#2772

Medium

LeetCode Problem Workspace

Apply Operations to Make All Array Elements Equal to Zero

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Determine if all elements in a given array can be reduced to zero using repeated k-length prefix operations efficiently.

array prefix sum

Problem Statement

You are given a 0-indexed array nums of non-negative integers and a positive integer k. You may perform operations that subtract 1 from each element of any contiguous subarray of length k any number of times. Determine if it is possible to make all elements equal to 0.

Return true if all elements in the array can be reduced to zero through these operations, otherwise return false. Consider how the order of choosing subarrays and cumulative effects influence whether the array can be fully zeroed.

Examples

Example 1

Input: nums = [2,2,3,1,1,0], k = 3

Output: true

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

It is not possible to make all the array elements equal to 0.

Constraints

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

Solution Approach

Use Prefix Sum to Track Operations

Maintain a difference array to record the net effect of operations. For each index, apply the cumulative effect from previous operations and determine how many more are needed to reduce nums[i] to zero. This ensures efficient O(n) processing without repeated subarray traversals.

Greedy Element Reduction

Process the array from left to right. At each element, if the current value after applying prefix sums is positive, apply enough operations on the k-length subarray starting at that index. This greedy approach aligns with the array plus prefix sum pattern and prevents over-subtracting.

Validate Final Array State

After applying operations, check the last n-k+1 elements for zero. Any remaining positive value means the array cannot be fully reduced. This final validation ensures correctness and accounts for the boundary effects in the prefix sum tracking.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is O(n) due to single-pass prefix sum updates, and space complexity is O(n) for the difference array storing operation effects. The greedy subarray reduction avoids nested loops.

What Interviewers Usually Probe

Ask for an O(n) solution using a prefix sum or difference array to avoid TLE on large arrays.
Probe how the order of applying operations affects the ability to reach zero, especially at boundaries.
Check if candidates handle cases where k equals array length or when elements exceed available operations.

Common Pitfalls or Variants

Common pitfalls

Failing to account for cumulative operations when reducing elements leads to over- or under-subtracting.
Attempting to brute-force every subarray results in O(n*k) time and may time out on large inputs.
Ignoring boundary elements beyond n-k+1 can cause incorrect false negatives even when zeroing is possible.

Follow-up variants

Modify the problem to maximize operations applied before reaching zero, focusing on order selection.
Allow negative numbers in nums and track net operations needed using a signed prefix sum array.
Change k dynamically per operation and determine feasibility using a sliding window difference approach.

FAQ

What is the core pattern in Apply Operations to Make All Array Elements Equal to Zero?

The core pattern is combining greedy subarray reduction with a prefix sum to track cumulative operations efficiently.

How does the prefix sum approach optimize this array problem?

It reduces repeated subarray updates to a single cumulative operation, allowing O(n) time instead of O(n*k).

Can this method handle k equal to the array length?

Yes, the approach correctly applies operations once across the full array and checks if all elements reach zero.

What happens if some elements are initially zero?

Prefix sum tracking accounts for existing zeros, skipping unnecessary operations while still reducing others to zero.

Is there a risk of negative numbers with this approach?

Since the problem restricts nums to non-negative values, the difference array only tracks positive operations, avoiding negative results.

terminal

Solution

Solution 1: Difference Array + Prefix Sum

First, let's consider the first element of $nums$, $nums[0]$:

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class Solution:
    def checkArray(self, nums: List[int], k: int) -> bool:
        n = len(nums)
        d = [0] * (n + 1)
        s = 0
        for i, x in enumerate(nums):
            s += d[i]
            x += s
            if x == 0:
                continue
            if x < 0 or i + k > n:
                return False
            s -= x
            d[i + k] += x
        return True

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