平面图中某些问题的同时时空上界
Simultaneous Time-Space Upper Bounds for Certain Problems in Planar Graphs
摘要：
在本文中，我们证明了，给定一个加权的，有向平面图$G$，和任何$ε>0$，存在一个多项式时间和$O(n^{\frac{1}{2}+ε})$空间算法，计算两个固定顶点之间的最短路径$G$。我们还考虑了{ \RB }问题，即给定一个图$G$，其边被着色为红色或蓝色，并且在$G$中有两个固定顶点$s$和$t$，是否存在从$s$到$G$中的$t$的路径，该路径在红色和蓝色边之间交替。平面DAG中的{ \RB }问题是{ \NL }完全的。我们展示了一个多项式时间和$O(n^{\frac{1}{2}+ε})$空间算法（对于任何$ε>0$）的{ \RB }问题的平面DAG。在本文的最后一部分中，我们考虑了平面二部图中存在的完美匹配的判定和构造问题，以及一个类似的问题，即在平面二部图中找到一个Hall障碍。证明了这两个问题的时空界与平面有向图中最短路问题的时空界相同。
英文摘要：
In this paper, we show that given a weighted, directed planar graph $G$, and any $ε>0$, there exists a polynomial time and $O(n^{\frac{1}{2}+ε})$ space algorithm that computes the shortest path between two fixed vertices in $G$. We also consider the {\RB} problem, which states that given a graph $G$ whose edges are colored either red or blue and two fixed vertices $s$ and $t$ in $G$, is there a path from $s$ to $t$ in $G$ that alternates between red and blue edges. The {\RB} problem in planar DAGs is {\NL}-complete. We exhibit a polynomial time and $O(n^{\frac{1}{2}+ε})$ space algorithm (for any $ε>0$) for the {\RB} problem in planar DAG. In the last part of this paper, we consider the problem of deciding and constructing the perfect matching present in a planar bipartite graph and also a similar problem which is to find a Hall-obstacle in a planar bipartite graph. We show the time-space bound of these two problems are same as the bound of shortest path problem in a directed planar graph.
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