Stefan Funke

Topological hole detection in wireless sensor networks and its applications

The identification of holes in a wireless sensor network is of primary interest since the breakdown of sensor nodes in a larger area often indicates one of the special events to be monitored by the network in the first place (e.g. outbreak of a fire, destruction by an earthquakes etc.). This task of identifying holes is especially challenging since typical wireless sensor networks consist of lightweight, low-capability nodes that are unaware of their geographic location.But there is also a secondary interest in detecting holes in a network: recently routing schemes have been proposed that do not assume knowledge of the geographic location of the network nodes but rather perform routing decisions based on the topology of the communication graph. Holes are salient features of the topology of a communication graph.In the first part of this paper we propose a simple distributed procedure to identify nodes near the boundary of the sensor field as well as near hole boundaries. Our hole detection algorithm is based purely on the topology of the communication graph, i.e. the only information available is which nodes can communicate with each other. In the second part of this paper we illustrate the secondary interest of our hole detection procedure using several examples.

Hole detection or: "how much geometry hides in connectivity?"

Wireless sensor networks typically consist of small, very simple network nodes without any positioning device like GPS. After an initialization phase, the nodes know with whom they can talk directly, but have no idea about their relative geographic locations. We examine how much geometry information is nevertheless hidden in the communication graph of the network: Assuming that the connectivity is determined by the well-known unit-disk graph model, we prove that using an extremely simple linear-time algorithm one can identify nodes on the boundaries of holes of the network. That is, there is enough geometry information hidden in the connectivity structure to identify topological features—in our example the holes in the network. While the theoretical analysis turns out to be quite conservative,an actual implementation shows that the algorithm works well under less stringent conditions.

A Separation Bound for Real Algebraic Expressions

Abstract Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda::real.

Improved approximation algorithms for connected sensor cover

Wireless sensor networks have recently posed many new system building challenges. One of the main problems is energy conservation since most of the sensors are devices with limited battery life and it is infeasible to replenish energy via replacing batteries. An effective approach for energy conservation is scheduling sleep intervals for some sensors, while the remaining sensors stay active providing continuous service. In this paper we consider the problem of selecting a set of active sensors of minimum cardinality so that sensing coverage and network connectivity are maintained. We show that the greedy algorithm that provides complete coverage has an approximation factor no better than Ω(log n), where n is the number of sensor nodes. Then we present algorithms that provide approximate coverage while the number of nodes selected is a constant factor far from the optimal solution. Finally, we show how to connect a set of sensors that already provides coverage.

Guaranteed-Delivery Geographic Routing Under Uncertain Node Locations

Geographic routing protocols like GOAFR or GPSR rely on exact location information at the nodes, as when the greedy routing phase gets stuck at a local minimum, they require, as a fallback, a planar subgraph whose identification, in all existing methods, depends on exact node positions. In practice, however, location information at the network nodes is hardly precise; be it because the employed location hardware, such as GPS, exhibits an inherent measurement imprecision, or because the localization protocols which estimate positions of the network nodes cannot do so without errors. In this paper we propose a novel naming and routing scheme that can handle the uncertainty in location information. It is based on a macroscopic variant of geographic greedy routing, as well as a macroscopic planarization of the communication graph. If an upper bound on the deviation from true node locations is available, our routing protocol guarantees delivery of messages. Due to its macroscopic view, our routing scheme also produces shorter and more load-balanced paths than common geographic routing schemes, in particular in sparsely connected networks or in the presence of obstacles.

Approximating k-hop minimum-spanning trees

Given a complete graph on~n nodes with metric edge costs, the minimum-costk-hop spanning tree (kHMST) problem asks for a spanning tree of minimum total cost such that the longest root-leaf-path in the tree has at most k edges. We present an algorithm that computes such a tree of total expected cost O(logn) times that of a minimum-cost k-hop spanning-tree.

Curve reconstruction from noisy samples

We present an algorithm to reconstruct a collection of disjoint smooth closed curves from n noisy samples. Our noise model assumes that the samples are obtained by first drawing points on the curves according to a locally uniform distribution followed by a uniform perturbation of each point in the normal direction with a magnitude smaller than the minimum local feature size. The reconstruction is faithful with a probability that approaches 1 as n increases.We expect that our approach can lead to provable algorithms under less restrictive noise models and for handling non-smooth features.

Exact geometric predicates using cascaded computation

In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing eometric algorithms. Using various floating-point filters together with arbitrary precision packages, we develop an easy-to-use xpression compiler called EXPCOMP. EXPCOMP supports all common operations +Ibines , ‘, /, ,/, Applying a new semi-static filter, EXPCOMP comthe speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floating-point exceptions, including underflow The resulting programs how a very good runtime behaviour.

Exact geometric computation using Cascading

In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various floating-point filters together with arbitrary precision packages, we develop an easy-to-use expression compiler called EXPCOMP. EXPCOMP supports all common operations . Applying a new semi-static filter, EXPCOMP combines the speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floating-point exceptions, including underflow. The resulting programs show a very good runtime behaviour.

Look — a Lazy Object-Oriented Kernel for geometric computation

In this paper we describe and discuss a new kernel design for g eometric computation in the plane. It combines different kin ds of floating-point filter techniques and a lazy evaluation schem e with the exact number types provided by LEDA allowing for efficien t and exact computation with rational and algebraic geometri c objects. It is the first kernel design which uses floating-point filter t chniques on the level of geometric constructions. The experiments we present – partly using the CGAL framework – show a great improvement in speed and – maybe even more important for practical applications – memory consumption when dealing with more complex geometric computations.