三维空间中的最短路问题

The Problem of Shortest Path in 3D Space

在包含一组相互分离凸多面体的三维空间中为任意两点寻找最短路的问题是ＮＰ问题．当凸多面体的个数ｋ 任意时，它为指数时间复杂度；而当ｋ＝ １时，为Ｏ（ｎ２）（ｎ 为凸多面体的顶点数）．文章主要研究了ｋ＝ ２情形下的最短路问题，提出一个在Ｏ（ｎ２）时间内解决该问题的算法．所得结果大大优于此情形下迄今为止最好的结果——Ｏ（ｎ３ｌｏｇｎ）．另外，将此结果应用到ｋ＞ ２的情形后，获得的结果为Ｏ（１２ｉ－ １ｎ２ｉ）．

The problem of computing the euclidean shortest path between two points in the three dimensional space bounded by a collection of convex disjoint polyhedral obstacles is known to be NP hard and in exponential time for arbitrarily many obstacles. It can be solved in O(n 2) time for single convex polyhedron obstacle (here n is the total number of vertices of polyhedron). In this paper, the author mainly researchs the shortest problem of the case of two convex polyhedral obstacles, and presents an algorithm that solves this problem in O(n 2) time, and improves improving significantly previous best result O(n 3 log n ) for this special case. On the other hand, the author also presents a better result O(? 苮12 i-1 n 2 i) for the problem of shortest path amidst a fixed number of convex polyhedral obstacles.