In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points $S\subset \mathbb{R}^2$ and an angle $0 < θ\leq 2π$, we define the continuous Yao graph $cY(θ)$ with vertex set $S$ and angle $θ$ as follows. For each $p,q\in S$, we add an edge from $p$ to $q$ in $cY(θ)$ if there exists a cone with apex $p$ and aperture $θ$ such that $q$ is the closest point to $p$ inside this cone.
We study the spanning ratio of $cY(θ)$ for different values of $θ$. Using a new algebraic technique, we show that $cY(θ)$ is a spanner when $θ\leq 2π/3$. We believe that this technique may be of independent interest. We also show that $cY(π)$ is not a spanner, and that $cY(θ)$ may be disconnected for $θ> π$.
It is a long standing open problem whether Yao-Yao graphs $\mathsf{YY}_{k}$ are all spanners [li2002sparse]. Bauer and Damian [bauer2013infinite] showed that all $\mathsf{YY}_{6k}$ for $k \geq 6$ are spanners. Li and Zhan [li2016almost] generalized their result and proved that all even Yao-Yao graphs $\mathsf{YY}_{2k}$ are spanners (for $k\geq 42$). However, their technique cannot be extended to odd Yao-Yao graphs, and whether they are spanners are still elusive. In this paper, we show that, surprisingly, for any integer $k \geq 1$, there exist odd Yao-Yao graph $\mathsf{YY}_{2k+1}$ instances,…
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 v…
For a set of points in the plane and a fixed integer $k > 0$, the Yao graph $Y_k$ partitions the space around each point into $k$ equiangular cones of angle $θ=2π/k$, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of $Y_5$, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd $k \geq 5$, the spanning ratio of $Y_k$ is at most $1/(1-2\sin(3θ/8))$, which gives the first constant upper bound for $Y_5$, and is an improvement over the previous bound of $1/(1-2\sin(θ/2))$ for odd $k \g…
Yao graphs are geometric spanners that connect each point of a given point set to its nearest neighbor in each of $k$ cones drawn around it. Yao graphs were introduced to construct minimum spanning trees in $d$ dimensional spaces. Moreover, they are used for instance in topology control in wireless networks. An optimal \Onlogn time algorithm to construct Yao graphs for given point set has been proposed in the literature but -- to the best of our knowledge -- never been implemented. Instead, algorithms with a quadratic complexity are used in popular packages to construct these graphs. In this p…
We show that, unlike the Yao-Yao graph $YY_6$, the Theta-Theta graph $ΘΘ_6$ defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of $ΘΘ_6$ is unbounded.
Ordered Yao graphs: maximum degree, edge density, and clique numbers
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Charles University（查理大学）；Algoresearch L.L.C.（Algoresearch有限责任公司）；Research Institute of the University of Bucharest（布加勒斯特大学研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Indraprastha Institute of Information Technology, Delhi（德里因德拉普拉斯塔信息技术学院）；University of California San Diego（加州大学圣地亚哥分校）
We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a natural measure of the compactness of the graph, and has been intensively studied, together with several variants, including its version for weighted graphs. The continuous analog of the (discrete) mean distance is the mean of the distances between all pairs of points on the edges of the graph. Despite being a very natural generalization, to the best of our kno…
Algorithms for Distance Problems in Continuous Graphs
University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Seville（塞维利亚大学）；Technical University of Dortmund（多特蒙德工业大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
研究连续图直径和平均距离计算问题，利用几何技术为两类稀疏图设计次二次时间算法，树宽至多为\(k\)的连续图计算时间为\(O(n\log^{O(k)} n)\)，\(n\)个顶点\(F\)个面的连续平面图计算时间为\(O(n F \log n)\)。
