# A fast planar partition algorithm. I

@article{Mulmuley1988AFP, title={A fast planar partition algorithm. I}, author={Ketan Mulmuley}, journal={[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science}, year={1988}, pages={580-589} }

A fast randomized algorithm is given for finding a partition of the plane induced by a given set of linear segments. The algorithm is ideally suited for a practical use because it is extremely simple and robust, as well as optimal; its expected running time is O(m+n log n) where n is the number of input segments and m is the number of points of intersection. The storage requirement is O(m+n). Though the algorithm itself is simple, the global evolution of the partition is complex, which makes…

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## 68 Citations

A Fast Planar Partition Algorithm, I

- Mathematics, Computer ScienceJ. Symb. Comput.
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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.

A fast planar partition algorithm, II

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Randomized, optimal algorithms to find a partition of the plane induced by a set of algebraic segments of a bounded degree, and a set of linear chains of a bounded degree, are given. This paper also…

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The distribution of the running time of Mulmuley's randomized algorithm for computing the intersections of n given segments in the plane is analyzed and it is shown that for values of k not too close to n (k≥Cn log 15n for a large enough constant C), theRunning time is sharply concentrated around the expected value.

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## References

SHOWING 1-6 OF 6 REFERENCES

An optimal algorithm for intersecting line segments in the plane

- Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

The authors present the first optimal algorithm for the following problem: given n line segments in the plane, compute all k pairwise intersections in O(n log n+k) time. Within the same asymptotic…

Algorithms for Reporting and Counting Geometric Intersections

- Computer ScienceIEEE 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.

Constructing Arrangements of Lines and Hyperplanes with Applications

- Mathematics, Computer ScienceSIAM J. Comput.
- 1986

An algorithm is presented that constructs a representation for the cell complex defined by n hyperplanes in optimal $O(n^d )$ time in d dimensions, which is shown to lead to new methods for computing $\lambda $-matrices, constructing all higher-order Voronoi diagrams, halfspatial range estimation, degeneracy testing, and finding minimum measure simplices.

Applications of random sampling in computational geometry, II

- Computer Science, MathematicsSCG '88
- 1988

Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.

Intersecting is easier than sorting

- MathematicsSTOC '84
- 1984

This paper settles a long-standing open question of computational geometry: <italic>Is it possible to compute all k intersections between n arbitrary line segments in time linear in k?</italic> We…