半平面交的新算法及其实用价值.docx

半平面交的新算法及其实用价值 Keywords: Half-plane, Intersection, Feasible Region, Algorithm, Polygon, PracticalAbstract主旨:半平面的交是当今学术界热烈讨论的问题之一，本文将介绍一个全新的O(nlogn)半平面交算法，强调它在实际运用中的价值，并且在某种程度上将复杂度下降至O(n)线性。最重要的是，我将介绍的算法非常便于实现.§1 introduces what half-plane intersection is. §2 prepares a linear algorithm for convex polygon intersection (abbr. CPI). Equipped with such knowledge, a common solution for HPI is briefly discussed in §3. Then, my new algorithm emerges in §4 detailedly. Not only as a conclusion of the whole paper, §5 also discuss its further usage practically and compares it with the older algorithm described in §3.§1什么是半平面交. §2凸多边形交预备知识. §3简要介绍旧DC算法. §4揭开我的新算法SI神秘面纱. §5总结和实际运用.Timestamps: Came up with it in April 2005; implemented partly in June 2005; problem set in July 2005; publicized as a post in USENET, November 6, 2005.IntroductionA line in plane is usually represented as ax+by=c. Similarly, its inequality form ax+by≤c represents a half-plane (also named h-plane for short) as one side of this line. Notice that ax+by≤c and -ax-by≤-c show opposite h-planes unlike ax+by=c and -ax-by=-c. Half-Plane Intersection (abbr. HPI) considers the following problem:众所周知，直线常用ax+by=c表示，类似地半平面以ax+by≤(≥)c为定义。Given n half-planes, aix+biy≤ci (1≤i≤n), you are to determine the set of all points that satisfying all the n inequations. 给定n个形如aix+biy≤ci的半平面，找到所有满足它们的点所组成的点集As Figure iTwo examples of HPI. (b) gives an example of unbounded intersection area. describes, the feasible region, which is the intersection, forms a shape of convex hull but possibly unbounded. However, we shall add four h-planes forming a rectangle, which is large enough to make sure the area after intersections finite. In the following sections, we suppose the feasible region is bounded with a finite area.合并后区域形如凸多边形，可能无界。此时增加4个半平面保证面积有限??! ??? ???????. (a) (b) Each h-plane builds up at most one side of the convex polygon, hence, the convex region will be bounded by at most n edges. Pay attention the intersection somet