Competitive online routing in geometric graphs

AbstractWe consider online routing algorithms for finding paths between the vertices of plane graphs. Although it has been shown in Bose et al. (Internat. J. Comput. Geom. 12(4) (2002) 283) that there exists no competitive routing scheme that works on all triangulations, we show that there exists a simple online O(1)-memory c-competitive routing strategy that approximates the shortest path in triangulations possessing the diamond property, i.e., the total distance travelled by the algorithm to route a message between two vertices is at most a constant c times the shortest path. Our results imply a competitive routing strategy for certain classical triangulations such as the Delaunay, greedy, or minimum-weight triangulation, since they all possess the diamond property. We then generalize our results to show that the O(1)-memory c-competitive routing strategy works for all plane graphs possessing both the diamond property and the good convex polygon property

An approximative calculation of the fractal structure in self-similar tilings

Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the fractal dimension by using the distribution without huge computations. This method can be applied to self-similar tilings based on a stochastic process.Comment: 5 pages, 6 figure

The Stretch Factor of L1L_1- and L∞L_\infty-Delaunay Triangulations

In this paper we determine the stretch factor of the $L_1$-Delaunay and $L_\infty$-Delaunay triangulations, and we show that this stretch is $\sqrt{4+2\sqrt{2}} \approx 2.61$. Between any two points $x,y$ of such triangulations, we construct a path whose length is no more than $\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this bound is best possible. This definitively improves the 25-year old bound of $\sqrt{10}$ by Chew (SoCG '86). To the best of our knowledge, this is the first time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for any real $p\ge 1$, is determined exactly

Routing with Guaranteed Delivery on Virtual Coordinates

We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.Engineering and Applied Science

The world's colonization and trade routes formation as imitated by slime mould

The plasmodium of Physarum polycephalum is renowned for spanning sources of nutrients with networks of protoplasmic tubes. The networks transport nutrients and metabolites across the plasmodium's body. To imitate a hypothetical colonization of the world and the formation of major transportation routes we cut continents from agar plates arranged in Petri dishes or on the surface of a three-dimensional globe, represent positions of selected metropolitan areas with oat flakes and inoculate the plasmodium in one of the metropolitan areas. The plasmodium propagates towards the sources of nutrients, spans them with its network of protoplasmic tubes and even crosses bare substrate between the continents. From the laboratory experiments we derive weighted Physarum graphs, analyze their structure, compare them with the basic proximity graphs and generalized graphs derived from the Silk Road and the Asia Highway networks. © 2012 World Scientific Publishing Company

Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201

Routing in Histograms

Let $P$ be an $x$-monotone orthogonal polygon with $n$ vertices. We call $P$ a simple histogram if its upper boundary is a single edge; and a double histogram if it has a horizontal chord from the left boundary to the right boundary. Two points $p$ and $q$ in $P$ are co-visible if and only if the (axis-parallel) rectangle spanned by $p$ and $q$ completely lies in $P$. In the $r$-visibility graph $G(P)$ of $P$, we connect two vertices of $P$ with an edge if and only if they are co-visible. We consider routing with preprocessing in $G(P)$. We may preprocess $P$ to obtain a label and a routing table for each vertex of $P$. Then, we must be able to route a packet between any two vertices $s$ and $t$ of $P$, where each step may use only the label of the target node $t$, the routing table and neighborhood of the current node, and the packet header. We present a routing scheme for double histograms that sends any data packet along a path whose length is at most twice the (unweighted) shortest path distance between the endpoints. In our scheme, the labels, routing tables, and headers need $O(\log n)$ bits. For the case of simple histograms, we obtain a routing scheme with optimal routing paths, $O(\log n)$-bit labels, one-bit routing tables, and no headers.Comment: 18 pages, 11 figure

Efficiently Navigating a Random Delaunay Triangulation

International audiencePlanar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. Whilst many algorithms have been proposed, very little theoretical analysis is available for the properties of the paths generated or the computational resources required to generate them. In this work, we propose and analyse a new planar navigation algorithm for the Delaunay triangulation. We then demonstrate a number of strong theoretical guarantees for the algorithm when it is applied to a random set of points in a convex region