Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this paper, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, including:

- an algorithm for the 2D maxima problem that uses n lg h + O(n sqrt{lg h}) comparisons, where h denotes the output size;
- a randomized algorithm for the 3D maxima problem that uses n lg h + O(n lg^{2/3} h) expected number of comparisons;
- a randomized algorithm for detecting intersections among a set of orthogonal line segments that uses n lg n + O(n sqrt{lg n}) expected number of comparisons;
- a data structure for point location among 3D disjoint axis-parallel boxes that can answer queries in (3/2)lg n + O(lg lg n) comparisons;
- a data structure for point location in a 3D box subdivision that can answer queries in (4/3)lg n + O(sqrt{lg n}) comparisons.

Our algorithms and data structures use a variety of techniques, including Seidel and Adamy's planar point location method, weighted binary search, and height-optimal BSP trees.

- PDF file
- Discrete and Computational Geometry, 53:489-513, 2015 (SoCG special issue)
- In Proc. 30th Symposium on Computational Geometry (SoCG), pages 40-49, 2014

We prove the existence of an algorithm A for computing 2-d or 3-d
convex hulls that is optimal for *every point set* in the
following sense: for every set S of n points and for every
algorithm A' in a certain class C, the maximum running time of
A on input s_1,...,s_n is at most a constant factor
times the maximum running time of A' on s_1,...,s_n,
where the maximum is taken over all permutations s_1,...,s_n of S.
In fact, we can establish a stronger property:
for every S and A', the maximum running time of A is at most a
constant factor times the average running time of A' over all
permutations of S. We call algorithms satisfying these properties
*instance-optimal* in the *order-oblivious* and
*random-order* setting. Such instance-optimal algorithms
simultaneously subsume output-sensitive algorithms and
distribution-dependent average-case algorithms, and all algorithms
that do not take advantage of the order of the input or that assume
the input is given in a random order.

The class C under consideration consists of all algorithms in a
decision tree model where the tests involve only *multilinear*
functions with a constant number of arguments. To establish an
instance-specific lower bound, we deviate from traditional
Ben-Or-style proofs and adopt an interesting adversary argument. For
2-d convex hulls, we prove that a version of the well known algorithm
by Kirkpatrick and Seidel (1986) or Chan,
Snoeyink, and Yap (1995)
already attains this lower bound. For 3-d convex hulls, we propose a
new algorithm.

To demonstrate the potential of the concept, we further obtain instance-optimal results for a few other standard problems in computational geometry, such as maxima in 2-d and 3-d, orthogonal line segment intersection in 2-d, finding bichromatic L_infty-close pairs in 2-d, off-line orthogonal range searching in 2-d, off-line dominance reporting in 2-d and 3-d, off-line halfspace range reporting in 2-d and 3-d, and off-line point location in 2-d.

- PostScript file (preliminary version)
- Journal of the ACM, 64(1): 3:1-3:38, 2017
- In
*Proc. 50th IEEE Symposium on Foundations of Computer Science (FOCS)*, pages 129-138, 2009

- Gzipped postscript file
- Combines papers:

- Postscript file
- Discrete & Computational Geometry, 16:361-368, 1996 (SoCG special issue)

- Postscript file
- Discrete & Computational Geometry, 16:369-387, 1996 (SoCG special issue)
- Preliminary version in Proc. 11th ACM Symposium on Computational Geometry (SoCG), pages 10-19, 1995
- Specialization of results to 2-d and 3-d in separate paper

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E^4. Our algorithm runs in O((n+f) log^2 f) time and uses O(n+f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E^3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E^2 and also leads to improved output-sensitive results on constructing convex hulls in E^d for any even constant d > 4.

- Postscript file
- Discrete & Computational Geometry, 18:433-454, 1997
- Preliminary "dual" version in Proc. 6th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 282-291, 1995

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Timothy Chan (Last updated September 2018)