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2009 50th Annual IEEE Symposium on Foundations of Computer Science (2009)
Atlanta, Georgia
Oct. 25, 2009 to Oct. 27, 2009
ISSN: 0272-5428
ISBN: 978-0-7695-3850-1
pp: 129-138
ABSTRACT
We prove the existence of an algorithm $A$ for computing 2-d or 3-dconvex hulls that is optimal for {\em every point set\/} in the following sense: %for every set $S$ of $n$ points and for every algorithm $A'$ in a certain class $\A$, the running time of $A$ on the worst permutation of $S$ for $A$ is at most a constant factor times the running time of $A'$ on the worst permutation of $S$ for $A'$.%In fact, we can establish a stronger property: for every $S$ and $A'$, the running time of $A$ on $S$ is at most a constant factor times the average running time of $A'$ over all permutations of $S$. %We call algorithms satisfying these properties {\em instance-optimal\/} in the {\em order-oblivious\/} and {\em 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 $\A$ under consideration consists of all algorithms in a decision tree model where the tests involve only {\em 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. 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. The theory we develop also neatly reveals connections to entropy-dependent data structures, and yields as a byproduct new expected-case results, e.g., for on-line orthogonal range counting in 2-d.
INDEX TERMS
instance optimality, computational geometry, adaptive algorithms, output-sensitive algorithms, entropy-sensitive data structures, decision trees, lower bounds, convex hull, maxima, orthogonal segment intersection, point location
CITATION

P. Afshani, J. Barbay and T. M. Chan, "Instance-Optimal Geometric Algorithms," 2009 50th Annual IEEE Symposium on Foundations of Computer Science(FOCS), Atlanta, Georgia, 2009, pp. 129-138.
doi:10.1109/FOCS.2009.34
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