#3356

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Zero Array Transformation II

Zero Array Transformation II requires determining the minimum query sequence to convert all array elements to zero using binary search over valid answer space.

array binary search prefix sum

Problem Statement

You are given an integer array nums of length n and a list of queries where each query contains three integers [li, ri, vali]. Each query represents adding vali to every element from index li to ri inclusive in nums. Your goal is to determine the minimum number of queries needed to transform nums into a Zero Array where all elements are zero.

If it is impossible to achieve a Zero Array with the given queries, return -1. Otherwise, return the smallest number of queries that will result in all elements being zero. Consider the constraints of array length and query sizes while applying an efficient method rather than simulating every combination.

Examples

Example 1

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

Output: 2

Example 2

Input: nums = [4,3,2,1], queries = [[1,3,2],[0,2,1]]

Output: -1

Constraints

1 <= nums.length <= 105
0 <= nums[i] <= 5 * 105
1 <= queries.length <= 105
queries[i].length == 3
0 <= li <= ri < nums.length
1 <= vali <= 5

Solution Approach

Binary Search Over Query Count

Apply binary search on the range from 1 to the total number of queries. For each midpoint, simulate the effect of the first mid queries using prefix sums. If the array can be zeroed, move the search to the left half to find a smaller valid count; otherwise, move to the right half.

Prefix Sum Simulation

Use a prefix sum array to accumulate the effects of queries efficiently. For each query, add the value at the start index and subtract it at the index after the end. Compute running sums to check if all elements reach zero, avoiding O(N*M) brute-force updates.

Validation Check

After applying prefix sums for a candidate number of queries, check every element. If any element is non-zero, the current midpoint fails, and binary search continues. This ensures that only the minimal number of queries is selected without unnecessary computation.

Complexity Analysis

Metric	Value
Time	O(N + M)
Space	O(N)

Time complexity is O(N + M) because each prefix sum simulation runs in O(N) and binary search iterates log(M) times, but combined efficiently with cumulative sums. Space complexity is O(N) for the prefix sum array to track cumulative additions.

What Interviewers Usually Probe

Consider if binary search can reduce the number of simulations instead of checking each query sequence.
Using prefix sums indicates you understand efficient range update techniques in arrays.
Identifying a Zero Array requirement shows awareness of validation failure modes and cumulative effects.

Common Pitfalls or Variants

Common pitfalls

Simulating each query individually without prefix sums can lead to TLE for large N and M.
Forgetting to handle queries that extend beyond the array bounds can cause index errors.
Misinterpreting the requirement to return -1 when no sequence can zero the array leads to incorrect outputs.

Follow-up variants

Zero Array Transformation I with fixed-length queries for simplified index management.
Negative Adjustment Queries where queries can subtract instead of add, changing prefix sum signs.
Zero Array Transformation III where queries can overlap and must be applied in a constrained order.

FAQ

What is the key pattern used in Zero Array Transformation II?

The main pattern is binary search over the valid answer space combined with prefix sums to efficiently simulate cumulative query effects.

Why are prefix sums important for this problem?

Prefix sums allow range updates from queries to be applied in O(1) per query and checked in O(N), avoiding brute-force O(N*M) complexity.

Can this approach handle the largest constraints efficiently?

Yes, using binary search and prefix sums ensures the solution works within O(N + M) time, handling arrays and queries up to 10^5 elements.

What should I return if no queries can zero the array?

Return -1 to indicate that it is impossible to transform the array into a Zero Array with the given queries.

How do I verify the minimal number of queries needed?

Use binary search over query counts and validate with prefix sums, moving the search left when a candidate count succeeds, to find the minimal number.

terminal

Solution

Solution 1: Difference Array + Binary Search

We notice that the more queries we use, the easier it is to turn the array into a zero array, which shows monotonicity. Therefore, we can use binary search to enumerate the number of queries and check whether the array can be turned into a zero array after the first $k$ queries.

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class Solution:
    def minZeroArray(self, nums: List[int], queries: List[List[int]]) -> int:
        def check(k: int) -> bool:
            d = [0] * (len(nums) + 1)
            for l, r, val in queries[:k]:
                d[l] += val
                d[r + 1] -= val
            s = 0
            for x, y in zip(nums, d):
                s += y
                if x > s:
                    return False
            return True

        m = len(queries)
        l = bisect_left(range(m + 1), True, key=check)
        return -1 if l > m else l

Continue Practicing

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