Timothy M. Chan's Publications: Convex hulls


A simpler linear-time algorithm for intersecting two convex polyhedra in three dimensions

Chazelle [FOCS'89] gave a linear-time algorithm to compute the intersection of two convex polyhedra in three dimensions. We present a simpler algorithm to do the same.


On constant factors in comparison-based geometric algorithms and data structures

(with Patrick Lee)

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:

Some of the results can be adapted to solve nonorthogonal problems, such as 2D convex hulls and general line segment intersection.

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.


Three problems about dynamic convex hulls

We present three results related to dynamic convex hulls:


Instance-optimal geometric algorithms

(with
Peyman Afshani and Jérémy Barbay)

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.


Optimal in-place and cache-oblivious algorithms for 3-d convex hulls and 2-d segment intersection

(with Eric Y. Chen)

We describe the first optimal randomized in-place algorithm for the basic 3-d convex hull problem (and, in particular, for 2-d Voronoi diagrams). The algorithm runs in O(n log n) expected time using only O(1) extra space; this improves the previous O(n log^3 n) bound by Bronnimann, Chan, and Chen [SoCG'04]. The same approach leads to an optimal randomized in-place algorithm for the 2-d line segment intersection problem, with O(n log n + K) expected running time for output size K, improving the previous O(n log^2 n + K) bound by Vahrenhold [WADS'05]. As a bonus, we also point out a simplification of a known optimal cache-oblivious (non-in-place) algorithm by Kumar and Ramos (2002) for 3-d convex hulls, and observe its applicability to 2-d segment intersection, extending a recent result for red/blue segment intersection by Arge, Molhave, and Zeh [ESA'08]. Our results are all obtained by standard random sampling techniques, with some interesting twists.


Transdichotomous results in computational geometry, II: offline search

(with
Mihai Patrascu)

We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a two-dimensional analog of sorting where one needs to order points with respect to segments. This result implies, for example, that the Voronoi diagram of n points in the plane can be constructed in (randomized) time n . 2^{O(\sqrt{lg lg n})}. Similar bounds hold for numerous other geometric problems, such as three-dimensional convex hulls, planar Euclidean minimum spanning trees, line segment intersection, and triangulation of non-simple polygons.

In FOCS'06, we developed a data structure for online point location, which implied a bound of O(n lg n / lg lg n) for Voronoi diagrams and the other problems. Our current bounds are dramatically better, and a convincing improvement over the classic O(n lg n) algorithms. As in the field of integer sorting, the main challenge is to find ways to manipulate information, while avoiding the online problem (in that case, predecessor search).


Transdichotomous results in computational geometry, I: Point location in sublogarithmic time

(with
Mihai Patrascu)

Given a planar subdivision whose coordinates are integers bounded by U <= 2^w, we present a linear-space data structure that can answer point location queries in O(min{ lg n/lglg n, sqrt{lg U/lglg U} }) time on the unit-cost RAM with word size w. This is the first result to beat the standard Theta(lg n) bound for infinite precision models.

As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed.

Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA'92).


A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries

We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log^3 n) expected amortized time, deletions take O(log^6 n) expected amortized time, and extreme-point queries take O(log^2 n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n^epsilon)-time method by Agarwal and Matousek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).


Multi-pass geometric algorithms

(with Eric Y. Chen)

We initiate the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input. Fundamental problems such as low-dimensional linear programming and convex hulls are considered.


Three problems about simple polygons

We give three related algorithmic results concerning a simple polygon P:


Towards in-place geometric algorithms and data structures

(with
Hervé Brönnimann, and Eric Y. Chen)

For many geometric problems, there are efficient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are efficient data structures that need no extra space at all other than an array holding a permutation of the input. In this paper, we obtain the first such space-economical solutions for a number of fundamental problems, including three-dimensional convex hulls, two-dimensional Delaunay triangulations, fixed-dimensional range queries, and fixed-dimensional nearest neighbor queries.


Space-efficient algorithms for computing the convex hull of a simple polygonal line in linear time

(with
Hervé Brönnimann)

We present space-efficient algorithms for computing the convex hull of a simple polygonal line in-place, in linear time. It turns out that the problem is as hard as stable partition, i.e., if there were a truly simple solution then stable partition would also have a truly simple solution, and vice versa. Nevertheless, we present a simple self-contained solution that uses O(log n) space, and indicate how to improve it to O(1) space with the same techniques used for stable partition. If the points inside the convex hull can be discarded, then there is a truly simple solution that uses a single call to stable partition, and even that call can be spared if only extreme points are desired (and not their order). If the polygonal line is closed, then the problem admits a very simple solution which does not call for stable partitioning at all.


A minimalist's implementation of the 3-d divide-and-conquer convex hull algorithm

We give a simple interpretation and a simple implementation of the classical divide-and-conquer algorithm for computing 3-d convex hulls (and in particular, 2-d Delaunay triangulations and Voronoi diagrams). The entire C++ code is under 100 lines long, requires no special data structures, and uses only 6n pointers for space.


Dynamic planar convex hull operations in near-logarithmic amortized time

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P, such as membership and tangent-finding. Updates take O(log^{1+eps} n) amortized time and queries take O(log n) time each, where n is the maximum size of P and eps is any fixed positive constant. For some advanced queries such as bridge-finding, both our bounds increase to O(log^{3/2} n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log^2 n) time per update and O(log n) time per query.


Output-Sensitive Construction of Convex Hulls

(Ph.D. thesis,
Department of Computer Science, University of British Columbia, November 1995)

Optimal output-sensitive convex hull algorithms in two and three dimensions

We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.


Output-sensitive results on convex hulls, extreme points, and related problems

We use known data structures for ray shooting and linear programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that the f-face convex hull of an n-point set P in a fixed dimension d >= 2 can be constructed in O(n log f + (nf)^{1-1/(floor(d/2)+1)} polylog n) time; this is optimal if f = O(n^{1/floor(d/2)} / log^K n) for some sufficiently large constant K. We also show that the h extreme points of P can be computed in O(n polylog h + (nh)^{1-1/(floor(d/2)+1)} polylog n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n^{2-gamma}) time for any constant gamma < 2/((floor(d/2))^2+1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms, the input is assumed to be in general position.


Primal dividing and dual pruning: output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams

(with
Jack Snoeyink, and Chee-Keng Yap)

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.


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Timothy Chan (Last updated April 2018)