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:
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.
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.
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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