In this research paper, the authors aim to improve the efficiency of algorithms used to find the closest point to a query segment in a set of lines. They present two new deterministic and randomized algorithms that significantly reduce the preprocessing time and query time for various applications.
Deterministic Algorithms
The authors construct a new data structure with a complexity of O(nm(n/m)δ), where n is the number of lines, m is the number of points in the query segment, δ is a small positive parameter, and nm is the total number of points. This data structure can be used to find the closest point to a query segment in O(n1.5, n, n log n) time.
Randomized Algorithms
The authors also propose a faster randomized algorithm with a complexity of O(n log n, n, n), which is an improvement over previous work. This algorithm uses a combination of ray-shooting and nearest neighbor search techniques to find the closest point to a query segment.
Preprocessing Time
The preprocessing time for constructing the cutting takes O(nr) time, where r is the radius of the query segment. Constructing the BS data structure for the closest points takes O(nm log nm) time. The total preprocessing time is O(nr log n + nm log nm).
Comparison with Previous Work
The authors compare their algorithms with previous work, showing that they have faster query times and lower preprocessing times. They also demonstrate the effectiveness of their algorithms through experimental results.
Conclusion
In summary, this research paper presents two fast algorithms for finding the closest point to a query segment in a set of lines. The algorithms have improved efficiency over previous work, making them suitable for real-world applications. The authors provide a detailed analysis of the time complexity and preprocessing time for their algorithms, showing their effectiveness and efficiency.
