Sensor networks often involve the monitoring of mobile phenomena, which can be facilitated by a spatiotemporal multicast protocol we call "mobicast". Mobicast is a novel spatiotemporal multicast protocol featuring a delivery zone that evolves over time. Mobicast can in theory provide absolute spatiotemporal delivery guarantees by limiting communication to a mobile forwarding zone whose size is determined by the global worst-case value associated with a compactness metric defined over the geometry of the network.In this work, we first studied the compactness properties of sensor networks with uniform distribution. The results of this study motivate three approaches for improving the e#ciency of spatiotemporal multicast in such networks. First, one may achieve high savings on communication overhead by slightly relaxing spatiotemporal delivery guarantees. Second, spatiotemporal multicast may exploit local compactness values for higher e#ciency for networks with non uniform spatial distribution of compactness. Third, for random uniformly distributed sensor network deployment, one may choose a deployment density to best support spatiotemporal communication. We also explored all these directions via mobicast simulation and results are presented in this paper.

In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general real-weighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs. 1

Given a geometric t-spanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+")- approximate shortest path queries in O(1) time. The data structure uses O(n log n) space.

For a set S of points in R d,ans-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers. 1

We study the emerging phenomenon of ad hoc, sensor-based communication networks. The communication is modeled by the random geometric graph model G(n, r, ℓ) where n points randomly placed within [0, ℓ] d form the nodes, and any two nodes that correspond to points at most distance r away from each other are connected. We study fundamental properties of G(n, r, ℓ) of interest: connectivity, coverage, and routing-stretch. Our main contribution is a simple analysis technique we call bin-covering that we apply uniformly to get (asymptotically) tight thresholds for each of these properties. Typically, in the past, random geometric graph analyses involved sophisticated methods from continuum percolation theory; on contrast, our bin-covering approach is discrete and very simple, yet it gives us tight threshold bounds. The technique also yields algorithmic benefits as illustrated by a simple local routing algorithm for finding paths with low stretch. Our specific results should also prove interesting to the networking community that has seen a recent increase in the study of random geometric graphs motivated by engineering ad hoc networks.

Let G = (V, E) be an embedded connected graph with n vertices and m edges. Specifically, the vertex set V consists of points in R 2, and E consists

Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that #/2 1.570 ... is sometimes necessary in order to accommodate a finite set of points.

We address the problem of designing compact routing tables for an unlabeled connected n-node planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.

Let P be a simple polygonal chain in E with n edges. The detour of P between two points, x and y, is the ratio between the length of P between x any y and their Euclidean distance. The detour

Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting in a trivial O(mn 3 + n 4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n 4) time using O(n 2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2 + ε)-approximation algorithm with running time O(nm + n 2 (log n +1/ε 3d)) using O(n 2)space.