Timothy M. Chan's Publications: Line segment intersection


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.


Bichromatic line segment intersection counting in O(n sqrt{log n}) time

(with
Bryan T. Wilkinson)

We give an algorithm for bichromatic line segment intersection counting that runs in O(n sqrt{log n}) time under the word RAM model via a reduction to dynamic predecessor search, offline point location, and offline dynamic ranking. This algorithm is the first to solve bichromatic line segment intersection counting in o(n log n) time.


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 space-efficient algorithm for segment intersection

(with Eric Y. Chen)

We examine the space requirement for the classic line-segment intersection problem. Using so-called implicit data structures, we show how to make the standard sweep-line algorithm run in O((n+k)log^2 n) time with only O(log^2 n) extra space, where n is the number of line segments and k is the number of intersections. If division is allowed and input can be destroyed, the algorithm can run in O((n+k)log n) time with O(1) extra space.


Reporting curve segment intersections using restricted predicates

We investigate how to report all k intersecting pairs among a collection of n x-monotone curve segments in the plane, using only predicates of the following forms: is an endpoint to the left of another? is an endpoint above a segment? do two segments intersect? By studying the intersection problem in an abstract setting that assumes the availability of certain "detection oracles," we obtain a near-optimal randomized algorithm that runs in O(n log n +n sqrt{k log (n^2/k)}) expected time. In the bichromatic case (where segments are colored red or blue with no red/red or blue/blue intersections), we find a better algorithm that runs in O((n+k) log_{2+k/n} n) worst-case time, by modifying a known segment-tree method. Two questions of Boissonnat and Snoeyink are thus answered to within logarithmic factors.


A simple trapezoid sweep algorithm for reporting red/blue segment intersections

We present a new simple algorithm for computing all intersections between two collections of disjoint line segments. The algorithm runs in O(n log n + k) time and O(n) space, where n and k are the number of segments and intersections respectively. We also show that the algorithm can be extended to handle single-valued curve segments with the same time and space bound.


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Timothy Chan (Last updated September 2018)