positive_number = min(positive_numbers)

        # Subtract the smallest positive number from all positive numbers
        for index in range(len(numbers)):
            if numbers[index] > 0:
                numbers[index] -= smallest_positive_number

        # Increment the operation count, since one subtraction occured
        operations_count += 1

    return operations_count

Big(O) Analysis

Time Complexity

O(n²) – The algorithm iterates until all numbers in the input array of size n are zero. In each iteration, we find the smallest positive number, which takes O(n) time in the worst case. Then, we iterate through the array again to subtract this smallest positive number from all other positive numbers, also taking O(n) time. Since, in the worst-case scenario, we might need to perform these operations for each distinct positive number in the array (which can be up to n times), the overall time complexity is approximately n * n. Therefore, the time complexity is O(n²).

Space Complexity

O(1) – The described algorithm operates directly on the input array and only requires a few variables to store temporary values, such as the smallest positive number found in each iteration. No additional data structures that scale with the input size N (the number of elements in the array) are used. Therefore, the auxiliary space complexity is constant.

Edge Cases

Case	How to Handle
Null or empty input array	Return 0, as no operations are needed to make an empty array zero.
Array containing only zeros	Return 0, because all elements are already zero.
Array with a single non-zero element	Return 1, as one operation is needed to make the single element zero.
Array with duplicate non-zero values	The algorithm should count each unique non-zero value as one operation.
Array with a large number of distinct non-zero elements	The solution should scale efficiently, preferably with O(N) or O(N log N) time complexity, where N is the length of the array.
Array with very large integer values (approaching maximum integer value)	Ensure the subtraction operations don't cause integer overflow, using appropriate data types if necessary.
Array with a mix of many zeros and a few large positive numbers	The solution should efficiently skip the zeros and focus on the positive numbers to minimize operations.
Array already sorted in ascending or descending order	The algorithm should correctly handle pre-sorted arrays without any performance degradation.

Null or empty input array

How to Handle:

Return 0, as no operations are needed to make an empty array zero.

Array containing only zeros

How to Handle:

Return 0, because all elements are already zero.

Array with a single non-zero element

How to Handle:

Return 1, as one operation is needed to make the single element zero.

Array with duplicate non-zero values

How to Handle:

The algorithm should count each unique non-zero value as one operation.

Array with a large number of distinct non-zero elements

How to Handle:

The solution should scale efficiently, preferably with O(N) or O(N log N) time complexity, where N is the length of the array.

Array with very large integer values (approaching maximum integer value)

How to Handle:

Ensure the subtraction operations don't cause integer overflow, using appropriate data types if necessary.

Array with a mix of many zeros and a few large positive numbers

How to Handle:

The solution should efficiently skip the zeros and focus on the positive numbers to minimize operations.

Array already sorted in ascending or descending order

How to Handle:

The algorithm should correctly handle pre-sorted arrays without any performance degradation.

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 maximum size of the input array `nums`? Are there any memory constraints I should be aware of?
The problem states non-negative integers, but is it possible for the input array `nums` to be null or empty?
If all the elements in `nums` are initially zero, what should the return value be?
Are there any specific data type constraints on the integers in `nums` beyond being non-negative (e.g., can they be very large, potentially causing integer overflow issues if not handled carefully)?
If there are multiple identical non-zero minimum values in `nums`, does the order in which I conceptually subtract `x` from the positive elements matter, or can I assume any of them can be chosen as `x`?

Brute Force Solution

Approach

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

Look at all the positive numbers in the group.
Find the smallest positive number.
Subtract that smallest number from every other positive number in the group.
Count this as one operation.
Repeat this process of finding the smallest positive number and subtracting until there are no more positive numbers left.
The number of times you subtract is the answer.

Code Implementation

def make_array_zero(numbers):
    operation_count = 0

    while True:
        positive_numbers = [number for number in numbers if number > 0]

        # If there are no positive numbers, we're done.
        if not positive_numbers:
            break

        smallest_positive = min(positive_numbers)

        # Subtract the smallest positive number from all positive numbers.
        for index in range(len(numbers)):
            if numbers[index] > 0:
                numbers[index] -= smallest_positive

        operation_count += 1

    return operation_count

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Find the smallest positive number in the list.
If there are no positive numbers, you're done.
Subtract this smallest positive number from every positive number in the list.
Count this subtraction as one operation.
Repeat this process until all numbers in the list are zero.

Code Implementation

def make_array_zero(numbers):
    operations_count = 0

    while True:
        positive_numbers = [number for number in numbers if number > 0]

        if not positive_numbers:
            break

        smallest_positive_number = min(positive_numbers)

        # Subtract the smallest positive number from all positive numbers
        for index in range(len(numbers)):
            if numbers[index] > 0:
                numbers[index] -= smallest_positive_number

        # Increment the operation count, since one subtraction occured
        operations_count += 1

    return operations_count

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input array	Return 0, as no operations are needed to make an empty array zero.
Array containing only zeros	Return 0, because all elements are already zero.
Array with a single non-zero element	Return 1, as one operation is needed to make the single element zero.
Array with duplicate non-zero values	The algorithm should count each unique non-zero value as one operation.
Array with a large number of distinct non-zero elements	The solution should scale efficiently, preferably with O(N) or O(N log N) time complexity, where N is the length of the array.
Array with very large integer values (approaching maximum integer value)	Ensure the subtraction operations don't cause integer overflow, using appropriate data types if necessary.
Array with a mix of many zeros and a few large positive numbers	The solution should efficiently skip the zeros and focus on the positive numbers to minimize operations.
Array already sorted in ascending or descending order	The algorithm should correctly handle pre-sorted arrays without any performance degradation.

Null or empty input array

How to Handle:

Return 0, as no operations are needed to make an empty array zero.

Array containing only zeros

How to Handle:

Return 0, because all elements are already zero.

Array with a single non-zero element

How to Handle:

Return 1, as one operation is needed to make the single element zero.

Array with duplicate non-zero values

How to Handle:

The algorithm should count each unique non-zero value as one operation.

Array with a large number of distinct non-zero elements

How to Handle:

The solution should scale efficiently, preferably with O(N) or O(N log N) time complexity, where N is the length of the array.

Array with very large integer values (approaching maximum integer value)

How to Handle:

Ensure the subtraction operations don't cause integer overflow, using appropriate data types if necessary.

Array with a mix of many zeros and a few large positive numbers

How to Handle:

The solution should efficiently skip the zeros and focus on the positive numbers to minimize operations.

Array already sorted in ascending or descending order

How to Handle:

The algorithm should correctly handle pre-sorted arrays without any performance degradation.