#3275

Medium

auto_awesomeArray plus Heap (Priority Queue)

LeetCode Problem Workspace

K-th Nearest Obstacle Queries

Solve the K-th nearest obstacle query problem using array and heap (priority queue) techniques to find distances efficiently.

array heap priority queue

Problem Statement

You are given a 2D infinite plane and a set of queries. Each query is a coordinate in the plane, and your task is to find the distance to the k-th nearest obstacle from the origin after each query. The distance between any two points is measured using the Euclidean distance formula.

You are also given a positive integer k and a series of queries. After each query, compute the k-th nearest obstacle from the origin. The queries involve determining the appropriate k-th distance from the origin to obstacles that are processed dynamically, requiring an efficient data structure for optimal performance.

Examples

Example 1

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

Output: [-1,7,5,3]

Example 2

Input: queries = [[5,5],[4,4],[3,3]], k = 1

Output: [10,8,6]

Constraints

1 <= queries.length <= 2 * 105
All queries[i] are unique.
-109 <= queries[i][0], queries[i][1] <= 109
1 <= k <= 105

Solution Approach

Array and Heap-Based Distance Calculation

To efficiently solve this problem, store the distances of all obstacles in an array. For each query, calculate the distance from the origin and insert it into a heap. Use the heap to maintain the k-th smallest distance, ensuring efficient retrieval of the k-th nearest obstacle.

Heap for Efficient k-th Nearest Element

A max-heap (priority queue) can be used to store the smallest k distances. As each new distance is computed, if the heap contains fewer than k elements, insert the new distance. If the heap is full, compare the new distance with the largest distance in the heap and replace it if necessary, ensuring that the heap always contains the k closest distances.

Optimizing with Sorting

Alternatively, sorting all distances after each query and picking the k-th smallest distance is another approach, though less efficient than the heap-based method. Sorting ensures that the distances are ordered, but it comes with a higher computational cost, especially for large queries.

Complexity Analysis

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

The time complexity primarily depends on the heap operations. Inserting into a heap takes O(log k), and since there are up to 2 * 10^5 queries, the overall time complexity can be O(n log k), where n is the number of queries. For sorting-based solutions, the complexity is O(n log n), which is less efficient for large datasets. The space complexity is O(k) for the heap-based solution, while a sorting solution would have a space complexity of O(n).

What Interviewers Usually Probe

Does the candidate consider heap-based solutions to optimize the k-th nearest distance?
Does the candidate evaluate the trade-offs between heap and sorting approaches?
How efficiently does the candidate handle multiple queries with large inputs?

Common Pitfalls or Variants

Common pitfalls

Not using the heap efficiently, leading to suboptimal performance for large inputs.
Incorrectly assuming that sorting all obstacles will be faster than using a heap for k-th nearest calculations.
Overcomplicating the solution by not considering that only the k closest obstacles matter for each query.

Follow-up variants

What if the value of k is very large? Can the approach handle it efficiently?
How would the solution change if obstacles were added dynamically after each query?
What if the query coordinates themselves were to change during the problem?

FAQ

What is the K-th Nearest Obstacle Queries problem?

The K-th Nearest Obstacle Queries problem involves finding the distance of the k-th nearest obstacle from the origin in an infinite 2D plane, given multiple queries.

How does a heap help with the K-th Nearest Obstacle Queries?

A heap (priority queue) helps efficiently track the k-th nearest distance by storing the smallest k distances and quickly updating them as new queries arrive.

What is the time complexity of solving this problem?

The time complexity is O(n log k), where n is the number of queries, and k is the number of closest obstacles to track.

Why not use sorting for K-th Nearest Obstacle Queries?

Sorting all distances after each query is less efficient than using a heap because it has a time complexity of O(n log n), whereas heap operations can be performed in O(log k) time.

Can GhostInterview help optimize solutions for this problem?

Yes, GhostInterview provides guidance on selecting the optimal data structure and method, focusing on performance improvements through heap-based solutions for the K-th Nearest Obstacle Queries.

terminal

Solution

Solution 1: Priority Queue (Max-Heap)

We can use a priority queue (max-heap) to maintain the $k$ obstacles closest to the origin.

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class Solution:
    def resultsArray(self, queries: List[List[int]], k: int) -> List[int]:
        ans = []
        pq = []
        for i, (x, y) in enumerate(queries):
            heappush(pq, -(abs(x) + abs(y)))
            if i >= k:
                heappop(pq)
            ans.append(-pq[0] if i >= k - 1 else -1)
        return ans

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