# Self-improving algorithms for coordinate-wise maxima

@article{Clarkson2012SelfimprovingAF, title={Self-improving algorithms for coordinate-wise maxima}, author={Kenneth L. Clarkson and Wolfgang Mulzer and C. Seshadhri}, journal={ArXiv}, year={2012}, volume={abs/1211.0952} }

Computing the coordinate-wise maxima of a planar point set is a classic and well-studied problem in computational geometry. We give an algorithm for this problem in the self-improving setting. We have n (unknown) independent distributions cD1, cD2, ..., cDn of planar points. An input pointset (p1, p2, ..., pn) is generated by taking an independent sample pi from each cDi, so the input distribution cD is the product prodi cDi. A self-improving algorithm repeatedly gets input sets from the…

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## References

SHOWING 1-10 OF 39 REFERENCES

### Self-improving algorithms for delaunay triangulations

- Computer ScienceSCG '08
- 2008

The running time of the algorithm matches the information-theoretic lower bound for the given input distribution, implying that if the input distribution has low entropy, then the algorithm beats the standard Ω(n log n) bound for computing Delaunay triangulations.

### Applications of random sampling in computational geometry, II

- Computer Science, MathematicsSCG '88
- 1988

Asymptotically tight bounds for (≤k)-sets are given, which are certain halfspace partitions of point sets, and a simple proof of Lee's bounds for high-order Voronoi diagrams is given.

### New applications of random sampling in computational geometry

- Computer ScienceDiscret. Comput. Geom.
- 1987

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by…

### Further applications of random sampling to computational geometry

- Computer ScienceSTOC '86
- 1986

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.

### Adaptive ( Analysis of ) Algorithms for Convex Hulls and Related Problems

- Computer Science
- 2008

This work presents two adaptive techniques for the computation of the convex hull in two and three dimensions and related problems, and yields adaptive algorithms which perform faster on many classes of instances, while performing asymptotically no worse in the worst case over all instances of xed size.

### Convex hull of points lying on lines in time after preprocessing

- Mathematics, Computer ScienceComput. Geom.
- 2013

### A provably fast linear-expected-time maxima-finding algorithm

- Computer ScienceAlgorithmica
- 2005

This paper proves Bentleyet al.'s conjecture that the Move-To-Front (MTF) algorithm runs inN+o(N) expected time, and mixes probabilistic and amortized techniques.

### Convex hull of imprecise points in o(n log n) time after preprocessing

- Computer Science, MathematicsSoCG '11
- 2011

The convex hull problem under this setting is easier than sorting, contrary to the "standard" convex Hull and sorting problems, in which the two problems require Theta(n log n) steps in the worst case (under the algebraic computation tree model).

### The Ultimate Planar Convex Hull Algorithm?

- Computer ScienceSIAM J. Comput.
- 1986

We present a new planar convex hull algorithm with worst case time complexity $O(n \log H)$ where $n$ is the size of the input set and $H$ is the size of the output set, i.e. the number of vertices…

### Cutting hyperplanes for divide-and-conquer

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1993

A deterministic algorithm for computing a (1/r)-cutting ofO(rd) size inO(nrd−1) time is presented, based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes.