In this paper, we propose a new dynamic and efficient bounding volume hierarchy for breakable objects undergoing structured and/or unstructured motion. Our object–space method is based on different ways to incrementally update the hierarchy during simulation by exploiting temporal coherence and lazy evaluation techniques. This leads to significant advantages in terms of execution speed. Furthermore, we also show how our method lends itself naturally for an adaptive low memory cost implementation, which may be of critical importance in some applications. Finally, we propose two different techniques for detecting self-intersections, one using our hierarchical data structure, and the other is an improved sorting-based method.

We investigate the problem of optimizing the routing performances of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1 − O(1/n). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node u and any radius r the number of nodes within distance 2r from u is at most a constant times the number of nodes within distance r, our augmentation process proceeds with high probability in O(log n log D) communication rounds, with O(log n log D) messages of size O(log n) bits sent per node and requiring only O(log n log D) bit space in each node, where n is the number of nodes, and D the diameter. In particular, with the only knowledge of original distances, greedy routing computes,

We present an efficient distributed data structure, called the D-SPANNER, for maintaining proximity information among communicating mobile nodes. The D-SPANNER is a kinetic sparse graph spanner on the nodes that allows each node to quickly determine which other nodes are within a given distance of itself, to estimate an approximate nearest neighbor, and to perform a variety of other proximity related tasks. A lightweight and fully distributed implementation is possible, in that maintenance of the proximity information only requires each node to exchange a modest number of messages with a small number of mostly neighboring nodes. The structure is based on distance information between communicating nodes that can be derived using ranging or localization methods and requires no additional shared infrastructure other than an underlying communication network. Its modest requirements make it scalable to networks with large numbers of nodes.

For a set S of points in R d, a t-spanner is a sparse graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the Euclidean distance between the points. In this paper, we show how to construct a (1 + ε)-spanner with O(n/ε d) edges and maximum degree O(1/ε d) in time O(n log n). A spanner with similar properties was previously presented in [6, 8]. However, using our new construction (coupled with several other innovations) we obtain new results for two fundamental problems for constant doubling dimension metrics: The first result is an essentially optimal compact routing scheme. In particular, we show how to perform routing with a stretch of 1 + ɛ, where the label size is ⌈log n ⌉ and the size of the table stored at each point is only O(log n/ε d). This routing problem was first considered by Peleg and Hassin [11], who presented a routing scheme in the plane. Later, Chan et al. [6] and Abraham et al. [1] considered this problem for doubling dimension metric spaces. Abraham et al. [1] were the first to present a (1 + ɛ) routing scheme where the label size depends solely on the number of points. In their scheme labels are of size of ⌈log n⌉, and each point stores a table of size O(log 2 n/ε d). In our routing scheme, we achieve routing tables of size O(log n/ε d), which is essentially the same size as a label (up to the factor of 1/ε d). The second and main result of this paper is the first fully dynamic geometric spanner with polylogarithmic update time for both insertions and deletions. We present an algorithm that allows points to be inserted into and deleted from S with an amortized update time of O(log 3 n).

Abstract — In a sensor network information from multiple nodes must usually be aggregated in order to accomplish a certain task. A natural way to view this information gathering is in terms of interactions between nodes that are producers of information, e.g., those that have collected data, detected events, etc., and nodes that are consumers of information, i.e., nodes that seek data of certain types. Our overall goal in this paper is to construct efficient schemes allowing consumer and producer nodes to discover each other so that the desired information can be sent quickly to those who seek it. Here, efficiency is an issue for both the producers (limiting the redundancy of where information is stored) as well as the consumers (keeping the query time low). We introduce the notion of distance-sensitive information brokerage and provide schemes for efficiently bringing together information producers and consumers at a cost proportional to the separation between them—even though the consumers do not know the locations of the producers they seek. Our algorithms rely purely on the communication topology of the sensor network and do not require any geographic location information. In the process we introduce a new routing scheme that is of interest in its own right because it provides constant-factor approximations to the optimal paths. We give theoretical proofs of the efficiency of our scheme, as well as experimental results that further demonstrate its performance and suggest its practicality even for large scale sensor networks. I.

We present a new shape-matching deformation model that allows for efficient handling of topological changes and dynamic adaptive selection of levels of detail. Similar to the recently presented Fast Lattice Shape Matching (FLSM), we compute the position of simulation nodes by convolution of rigid shape matching operators on many overlapping regions, but we rely instead on octree-based hierarchical sampling and an interval-based region definition. Our approach enjoys the efficiency and robustness of shape-matching deformation models, and the same algorithmic simplicity and linear cost as FLSM, but it eliminates its dense sampling requirements. Our method can handle adaptive spatial discretizations, allowing the simulation of more degrees of freedom in arbitrary regions of interest at little additional cost. The method is also versatile, as it can simulate elastic and plastic deformation, it can handle cuts interactively, and it reuses the underlying data structures for efficient handling of (self-)collisions. All this makes it especially useful for interactive applications such as videogames.

Two complications frequently arise in real-world applications, motion and the contamination of data by outliers. We consider a fundamental clustering problem, the k-center problem, within the context of these two issues. We are given a finite point set S of size n and an integer k. In the standard k-center problem, the objective is to compute a set of k center points to minimize the maximum distance from any point of S to its closest center, or equivalently, the smallest radius such that S can be covered by k disks of this radius. In the discrete k-center problem the disk centers are drawn from the points of S, and in the absolute k-center problem the disk centers are unrestricted. We generalize this problem in two ways. First, we assume that points are in continuous motion, and the objective is to maintain a solution over time. Second, we assume that some given robustness parameter 0 < t ≤ 1 is given, and the objective is to compute the smallest radius such that there exist k disks of this radius that cover at least ⌈tn ⌉ points of S. We present a kinetic data structure (in the KDS framework) that maintains a (3 + ε)-approximation for the robust discrete k-center problem and a (4 + ε)-approximation for the robust absolute k-center problem, both under the assumption that k is a constant. We also improve on a previous 8-approximation for the non-robust discrete kinetic k-center problem, for arbitrary k, and show that our data structure achieves a (4 + ε)-approximation. All these results hold in any metric space of constant doubling dimension, which includes Euclidean space of constant dimension.

We describe the 3-D structure of a protein using geometric spanners — geometric graphs with a sparse set of edges where paths approximate the n 2 inter-atom distances. The edges in the spanner pick out important proximities in the structure, labeling a small number of atom pairs or backbone region pairs as being of primary interest. Such compact multiresolution views of proximities in the protein can be quite valuable, allowing, for example, easy visualization of the conformation over the entire folding trajectory of a protein and segmentation of the trajectory. These visualizations allow one to easily detect formation of secondary and tertiary structures as the protein folds. 1

By exploiting basic geometry concepts, we present a lightweight, distance-sensitive, and tunable querying service, Glance, for dense wireless sensor networks. Glance ensures that a “query” operation invoked within d hops of an event intercepts the event’s “publish” operation within d ∗ s hops, where s is a “stretch-factor” tunable by the user. A significant feature of our service is that it can be implemented easily without any localization information.

We show how to greatly accelerate self-collision detection (SCD) for reduced deformable models. Given a triangle mesh and a set of deformation modes, our method precomputes Subspace Self-Collision Culling (SSCC) certificates which, if satisfied, prove the absence of self-collisions for large parts of the model. At runtime, bounding volume hierarchies augmented with our certificates can aggressively cull overlap tests and reduce hierarchy updates. Our method supports both discrete and continuous SCD, can handle complex geometry, and makes no assumptions about geometric smoothness or normal bounds. It is particularly effective for simulations with modest subspace deformations, where it can often verify the absence of self-collisions in constant time. Our certificates enable low amortized costs, in time and across many objects in multibody dynamics simulations. Finally, SSCC is effective enough to support self-collision tests at audio rates, which we demonstrate by producing the first sound simulations of clattering objects.