Memoryless Routing in Convex Subdivisions: Random Walks are Optimal

A memoryless routing algorithm is one in which the decision about the next edge on the route to a vertex t for a packet currently located at vertex v is made based only on the coordinates of v, t, and the neighbourhood, N(v), of v. The current paper explores the limitations of such algorithms by showing that, for any (randomized) memoryless routing algorithm A, there exists a convex subdivision on which A takes Omega(n^2) expected time to route a message between some pair of vertices. Since this lower bound is matched by a random walk, this result implies that the geometric information available in convex subdivisions is not helpful for this class of routing algorithms. The current paper also shows the existence of triangulations for which the Random-Compass algorithm proposed by Bose etal (2002,2004) requires 2^{\Omega(n)} time to route between some pair of vertices.Comment: 11 pages, 6 figure

The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered $\theta$-graphs. A $\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\theta = 2 \pi/m$. An ordered $\theta$-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 $\theta$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(\theta/2) / (\cos(\theta/2) - \sin(\theta/2))$. We also show that for any integer $k \geq 2$, ordered $\theta$-graphs with $4k + 2$ cones have a tight spanning ratio of $1 / (1 - 2 \sin(\theta/2))$. We provide lower bounds for ordered $\theta$-graphs with $4k + 3$ and $4k + 5$ cones. For ordered $\theta$-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 $\theta$-graphs have worse spanning ratios than unordered $\theta$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $\theta$-graphs with 4, 5, and 6 cones are not spanners

Optimal area triangulation

Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in $O(n^2log{n})$ time and $O(n^2)$ space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area

Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which is closer to $x(t)$ than $x(s)$. This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph $G = (V,E)$, an embedding x: V \to \bbbr^2 of $G$ is a planar convex greedy embedding if and only if, in the embedding $x$, weight of the maximum weight spanning tree ($T$) and weight of the minimum weight spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq (\card{V}-1)^{1 - \delta}, for some $0 < \delta \leq 1$.Comment: 19 pages, A short version of this paper has been accepted for presentation in FCT 2009 - 17th International Symposium on Fundamentals of Computation Theor

Building Robust Distributed Infrastructure Networks

Many competing designs for Distributed Hash Tables exist exploring multiple models of addressing, routing and network maintenance. Designing a general theoretical model and implementation of a Distributed Hash Table allows exploration of the possible properties of Distributed Hash Tables. We will propose a generalized model of DHT behavior, centered on utilizing Delaunay triangulation in a given metric space to maintain the networks topology. We will show that utilizing this model we can produce network topologies that approximate existing DHT methods and provide a starting point for further exploration. We will use our generalized model of DHT construction to design and implement more efficient Distributed Hash Table protocols, and discuss the qualities of potential successors to existing DHT technologies

On a family of strong geometric spanners that admit local routing strategies

We introduce a family of directed geometric graphs, denoted \paz, that depend on two parameters $\lambda$ and $\theta$. For $0\leq \theta<\frac{\pi}{2}$ and ${1/2} < \lambda < 1$, the \paz graph is a strong $t$-spanner, with $t=\frac{1}{(1-\lambda)\cos\theta}$. The out-degree of a node in the \paz graph is at most $\lfloor2\pi/\min(\theta, \arccos\frac{1}{2\lambda})\rfloor$. Moreover, we show that routing can be achieved locally on \paz. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $\lambda$ and $\theta$ indicate that for random point sets, the spanning ratio of \paz is better than the proven theoretical bounds