New Bounds for Facial Nonrepetitive Colouring
We prove that the facial nonrepetitive chromatic number of any outerplanar graph is at most 11 and of any planar graph is at most 22.
math.CO↗
arXiv subjects
Prosenjit Bose
Publications and source records attributed to Prosenjit Bose.
Flipping Edge-Labelled Triangulations
Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge $e$, which is the set of all edges reachable from $e$ via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight $Θ(n \log n)$ bounds for edge-labelled triangulations of $n$-vertex convex polygons and combinatorial triangulations, contrasting with the $Θ(n)$ bounds in their respective unlabelled settings. The $Ω(n \log n)$ lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to $O(\log^2 n)$, although we no longer have a matching lower bound. Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of $O(n^2)$ on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter $Ω(n^3)$.
cs.CG↗
The Price of Order
We present tight bounds on the spanning ratio of a large family of ordered $θ$-graphs. A $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ= 2 π/m$. An ordered $θ$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $k \geq 1$, ordered $θ$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(θ/2) / (\cos(θ/2) - \sin(θ/2))$. We also show that for any integer $k \geq 2$, ordered $θ$-graphs with $4k + 2$ cones have a tight spanning ratio of $1 / (1 - 2 \sin(θ/2))$. We provide lower bounds for ordered $θ$-graphs with $4k + 3$ and $4k + 5$ cones. For ordered $θ$-graphs with $4k + 2$ and $4k + 5$ cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered $θ$-graphs have worse spanning ratios than unordered $θ$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $θ$-graphs with 4, 5, and 6 cones are not spanners.
cs.CG↗
Plane Bichromatic Trees of Low Degree
Let $R$ and $B$ be two disjoint sets of points in the plane such that $|B|\leqslant |R|$, and no three points of $R\cup B$ are collinear. We show that the geometric complete bipartite graph $K(R,B)$ contains a non-crossing spanning tree whose maximum degree is at most $\max\left\{3, \left\lceil \frac{|R|-1}{|B|}\right\rceil + 1\right\}$; this is the best possible upper bound on the maximum degree. This solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996.
cs.CG↗
Flips in Edge-Labelled Pseudo-Triangulations
We show that $O(n^2)$ exchanging flips suffice to transform any edge-labelled pointed pseudo-triangulation into any other with the same set of labels. By using insertion, deletion and exchanging flips, we can transform any edge-labelled pseudo-triangulation into any other with $O(n \log c + h \log h)$ flips, where $c$ is the number of convex layers and $h$ is the number of points on the convex hull.
cs.CG↗
The Shadows of a Cycle Cannot All Be Paths
A "shadow" of a subset $S$ of Euclidean space is an orthogonal projection of $S$ into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$-sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).
cs.CG↗
Improved Spanning Ratio for Low Degree Plane Spanners
We describe an algorithm that builds a plane spanner with a maximum degree of 8 and a spanning ratio of approximately 4.414 with respect to the complete graph. This is the best currently known spanning ratio for a plane spanner with a maximum degree of less than 14.
cs.CG↗
Packing Plane Perfect Matchings into a Point Set
Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem.
cs.CG↗
Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations
Consider a weighted graph G where vertices are points in the plane and edges are line segments. The weight of each edge is the Euclidean distance between its two endpoints. A routing algorithm on G has a competitive ratio of c if the length of the path produced by the algorithm from any vertex s to any vertex t is at most c times the length of the shortest path from s to t in G. If the length of the path is at most c times the Euclidean distance from s to t, we say that the routing algorithm on G has a routing ratio of c.We present an online routing algorithm on the Delaunay triangulation with competitive and routing ratios of 5.90. This improves upon the best known algorithm that has competitive and routing ratio 15.48. The algorithm is a generalization of the deterministic 1-local routing algorithm by Chew on the L1-Delaunay triangulation. When a message follows the routing path produced by our algorithm, its header need only contain the coordinates of s and t. This is an improvement over the currently known competitive routing algorithms on the Delaunay triangulation, for which the header of a message must additionally contain partial sums of distances along the routing path.We also show that the routing ratio of any deterministic k-local algorithm is at least 1.70 for the Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case of the L1-Delaunay triangulation, this implies that even though there exists a path between two points x and y whose length is at most 2.61|[xy]| (where |[xy]| denotes the length of the line segment [xy]), it is not always possible to route a message along a path of length less than 2.70|[xy]|. From these bounds on the routing ratio, we derive lower bounds on the competitive ratio of 1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations.
cs.CG↗
A linear-time algorithm for the geodesic center of a simple polygon
Given two points in a simple polygon $P$ of $n$ vertices, its geodesic distance is the length of the shortest path that connects them among all paths that stay within $P$. The geodesic center of $P$ is the unique point in $P$ that minimizes the largest geodesic distance to all other points of $P$. In 1989, Pollack, Sharir and Rote [Disc. \& Comput. Geom. 89] showed an $O(n\log n)$-time algorithm that computes the geodesic center of $P$. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.
cs.CG↗
Optimal Data Structures for Farthest-Point Queries in Cactus Networks
Consider the continuum of points on the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network distance. Within this metric space, we study farthest points and farthest distances. We introduce optimal data structures supporting queries for the farthest distance and the farthest points on trees, cycles, uni-cyclic networks, and cactus networks.
cs.DS↗
Towards a General Framework for Searching on a Line and Searching on $m$ Rays
Consider the following classical search problem: given a target point $p\in \Re$, starting at the origin, find $p$ with minimum cost, where cost is defined as the distance travelled. Let $D$ be the distance of $p$ from the origin. When no lower bound on $D$ is given, no competitive search strategy exists. Demaine, Fekete and Gal (Online searching with turn cost, Theor. Comput. Sci., 361(2-3):342-355, 2006) considered the situation where no lower bound on $D$ is given but a fixed \emph{turn cost} $t>0$ is charged every time the searcher changes direction. When the total cost is expressed as $c D+ϕ$, where $c$ and $ϕ$ are positive constants, they showed that if $c$ is set to $9$, then the optimal search strategy has a cost of $9D+2t$. Although their strategy is optimal for $c=9$, we prove that the minimum cost in their framework is $5D+t+2\sqrt{2D(2D+t)} < 9D+2t$. Note that the minimum cost requires knowledge of $D$. However, given $D$, the optimal strategy has a smaller cost of $3D+t$. Therefore, this problem cannot be solved optimally and exactly when no lower bound on $D$ is given. To resolve this issue, we introduce a general framework where the cost of moving distance $x$ away from the origin is $α_1 x+β_1$ and the cost of moving distance $y$ towards the origin is $α_2 y+β_2$ for constants $α_1,α_2,β_1,β_2$. Given a lower bound $λ$ on $D$, we provide a provably optimal competitive search strategy when $α_1,α_2,β_1,β_2 \geq 0$ and $α_1+α_2 > 0$. Finally, we address the problem of searching for a target lying on one of $m$ rays extending from the origin where the cost is measured as the total distance travelled plus $t \geq 0$ times the number of turns. We provide a search strategy and compute its cost. We prove our strategy is optimal for small values of $t$ and conjecture it is always optimal.
cs.DS↗
Optimal local routing on Delaunay triangulations defined by empty equilateral triangles
We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$θ_6$-graph (the half-$θ_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$θ_6$-graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$θ_6$-graph using $O(\log n)$ bits per vertex coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$θ_6$-graph.
cs.CG↗
Continuous Yao Graphs
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 $θ> π$.
cs.CG↗
The $θ_5$-graph is a spanner
Given a set of points in the plane, we show that the $θ$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most $\sqrt{50 + 22 \sqrt{5}}$ times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of $\frac{1}{2}(11\sqrt{5} -17) \approx 3.798$.
cs.CG↗
Theta-3 is connected
In this paper, we show that the $θ$-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.
cs.CG↗
Towards Tight Bounds on Theta-Graphs
We present improved upper and lower bounds on the spanning ratio of $θ$-graphs with at least six cones. Given a set of points in the plane, a $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ=2π/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \geq 1$, $θ$-graphs with $4k+2$ cones have a spanning ratio of $1+2\sin(θ/2)$ and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer $k \geq 1$, $θ$-graphs with $4k+4$ cones have spanning ratio at most $1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2))$. We also show that $θ$-graphs with $4k+3$ and $4k+5$ cones have spanning ratio at most $\cos(θ/4)/(\cos(θ/2)-\sin(3θ/4))$. This is a significant improvement on all families of $θ$-graphs for which exact bounds are not known. For example, the spanning ratio of the $θ$-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the $θ$-routing algorithm. In particular, we show that the $θ$-routing algorithm is $(1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2)))$-competitive on $θ$-graphs with $4k+4$ cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of $θ$-graphs. In particular, we show that $θ$-graphs with $4k+4$ cones have spanning ratio at least $1+2\tan(θ/2)+2\tan^2(θ/2)$. This is somewhat surprising since, for equal values of $k$, the spanning ratio of $θ$-graphs with $4k+4$ cones is greater than that of $θ$-graphs with $4k+2$ cones, showing that increasing the number of cones can make the spanning ratio worse.
cs.CG↗
Computing Covers of Plane Forests
Let $ϕ$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-empty subset $ϕ(A)$ of $\mathbb{R}^2$. A $ϕ$-cover of a set $T=\{T_1, T_2, \dots, T_m\}$ of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that each tree $T_i$ is contained in some region of the cover, and each region of the cover is either (1) $ϕ(T_i)$ for some $i$, or (2) $ϕ(A \cup B)$, where $A$ and $B$ are constructed by either (1) or (2), and $A \cap B \neq \emptyset$. We present two properties for the function $ϕ$ that make the $ϕ$-cover well-defined. Examples for such functions $ϕ$ are the convex hull and the axis-aligned bounding box. For both of these functions $ϕ$, we show that the $ϕ$-cover can be computed in $O(n\log^2n)$ time, where $n$ is the total number of vertices of the trees in $T$.
cs.CG↗