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| Title | Instance-Optimal Geometric Algorithms |
| Authors | Peyman Afshani, Jérémy Barbay, Timothy Chan |
| Publication date | 2009 |
| Abstract | 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 or der. \n\n 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. \n\n 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. |
| Downloaded | 9 times |
| Pages | 129-138 |
| Conference name | IEEE Symposium on Foundations of Computer Science |
| Publisher | IEEE Computer Society Press (Los Alamitos, CA, USA) |
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