Tight Adaptive Bounds for Convex Hulls

Abstract

Adaptive sorting algorithms exploit existing order in the input to obtain better-than-worst-case running times. A classical example is sorting by runs: if the input can be partitioned into increasing runs of sizes $s_1, \ldots, s_k$ then the \emph{run-length entropy} is $O(\sum_i s_i \log \frac{n}{s_i})$ and there exist many Merge-sort algorithms which run in this time. One can show optimality of such algorithms, by showing that for a fixed sequence of run sizes $s_1, \ldots, s_k$ the worst-case running time of any algorithm lies in $\Omega(\sum_i s_i \log \frac{n}{s_i})$. Recently, Eppstein, Goodrich, Illickan, and To introduced algorithms for Pareto fronts, planar convex hulls, and related problems whose running times improve when the input order contains few sorted runs. They analyze the running time algorithm by defining a \emph{Range Partition Entropy} which is a function that depends both the order of the input and the geometric input points. They ask whether matching adaptive lower bounds analogous to those used for run-length entropy can be shown. We provide matching adaptive lower bounds for constructing a convex hull or a Pareto front.