Narrow Results

In this paper, we review some new distributed algorithms that construct sparse subgraphs with bounded degree of the unit disk graph efficiently for wireless ad hoc networks. They maintain a linear number of links while still preserving power-efficient routes for any pair of nodes. It was open whether the Yao plus reverse Yao graph and the symmetric Yao graph are spanners. We show that the Yao plus reverse Yao graph has a bounded power stretch factor 2 in civilized unit disk graph. In addition, we review a recent example by M. Grünewald et al. [6] to show that the symmetric Yao graph does not have a constant bounded stretch factor. Finally, we conduct simulations to study the practical performances of these structures. All structures have small power stretch factors for randomly generated unit disk graphs in our experiments.

A t-spanner of a network is a subnetwork in which every two nodes that were connected by an edge in the original network are connected by a path of at most t edges in the subnetwork. This guarantees that if nodes u and v are at distance d(u,v) in the original network, they are at distance no more than t·d(u,v) in the t-spanner. We introduce a more general definition of spanners. A special case of this more general definition is the additive t-spanner: a subnetwork in which every two nodes u and v that were separated by distance d(u,v) in the original network are at distance no more than t+d(u,v) in the subnetwork. We construct low-degree additive t-spanners for hypercubes.

Given a set V of n points in a two-dimensional plane, we give an O(nlogn)-time centralized algorithm that constructs a planar t-spanner for V, for such that the degree of each node is bounded from above by , where 0<α<π/2 is an adjustable parameter. Here C_del is the spanning ratio of the Delaunay triangulation, which is at most . We also show, by applying the greedy method in Ref. [14], how to construct a low weighted bounded degree planar spanner with spanning ratio ρ(α)²(1+∊) and the same degree bound, where ∊ is any positive real constant. Here, a structure is called low weighted if its total edge length is proportional to the total edge length of the Euclidean minimum spanning tree of V. Moreover, we show that our method can be extended to construct a planar bounded degree spanner for unit disk graphs with the adjustable parameter α satisfying 0<α<π/3. Previously, only centralized method⁶ of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t≃10.02. The distributed implementation of this centralized method takes O(n²) communications in the worst case. Our method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n).

Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected subgraph G' of G with vertex set V whose maximum degree is at most 14 + ⌈2π/γ⌉. If G is the Delaunay triangulation of V, and γ = 2π/3, we show that G' is a t-spanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is a triangulation satisfying the diamond property, then for a specific range of values of γ dependent on the angle of the diamonds, we show that G' is a t-spanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on γ. If G is the graph consisting of all Delaunay edges of length at most 1, and γ = π/3, we show that a modified version of the algorithm produces a plane subgraph G' of the unit-disk graph which is a t-spanner (for some constant t) of the unit-disk graph of V, whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25.

We show that the Yao graph Y₄ in the L₂ metric is a spanner with stretch factor . Enroute to this, we also show that the Yao graph in the L_∞ metric is a plane spanner with stretch factor 8.

Given a set S of points in the plane, a geometric network for S is a graph G with vertex set S and straight edges. We consider a broadcasting situation, where one point r ∊ S is a designated source. Given a dilation factor δ, we ask for a geometric network G such that for every point v ∊ S there is a path from r to v in G of length at most δ|rv|, and such that the total edge length is minimized. We show that finding such a network of minimum total edge length is NP-hard, and give an approximation algorithm.

We look at the problem of coloring locally specially constructed spanners of unit disk graphs. First we present a local approximation algorithm for the vertex coloring problem in Unit Disk Graphs (UDGs) which uses at most four times as many colors as an optimal solution requires. Next we look at the colorability of spanners of UDGs. In particular we present a local algorithm for constructing a 4-colorable spanner of a unit disk graph. The output consists of the spanner and the 4-coloring. The computed spanner also has the properties that it is planar, the degree of a vertex in the spanner is at most 5 and the angles between two edges are at least π/3. By enlarging the locality distance (i.e. the size of the neighborhood which a vertex has to explore in order to compute its color) we can ensure the total weight of the spanner to be arbitrarily close to the weight of a minimum spanning tree.

We prove that a local algorithm cannot compute a bipartite spanner of a unit disk graph and therefore our algorithm needs at most one color more than any local algorithm for the task requires. Moreover, we prove that there is no local algorithm for 3-coloring UDGs or spanners of UDGs, even if the 3-colorability of the graph (or the spanner respectively) is guaranteed in advance.

Yao and Theta graphs are defined for a given point set and a fixed integer k > 0. The space around each point is divided into k cones of equal angle, and each point is connected to a nearest neighbor in each cone. The difference between Yao and Theta graphs is in the way the nearest neighbor is defined: Yao graphs minimize the Euclidean distance between a point and its neighbor, and Theta graphs minimize the Euclidean distance between a point and the orthogonal projection of its neighbor on the bisector of the hosting cone. We prove that, corresponding to each edge of the Theta graph Θ₆, there is a path in the Yao graph Y₆ whose length is at most 8.82 times the edge length. Combined with the result of Bonichon et al., who prove an upper bound of 2 on the stretch factor of Θ₆, we obtain an upper bound of 17.64 on the stretch factor of Y₆.

Let S be a set of points in the plane, such that the unit disk graph with vertex set S is connected. We address the problem of finding orientations and a minimum radius for directional antennas of a fixed cone angle placed at the points of S, such that the induced communication graph G[S] is a hop t-spanner of the unit disk graph for S (meaning that G[S] is strongly connected, and contains a directed path with at most t edges between any pair of points within unit distance). We consider problem instances in which antenna angles are bounded below by 120° and 90°. We show that, in the case of 120° angles, a radius of 5 suffices to establish a hop 4-spanner; and in the case of 90° angles, a radius of 7 suffices to establish a hop 5-spanner.