Peter Brass

Mobility improves coverage of sensor networks

Previous work on the coverage of mobile sensor networks focuses on algorithms to reposition sensors in order to achieve a static configuration with an enlarged covered area. In this paper, we study the dynamic aspects of the coverage of a mobile sensor network that depend on the process of sensor movement. As time goes by, a position is more likely to be covered; targets that might never be detected in a stationary sensor network can now be detected by moving sensors. We characterize the area coverage at specific time instants and during time intervals, as well as the time it takes to detect a randomly located stationary target. Our results show that sensor mobility can be exploited to compensate for the lack of sensors and improve network coverage. For mobile targets, we take a game theoretic approach and derive optimal mobility strategies for sensors and targets from their own perspectives.

Bounds on coverage and target detection capabilities for models of networks of mobile sensors

In this article we analyze the capabilities of various models of sensor networks with the Boolean sensing model for mobile or stationary sensors and targets, under random or optimal placement, independent or globally coordinated search, and stealthy or visible sensors. For each model we give an upper bound for the capabilities under any strategy, and a search strategy which at least asymptotically matches that bound. To ensure comparability of these models, we present them using the same parameters: the sensing radius r, sensor placement density λ, as well as the travel distance l of each sensor and d of the target. By this we obtain a complete analysis of the geometric coverage and detection capabilities of the various models of sensor networks, where we abstract from issues like communication and power management.

Pseudotriangulations from Surfaces and a Novel Type of Edge Flip

We prove that planar pseudotriangulations have realizations as polyhedral surfaces in three-space. Two main implications are presented. The spatial embedding leads to a novel flip operation that allows for a drastic reduction of flip distances, especially between (full) triangulations. Moreover, several key results for triangulations, like flipping to optimality, (constrained) Delaunayhood, and a convex polytope representation, are extended to pseudotriangulations in a natural way.

On counting point-hyperplane incidences

In this paper we discuss three closely related problems on the incidence structure between n points and m hyperplanes in d-dimensional space: the maximal number of incidences if there are no big bipartite subconfigurations, a compressed representation for the incidence structure, and a lower bound for any algorithm that determines the number of incidences (counting version of Hopcrofts problem). For this we give a construction of a special point-hyperplane configuration, giving a lower bound, which almost meets the best upper bound known thus far.

Erdos distance problems in normed spaces

Abstract We study the problems of the maximum numbers of unit distances, largest distances and smallest distances among n points in a two-dimensional normed space. We determine the exact maximum numbers of smallest and largest distances for each normed space, the maximum number of unit distances for each normed space in which the unit sphere is not strictly convex, and show that the best known upper bound for the euclidean case applies also for each normed space with strictly convex unit sphere, thereby partially answering a question of Erdos and Ulam. The results on smallest distances give also the exact maximum number of touching pairs among n translates of a convex set in the plane, thereby generalizing the results on the translative kissing number by Hadwiger and Grunbaum.

On Simultaneous Planar Graph Embeddings

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. In particular, given a mapping, we show how to embed two paths on an n ×n grid, and two caterpillar graphs on a 3n ×3n grid. We show that it is not always possible to simultaneously embed three paths. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) ×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n 2) ×O(n 2) grid.

TESTING THE CONGRUENCE OF d-DIMENSIONAL POINT SETS

This paper presents an algorithm that tests the congruence of two sets of n points in d-dimensional space in time. This improves the previous best algorithm for dimensions d≥6.

On the maximum number of edges in a C 4 -free subgraph of Q n

For the maximum number f(n) of edges in a C4-free subgraph of the n-dimensional cube-graph Qn we prove f(n) ≥ 1/2(n +√n)2n−1 for n = 4r, and f(n) ≥ 1/2(n +0.9√n)2n−1 for all n ≥ 9. This disproves one version of a conjecture of P. Erdos.

Multi-robot tree and graph exploration

In this paper we present an algorithm for the exploration of an unknown graph with k robots, which is guaranteed to succeed on any graph, and which on trees we prove to be near-optimal for two robots, having optimal dependence on the size of the tree but not on its radius. We believe that the algorithm performs well on any graph, and this is substantiated by simulations. For trees with n edges and radius r, the exploration time is equation, improving a recent method with equation [1], and almost reaching the lower bound equation. The algorithm is meant to be used in indoor navigation or cave search scenarios where the environment can be modeled as a graph. In this scenario, communication is realized by the devices being dropped by the robots at explored vertices, and the states of which are read and changed by further visiting robots. Simulations on Player/Stage platform have been performed in both tree and graph exploration which corroborate the mathematical results.

Constructing optimal highways

For two points p and q in the plane, a (unbounded) line h, called a highway, and a real v > 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed v on h and with speed 1 in the underlying metric elsewhere. Given a set S of n points in the plane and a high-way speed v, we consider the problem of finding an axis-parallel line, the highway, that minimizes the maximum travel time over all pairs of points in S. We achieve a linear-time algorithm both for the L1- and the Euclidean metric as the underlying metric. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.