Start by applying a greedy approach to process queries in order and track the effect on nums. Validate the zero array invariant after each operation. The solution efficiently finds the minimal index to remove or returns -1 if impossible using sorting and prefix sums.
Problem Statement
You are given an integer array nums of length n and a list of queries where each query is a pair [li, ri]. Each query represents an operation that subtracts 1 from every element nums[j] where li <= j <= ri. A Zero Array is an array where all elements are zero.
Determine the minimal number of queries to remove so that after applying the remaining queries in order, nums becomes a zero array. If no such removal exists, return -1.
Examples
Example 1
Input: nums = [2,0,2], queries = [[0,2],[0,2],[1,1]]
Output: 1
After removing queries[2] , nums can still be converted to a zero array.
Example 2
Input: nums = [1,1,1,1], queries = [[1,3],[0,2],[1,3],[1,2]]
Output: 2
We can remove queries[2] and queries[3] .
Example 3
Input: nums = [1,2,3,4], queries = [[0,3]]
Output: -1
nums cannot be converted to a zero array even after using all the queries.
Constraints
- 1 <= nums.length <= 105
- 0 <= nums[i] <= 105
- 1 <= queries.length <= 105
- queries[i].length == 2
- 0 <= li <= ri < nums.length
Solution Approach
Sort Queries and Track Coverage
Sort the queries by their start index to facilitate a greedy approach. Use a prefix sum array to efficiently accumulate the effects of each query on nums and quickly check the cumulative changes.
Greedy Removal Check
Iterate through nums using the prefix sums to identify the first query that violates the zero array invariant. Remove this query and update the prefix sums, repeating the check to find the minimal removal index.
Validate Zero Array Completion
After removing candidate queries, verify if the remaining queries can transform nums to all zeros. If validation fails for all removals, return -1; otherwise, return the minimal query index found.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n + m \times \log{m}) |
| Space | O(n + m) |
Time complexity is O(n + m \times \log{m}) due to sorting m queries and computing prefix sums over n elements. Space complexity is O(n + m) for the prefix sum array and storing sorted queries.
What Interviewers Usually Probe
- Pay attention if a candidate sorts queries to apply the greedy choice efficiently.
- Watch for correct prefix sum usage to track cumulative operations on nums.
- Check if the candidate validates the zero array invariant at each step.
Common Pitfalls or Variants
Common pitfalls
- Failing to sort queries before applying the greedy removal can lead to incorrect minimal removal.
- Neglecting prefix sum updates after removing a query may break invariant checks.
- Returning a valid index without fully confirming nums becomes zero after all remaining queries.
Follow-up variants
- Allow queries to both increment and decrement, requiring signed prefix sums.
- Optimize for very large n with sparse queries using segment trees instead of full prefix sums.
- Find the maximal number of removable queries while still achieving a zero array.
How GhostInterview Helps
- GhostInterview simulates the prefix sum and greedy removal logic to pinpoint minimal removable queries.
- It checks candidate solutions against the zero array invariant to catch subtle failures.
- It explains failure modes when removing queries incorrectly or skipping invariant validation.
Topic Pages
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- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the primary strategy for solving Zero Array Transformation III?
The main strategy is greedy choice plus invariant validation using sorted queries and prefix sums to track cumulative changes.
Can all arrays always be converted to zero arrays?
No, some nums arrays cannot reach all zeros even after applying all queries; in such cases, the solution returns -1.
Why is sorting queries important in this problem?
Sorting ensures that greedy removal checks respect the left-to-right order, preventing premature violation of the zero array invariant.
How does the prefix sum optimize the solution?
Prefix sums allow O(1) range updates for queries and fast cumulative effect computation, avoiding repeated iteration over nums.
What common mistakes should I avoid in Zero Array Transformation III?
Avoid failing to validate the zero array after removals, neglecting prefix sum updates, and misordering queries during the greedy process.
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