Yao-图和$Θ$ -图在约束条件下的生成性质
Spanning Properties of Yao and $Θ$-Graphs in the Presence of Constraints
摘要：
给出了锥数至少为6的约束$θ$ -图和锥数至少为5或7的约束Yao-图的生成比的改进上界。给定平面中的一组点，Yao图将围绕每个顶点的平面划分为$m$不相交的圆锥，每个圆锥具有孔径$θ= 2 π/m$，并将边缘添加到每个圆锥中最近的顶点。约束Yao-图具有一个附加性质，即没有边与给定的线段约束正确相交。约束$θ$ -图类似于约束Yao-图，但使用不同的方法来确定最近的顶点。我们提出了一个大家庭的约束$θ$ -图的生成比的紧界。证明了具有$4k + 2$（$k \geq 1$和整数）锥的约束$θ$ -图具有紧生成比$1 + 2 \sin(θ/2)$，其中$θ$是$2 π/ (4k + 2)$ .我们还提出了改进的上界的其他家庭的约束$θ$ -图的生成比。这些边界与无约束设置中的当前上限匹配。我们还证明了具有偶数个锥的约束Yao-图（$m \geq 8$）的生成比至多为$1 / ( 1 - 2 \sin (θ/2) )$，具有奇数个锥的约束Yao-图（$m \geq 5$）的生成比至多为$1 / ( 1 - 2 \sin (3θ/8) )$ .与约束$θ$ -图的情况一样，这些边界与无约束设置中的当前上界匹配，这意味着像在无约束设置中一样，使用更多的锥可以使生成比更差。
英文摘要：
We present improved upper bounds on the spanning ratio of constrained $θ$-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ= 2 π/m$, and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained $θ$-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained $θ$-graphs. We show that constrained $θ$-graphs with $4k + 2$ ($k \geq 1$ and integer) cones have a tight spanning ratio of $1 + 2 \sin(θ/2)$, where $θ$ is $2 π/ (4k + 2)$. We also present improved upper bounds on the spanning ratio of the other families of constrained $θ$-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ($m \geq 8$) have spanning ratio at most $1 / ( 1 - 2 \sin (θ/2) )$ and constrained Yao-graphs with an odd number of cones ($m \geq 5$) have spanning ratio at most $1 / ( 1 - 2 \sin (3θ/8) )$. As is the case with constrained $θ$-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
