整体非正曲率矩形复形中的最短路问题
Shortest path problem in rectangular complexes of global nonpositive curvature
摘要：
CAT（0）度量空间是欧氏空间、双曲空间和简单多边形的一个广泛的普遍推广：CAT（0）度量空间中的任意两点x和y通过唯一的最短路径γ（x，y）相连。本文提出了一个在CAT（0）矩形复形及其两个子类中回答两点距离查询的有效算法，这两个子类分别是分支直线多边形（CAT（0）矩形复形，其中所有顶点的连接都是二分图）和正方形图（CAT（0）矩形复形，其中所有内顶点的度都是\geq 4）。即，我们表明，对于具有n个顶点的CAT（0）矩形复形K，可以构造大小为$O(n^2)$的数据结构D，使得给定K中的任何两个点x、y，可以在O（d（p，q））时间内计算x和y之间的最短路径γ（x，y），其中p和q分别是包含点x和y的K的两个面的顶点，使得γ（x，y）包含在K（I（p，q））中q））和d（p，q）是K的基础图中p和q之间的距离。如果K是一个分歧的直线多边形，则可以构造一个最优大小为O（n）的数据结构D，并在O（d（p，q）logΔ）时间内回答两点最短路径查询，其中Δ是G（K）的顶点的最大度.最后，如果K是一个平方图，那么可以构造一个大小为O（nlogn）的数据结构D，并在O（d（p，q））时间内回答两点最短路径查询。
英文摘要：
CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x,y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees \geq4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size $O(n^2)$ so that, given any two points x,y in K, the shortest path γ(x,y) between x and y can be computed in O(d(p,q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x,y) is contained in K(I(p,q)) and d(p,q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p,q)logΔ) time, where Δ is the maximal degree of a vertex of G(K). Finally, if K is a squaregraph, then one can construct a data structure D of size O(nlogn) and answer two-point shortest path queries in O(d(p,q)) time.
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