Whereas earlier work on spatio-temporal databases generally focused on geometries changing in discrete steps, the emerging area of moving objects databases supports geometries changing continuously. Two important abstractions are moving point and moving region, modeling objects for which only the time-dependent position, or also the shape and extent are relevant, respectively. Examples of the first kind of moving entity are all kinds of vehicles, aircraft, people, or animals; of the latter hurricanes, forest res, forest growth, or oil spills in the sea. The goal is to develop data models and query languages as well as DBMS implementations supporting such entities, enabling new kinds of database applications. In earlier work we have proposed an approach based on abstract data types. Hence, moving point or moving region are viewed as data types with suitable operations. For example, a moving point might be projected into the plane, yielding a curve, or a moving region be mapped to a function describing the development of its size, yielding a real-valued function. A careful design of a system of types and operations (an algebra) has been presented, emphasizing completeness, closure, consistency and genericity. This design was given at an abstract level, defining, for example, geometries in terms of infinite point sets. In the next step, a discrete model was presented, o ering nite representations and data structures for all the types of the abstract model. The present paper provides the final step towards implementation by studying and developing systematically algorithms for (a large subset of) the operations. Some of them are relatively straightforward; others are quite complex. Algorithms are meant to be used in a database context; we also address...

We consider online routing algorithms for finding paths between the vertices of plane graphs.

We are given a transportation line where displacements happen at a bigger speed than in the rest of the plane. A shortest time path is a path between two points which takes less or equal time than any other. We consider the time to follow a shortest time path to be the time distance between the two points. In this paper, we give a simple algorithm for computing the Time Voronoi Diagram, that is, the Voronoi Diagram of a set of points using the time distance.

It is well known that on a line, a target point in unknown position can be found by walking a path at most 9 times as long as the distance from the start to the target point, in the worst case. This competitive factor of 9 is optimal. We investigate the case where the target is known to be within a fixed distance, r, of the start point, and determine the optimum competitive factor, C(r) < 9, that can be achieved by a competitive strategy S(r), under this additional assumption.

In terms of synthetic differential geometry, we give a variational characterization of the connection (parallelism) associated to a pseudo-Riemannian metric on a manifold.

A conservative implementation of a predicate returns true only if the exact predicate is true. That is, we accept a one sided error for the implementation. For geometric predicates, such as orientation- or incircle-tests, this allows efficient floating point implementations of the predicates with rare occurrences of the one sided error. We discuss the use of such conservative implementations for convex hull and triangulation algorithms for point sets in the plane. The resulting programs show a minor slowdown compared to an implementation that completely ignores the finite precision issue. However, our programs always produce output that satisfies basic desirable properties. The output can be easily checked for correctness and -- if necessary -- it can be repaired with an exact implementation of the needed predicates. Although (or since?) conservative implementations of predicates cannot detect degeneracies, the programs work for degenerate input. In fact, in our experiments the advanta...

How e#ciently can we search an unknown environment for a goal in unknown position? How much would it help if the environment were known? We answer these questions for simple polygons and for general graphs, by providing online search strategies that are as good as the best o#ine search algorithms, up to a constant factor. For other settings we prove that no such online algorithms exist.

We consider the problem of searching on m current rays for a target of unknown location. If no upper bound on the distance to the target is known in advance, then the optimal competitive ratio is 1 + 2m m =(m 1) m 1 . We show that if an upper bound of D on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least 1 + 2m m =(m 1) m 1 O(1= log 2 D) which is again optimal|but in a stricter sense. To show the optimality of our lower bound we construct a search strategy that achieves this ratio. Surprisingly, our strategy does not need to know an upper bound on the distance to the target in advance; it achieves a competitive ratio of 1 + 2m m =(m 1) m 1 O(1= log 2 D) if the target is found at distance D. Finally, we also present an algorithm to compute the strategy that allows the robot to search the farthest for a given competitive ratio C. 1 Introduction Searching for a target is an important and well studied problem i...

We introduce a class of planar arcs and curves, called tame arcs, which is general enough to describe (parts of) the boundaries of planar real objects. A tame arc can have a smooth parts as well as sharp (non-differentiable) corners. Thus, a polygonal arc is tame. On the other hand, this class of arcs is restrictive enough to rule out pathological arcs which have infinitely many inflections or which turn infinitely often: A tame arc can have only finitely many inflections, and its total absolute turn must be finite.

In this paper we study the problem of a robot searching for a visually recognizable target in an unknown simple polygon. We present two algorithms. Both work for arbitrarily oriented polygons. The search cost is proportional to the distance traveled by the robot. We use competitive analysis to judge the performance of our strategies. The first one is a simple modification of Djikstra's shortest path algorithm and achieves a competitive ratio of 2n \Gamma 7 where n is the number of vertices of the polygon. The second strategy achieves a competitive ratio of 1 + 2 \Delta (2k) 2k =(2k \Gamma 1) 2k\Gamma1 if the polygon is k-spiral. This can be shown to be optimal. It is based on an optimal algorithm to search in geometric trees which is also presented in this paper. Consider a geometric tree with m leaves and a target hidden somewhere in the tree. The target can only be detected if the robot reaches the location of the target. We provide an algorithm that achieves a competitive ratio ...