Given a point set $P$ in the plane, we seek a subset $Q\subseteq P$, whose convex hull gives a smaller and thus simpler representation of the convex hull of $P$. Specifically, let $cost(Q,P)$ denote the Hausdorff distance between the convex hulls $\mathcal{CH}(Q)$ and $\mathcal{CH}(P)$. Then given a value $\varepsilon>0$ we seek the smallest subset $Q\subseteq P$ such that $cost(Q,P)\leq \varepsilon$. We also consider the dual version, where given an integer $k$, we seek the subset $Q\subseteq P$ which minimizes $cost(Q,P)$, such that $|Q|\leq k$. For these problems, when $P$ is in convex position, we respectively give an $O(n\log^2 n)$ time algorithm and an $O(n\log^3 n)$ time algorithm, where the latter running time holds with high probability. When there is no restriction on $P$, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an $O(n^{2.5302})$ time algorithm when minimizing $k$ and an $O(\min\{n^{2.5302}, kn^{2.376}\})$ time algorithm when minimizing $\varepsilon$, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.
翻译:考虑到飞机上设定的美元点数,我们寻求一个子子数 $ subseteq P$, 其软质船体的比价较小, 因而更简单。 具体地说, 美元( Q, P) 表示软质船体之间的豪斯多夫距离 $\ mathcal{CH}( Q) 美元 和$\ mathcal{cal} (P) 美元。 以美元( varepsilon) > 0美元的价值, 我们寻求最小的子数 $ subseteq P$ P, 其成本( Q, P)\ subseteq 美元( Q, Q, P) 美元( Q, Q) 和 mathcrdcal 美元( 美元) 。 对于这些问题, 当 美元( $( rq) 美元) 时间算法和 美元( 美元( ) (n\ log3 n) leq) 时间算法值( 美元) 美元) 值( 美元) 。 当我们开始调整时, 时间算算算算算算算算算算一个高时, 时, 。