# An optimal algorithm for finding segments intersections

@inproceedings{Balaban1995AnOA,
title={An optimal algorithm for finding segments intersections},
author={Ivan J. Balaban},
booktitle={SCG '95},
year={1995}
}
• I. J. Balaban
• Published in SCG '95 1 September 1995
• Computer Science, Mathematics
This paper deals with a new deterministic algorithm for finding intersecting pairs from a given set of N segments in the plane. The algorithm is asymptotically optimal and has time and space complexity O(AJ log N+ K) and 0( IV ) respectively, where K is the number of intersecting pairs. The algorithm may be used for finding intersections not only line segments but also curve segments.
202 Citations

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## References

SHOWING 1-10 OF 14 REFERENCES
Algorithms for Reporting and Counting Geometric Intersections
• Computer Science
IEEE Transactions on Computers
• 1979
Algorithms that count the number of pairwise intersections among a set of N objects in the plane and algorithms that report all such intersections are given.
A fast planar partition algorithm. I
• K. Mulmuley
• Mathematics, Computer Science
[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
• 1988
Though the algorithm itself is simple, the global evolution of the partition is complex, which makes the analysis of the algorithm theoretically interesting in its own right.
Reporting and Counting Segment Intersections
• B. Chazelle
• Computer Science, Mathematics
J. Comput. Syst. Sci.
• 1986
An algorithm is described whose running time is O(n(log2 nlog log n)+k, the first solution with a time bound linear in the size of the output, and a new hierarchical strategy for dealing with segments without reducing the dimensionality of the problem is introduced.
A Fast Planar Partition Algorithm, I
• K. Mulmuley
• Mathematics, Computer Science
J. Symb. Comput.
• 1990
Though the algorithm itself is simple, the global evolution of the underlying partition is non-trivial, which makes the analysis of the algorithm theoretically interesting in its own right.
Optimal Point Location in a Monotone Subdivision
• Mathematics, Computer Science
SIAM J. Comput.
• 1986
A substantial refinement of the technique of Lee and Preparata for locating a point in $\mathcal{S}$ based on separating chains is exhibited, which can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.
A data structure for orthogonal range queries
• G. S. Lueker
• Computer Science
19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
• 1978
It is shown that a decision tree of height O(dn log n) can be constructed to process n operations in d dimensions, suggesting that the standard decision tree model will not provide a useful method for investigating the complexity of orthogonal range queries.
Lower bounds for algebraic computation trees
• M. Ben-Or
• Computer Science, Mathematics
STOC
• 1983
All the apparently known lower bounds for linear decision trees are extended to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20].
Filtering search: A new approach to query-answering
• B. Chazelle
• Computer Science, Mathematics
24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
• 1983
We introduce a new technique for solving problems of the following form: preprocess a set of objects so that those satisfying a given property with respect to a query object can be listed very
Applications of random sampling in computational geometry, II
Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Computational geometry: an introduction
• Computer Science, Mathematics
• 1985
This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.