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Apply Operations to Make All Array Elements Equal to Zero

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ArraysGreedy Algorithms

You are given a 0-indexed integer array nums and a positive integer k.

You can apply the following operation on the array any number of times:

Choose any subarray of size k from the array and decrease all its elements by 1.

Return true if you can make all the array elements equal to 0, or false otherwise.

A subarray is a contiguous non-empty part of an array.

Example 1:

Input: nums = [2,2,3,1,1,0], k = 3
Output: true
Explanation: We can do the following operations:
- Choose the subarray [2,2,3]. The resulting array will be nums = [1,1,2,1,1,0].
- Choose the subarray [2,1,1]. The resulting array will be nums = [1,1,1,0,0,0].
- Choose the subarray [1,1,1]. The resulting array will be nums = [0,0,0,0,0,0].

Example 2:

Input: nums = [1,3,1,1], k = 2
Output: false
Explanation: It is not possible to make all the array elements equal to 0.

Constraints:

1 <= k <= nums.length <= 10⁵
0 <= nums[i] <= 10⁶

Try coding it in LeetCode

Apply Operations to Make All Array Elements Equal to Zero

Medium

Asked by:

11 views

Topics:

ArraysGreedy Algorithms

You are given a 0-indexed integer array nums and a positive integer k.

You can apply the following operation on the array any number of times:

Choose any subarray of size k from the array and decrease all its elements by 1.

Return true if you can make all the array elements equal to 0, or false otherwise.

A subarray is a contiguous non-empty part of an array.

Example 1:

Input: nums = [2,2,3,1,1,0], k = 3
Output: true
Explanation: We can do the following operations:
- Choose the subarray [2,2,3]. The resulting array will be nums = [1,1,2,1,1,0].
- Choose the subarray [2,1,1]. The resulting array will be nums = [1,1,1,0,0,0].
- Choose the subarray [1,1,1]. The resulting array will be nums = [0,0,0,0,0,0].

Example 2:

Input: nums = [1,3,1,1], k = 2
Output: false
Explanation: It is not possible to make all the array elements equal to 0.

Constraints:

1 <= k <= nums.length <= 10⁵
0 <= 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 elements in the input array? Can they be negative, zero, or floating-point numbers?
What operations are allowed? Is there a specific range or type of values that I can use as the operand to apply in the operations?
If it's impossible to make all elements zero with the allowed operations, what should I return? (e.g., boolean `false`, an error code, or an empty array)
Are the operations applied sequentially to the array, and if so, does the order in which I apply them matter?
Can you provide a few more examples of how the operations are to be performed on the array to produce an output of zero?

Brute Force Solution

Approach

The brute force approach to making all numbers in a list zero involves trying every possible combination of subtractions. We'll systematically explore each starting position and length for these subtractions to see if we can zero out the entire list.

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

Start by considering the first number in the list.
Try subtracting it from itself and the next number. Check if that makes those numbers zero.
Next, try subtracting the first number from itself and the next two numbers. Check if that makes them zero.
Continue extending the subtraction range until you reach the end of the list.
Repeat this process, starting with the second number in the list, then the third, and so on.
For each combination of subtractions you try, check if it eventually makes all the numbers in the list zero.
If you find any combination that works, you have a solution.

Code Implementation

def apply_operations_brute_force(numbers):
    list_length = len(numbers)

    for start_index in range(list_length):
        for subtraction_length in range(1, list_length - start_index + 1):
            temp_numbers = numbers[:]
            first_number = temp_numbers[start_index]

            # This loop mimics applying subtractions from the start_index
            for index_to_subtract in range(start_index, start_index + subtraction_length):
                temp_numbers[index_to_subtract] -= first_number

            # Check if all elements are now zero
            if all(number == 0 for number in temp_numbers):
                return True

    # No combination of subtractions worked to make all elements zero
    return False

Big(O) Analysis

Time Complexity

O(n³) – The algorithm iterates through each element of the array (n elements). For each element, it attempts to subtract it from subsequent elements with increasing range, which effectively becomes nested loops. In the worst case, for each starting position, we iterate through the remaining n elements up to n times. Therefore, this nested iteration pattern gives us n * n * n, or O(n³).

Space Complexity

O(1) – The described brute force approach operates directly on the input array and doesn't create any auxiliary data structures like temporary arrays, hash maps, or sets. It only uses a few variables to track indices and the current subtraction amount, which requires a constant amount of extra space regardless of the input size N (the number of elements in the array). Therefore, the space complexity is O(1).

Optimal Solution

Approach

The key to solving this problem efficiently lies in focusing on making changes from left to right. We use information about the changes we made previously to minimize the work required in future steps, ensuring we don't have to undo earlier work.

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

Start from the beginning of the collection of numbers.
Examine the first number. This number represents the amount we need to change all numbers within the specified range from that point onwards.
Apply this change to every number within the range; this operation affects the numbers that are within the range that starts at our current position.
Now, move to the next number *after* the start of the range we just adjusted.
Repeat the process. Each time, calculate the needed change based on the current number and apply it within its range.
Continue this until you have processed all the numbers that matter based on the range constraints.
If, at the end, every number is zero, then the task is possible; otherwise, it is not.

Code Implementation

def can_make_all_zeros(numbers, array_range):
    number_of_elements = len(numbers)
    change_applied = [0] * number_of_elements

    for index in range(number_of_elements):
        # Accumulate changes applied up to current index
        if index > 0:
            change_applied[index] += change_applied[index - 1]

        current_value = numbers[index] + change_applied[index]

        # If the current value is not zero, apply the operation.
        if current_value != 0:
            if index + array_range > number_of_elements:
                return False

            # Update the array with the required changes.
            change_value = -current_value
            change_applied[index] += change_value

            if index + array_range < number_of_elements:
                change_applied[index + array_range] -= change_value

    # Verify all elements are zero after the operations
    last_change = 0

    for index in range(number_of_elements):
        last_change += change_applied[index] if index == 0 else change_applied[index] - change_applied[index - 1]

    # If all numbers are not zero return false
    return last_change == 0

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the array once with a single loop of size n, where n is the number of elements in the array. For each element, a constant number of operations are performed to update subsequent elements within a fixed range. Therefore, the time complexity is directly proportional to the number of elements, resulting in O(n).

Space Complexity

O(1) – The provided solution operates directly on the input array. It only uses a few constant space variables to keep track of the current position and accumulated changes. No auxiliary data structures like arrays, hash maps, or stacks are created. Therefore, the space complexity is constant and independent of the input array size N.

Edge Cases

Case	How to Handle
Null or empty array input	Return True if the array is empty (vacuously true), or return False if the input is null.
Array with a single element that is non-zero	Return False since it's impossible to make it zero with the given operation.
Array with all elements already zero	Return True as the condition is already satisfied.
Array with mixed positive and negative numbers	The algorithm should correctly handle subtractions resulting in both positive and negative values.
Very large array (scalability considerations)	The algorithm should ideally have a linear time complexity or close to it, to avoid timeouts for large arrays.
Input array where no sequence of operations can result in all zeros.	The algorithm should detect this condition and return False when it's impossible to achieve all zeros.
Integer overflow during subtraction	Use appropriate data types (e.g., long) or check for potential overflow before performing the subtractions.
Consecutive identical numbers that when operated on creates a cascade that is not zero	Solution must ensure each range operated on results in zero, and is not blocked by a cascade.

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 elements in the input array? Can they be negative, zero, or floating-point numbers?
What operations are allowed? Is there a specific range or type of values that I can use as the operand to apply in the operations?
If it's impossible to make all elements zero with the allowed operations, what should I return? (e.g., boolean `false`, an error code, or an empty array)
Are the operations applied sequentially to the array, and if so, does the order in which I apply them matter?
Can you provide a few more examples of how the operations are to be performed on the array to produce an output of zero?

Brute Force Solution

Approach

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

Start by considering the first number in the list.
Try subtracting it from itself and the next number. Check if that makes those numbers zero.
Next, try subtracting the first number from itself and the next two numbers. Check if that makes them zero.
Continue extending the subtraction range until you reach the end of the list.
Repeat this process, starting with the second number in the list, then the third, and so on.
For each combination of subtractions you try, check if it eventually makes all the numbers in the list zero.
If you find any combination that works, you have a solution.

Code Implementation

def apply_operations_brute_force(numbers):
    list_length = len(numbers)

    for start_index in range(list_length):
        for subtraction_length in range(1, list_length - start_index + 1):
            temp_numbers = numbers[:]
            first_number = temp_numbers[start_index]

            # This loop mimics applying subtractions from the start_index
            for index_to_subtract in range(start_index, start_index + subtraction_length):
                temp_numbers[index_to_subtract] -= first_number

            # Check if all elements are now zero
            if all(number == 0 for number in temp_numbers):
                return True

    # No combination of subtractions worked to make all elements zero
    return False

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Start from the beginning of the collection of numbers.
Examine the first number. This number represents the amount we need to change all numbers within the specified range from that point onwards.
Apply this change to every number within the range; this operation affects the numbers that are within the range that starts at our current position.
Now, move to the next number *after* the start of the range we just adjusted.
Repeat the process. Each time, calculate the needed change based on the current number and apply it within its range.
Continue this until you have processed all the numbers that matter based on the range constraints.
If, at the end, every number is zero, then the task is possible; otherwise, it is not.

Code Implementation

def can_make_all_zeros(numbers, array_range):
    number_of_elements = len(numbers)
    change_applied = [0] * number_of_elements

    for index in range(number_of_elements):
        # Accumulate changes applied up to current index
        if index > 0:
            change_applied[index] += change_applied[index - 1]

        current_value = numbers[index] + change_applied[index]

        # If the current value is not zero, apply the operation.
        if current_value != 0:
            if index + array_range > number_of_elements:
                return False

            # Update the array with the required changes.
            change_value = -current_value
            change_applied[index] += change_value

            if index + array_range < number_of_elements:
                change_applied[index + array_range] -= change_value

    # Verify all elements are zero after the operations
    last_change = 0

    for index in range(number_of_elements):
        last_change += change_applied[index] if index == 0 else change_applied[index] - change_applied[index - 1]

    # If all numbers are not zero return false
    return last_change == 0

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty array input	Return True if the array is empty (vacuously true), or return False if the input is null.
Array with a single element that is non-zero	Return False since it's impossible to make it zero with the given operation.
Array with all elements already zero	Return True as the condition is already satisfied.
Array with mixed positive and negative numbers	The algorithm should correctly handle subtractions resulting in both positive and negative values.
Very large array (scalability considerations)	The algorithm should ideally have a linear time complexity or close to it, to avoid timeouts for large arrays.
Input array where no sequence of operations can result in all zeros.	The algorithm should detect this condition and return False when it's impossible to achieve all zeros.
Integer overflow during subtraction	Use appropriate data types (e.g., long) or check for potential overflow before performing the subtractions.
Consecutive identical numbers that when operated on creates a cascade that is not zero	Solution must ensure each range operated on results in zero, and is not blocked by a cascade.