Eric Y. Chen

Towards in-place geometric algorithms and data structures

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.

Multi-Pass Geometric Algorithms

We propose 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.

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

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(nlogn) expected time using only O(1) extra space; this improves the previous O(nlog^3n) bound by Bronnimann, Chan, and Chen (2004) [10]. The same approach leads to an optimal randomized in-place algorithm for the 2-d line segment intersection problem, with O(nlogn+K) expected running time for output size K, improving the previous O(nlog^2n+K) bound by Vahrenhold (2007) [42]. As a bonus, we also point out a simplification of a known optimal cache-oblivious (non-in-place) algorithm by Kumar and Ramos (2002) [33] 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 (2008) [3]. Our results are all obtained by standard random sampling techniques, with some interesting twists.

Multi-pass geometric algorithms

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.

Geometric streaming algorithms with a sorting primitive

We solve several fundamental geometric problems under a new streaming model recently proposed by Ruhl et al. [2,12]. In this model, in one pass the input stream can be scanned to generate an output stream or be sorted based on a user-defined comparator; all intermediate streams must be of size O(n). We obtain the following geometric results for any fixed constant Ɛ > 0: - We can construct 2D convex hulls in O(1) passes with O(nƐ) extra space. - We can construct 3D convex hulls in O(1) expected number of passes with O(nƐ) extra space. - We can construct a triangulation of a simple polygon in O(1) expected number of passes with O(nƐ) extra space, where n is the number of vertices on the polygon. - We can report all k intersections of a set of 2D line segments in O(1) passes with O(nƐ) extra space, if an intermediate stream of size O(n + k) is allowed. We also consider a weaker model, where we do not have the sorting primitive but are allowed to choose a scan direction for every scan pass. Here we can construct a 2D convex hull from an x-ordered point set in O(1) passes with O(nƐ) extra space.