平面直线障碍物间的双准则直线最短路径
Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane
摘要：
给定平面上$h$两两不相交的直线障碍物的直线区域$\mathcal{P}$，研究了在$\mathcal{P}$中计算两点$s$和$t$之间的双准则直线最短路问题.考虑了三种类型的双准则直线路径：最小连接最短路径、最短最小连接路径和最小代价路径，其中路径的代价是边数和路径长度的非减函数，还考虑了单点和两点路径查询。这些问题的算法以前已经给出。我们的贡献是三方面的。首先，我们发现一个关键错误，在所有以前的算法。其次，我们以一种不那么微不足道的方式纠正错误。第三，我们进一步改进的算法，使他们甚至比以前的（不正确的）算法时，$h$是相对较小的。例如，对于最小链路最短路径，我们得到以下结果。我们的算法在$O(n+h\log^{3/2} h)$时间内计算最小链路最短$s$ - $t$路径。对于单点查询，我们为源点$s$构建了一个在$O(n+h\log^{3/2} h)$时间内大小为$O(n+ h\log h)$的数据结构，使得给定任何查询点$t$，可以在$O(\log n)$时间内确定最小链路最短$s$ - $t$路径。对于两点查询，通过$O(n+h^2\log^2 h)$时间和空间预处理，可以在$O(\log n+\log^2 h)$时间内为任意两个查询点$s$和$t$确定一条最小链接最短$s$-$t$路径;或者，通过$O(n+h^2\cdot \log^{2} h \cdot 4^{\sqrt{\log h}})$时间和$O(n+h^2\cdot \log h \cdot 4^{\sqrt{\log h}})$空间预处理，可以在$O(\log n)$时间内回答每个两点查询。
英文摘要：
Given a rectilinear domain $\mathcal{P}$ of $h$ pairwise-disjoint rectilinear obstacles with a total of $n$ vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points $s$ and $t$ in $\mathcal{P}$. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when $h$ is relatively small. For example, for the minimum-link shortest paths, we obtain the following results. Our algorithm computes a minimum-link shortest $s$-$t$ path in $O(n+h\log^{3/2} h)$ time. For the one-point queries, we build a data structure of size $O(n+ h\log h)$ in $O(n+h\log^{3/2} h)$ time for a source point $s$, such that given any query point $t$, a minimum-link shortest $s$-$t$ path can be determined in $O(\log n)$ time. For the two-point queries, with $O(n+h^2\log^2 h)$ time and space preprocessing, a minimum-link shortest $s$-$t$ path can be determined in $O(\log n+\log^2 h)$ time for any two query points $s$ and $t$; alternatively, with $O(n+h^2\cdot \log^{2} h \cdot 4^{\sqrt{\log h}})$ time and $O(n+h^2\cdot \log h \cdot 4^{\sqrt{\log h}})$ space preprocessing, we can answer each two-point query in $O(\log n)$ time.
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