Instance-Optimal Geometric Algorithms

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 A, the maximum running time of A on input s1, . . . , sn is at most a constant factor times the maximum running time of A on s1, . . . , sn , where the maximum is taken over all permutations s1, . . . , sn 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 A...

• Type: Conference
• Year: 2009
• Where: FOCS
• Authors: Peyman Afshani, Jérémy Barbay, Timothy M. Chan
• Keywords: 3-d Convex Hulls, Algorithm, FOCS 2009, Maximum Running Time, Theoretical Computer Science