Hans Rohnert

Shortest paths in the plane with convex polygonal obstacles

Abstract An algorithm is presented which computes shortest paths in the Euclidean plane that do not cross given obstacles. The set of obstacles is assumed to consist of f disjoint convex polygons with n vertices in total. After preprocessing time O(n + f 2 log n), the shortest path between two arbitrary query points can be found in O(f 2 + n log n) time. The space complexity is O(n + f 2 ).

Dynamic perfect hashing: upper and lower bounds

A randomized algorithm is given for the dictionary problem with O(1) worst-case time for lookup and O(1) amortized expected time for insertion and deletion. An Omega (log n) lower bound is proved for the amortized worst-case time complexity of any deterministic algorithm in a class of algorithms encompassing realistic hashing-based schemes. If the worst-case lookup time is restricted to k, then the lower bound for insertion becomes Omega (kn/sup 1/k/).<<ETX>>

A probabilistic algorithm for vertex connectivity of graphs

Abstract A probabilistic algorithm is presented which computes the vertex connectivity of an undirected graph G = ( V , E ) in expected time O((-log e|V| 3 2 |E|) with error probability at most e provided that | E | frcase |1/2 d | V | 2 for some universal constant d

Upper and lower bounds for the dictionary problem

We give a randomized algorithm for the dictionary problem with O(1) worst case time for lookup and O(1) expected amortized time for insertion and deletion. We also prove an Ω(log n) lower bound on the amortized worst case time complexity of any deterministic algorithm based on hashing. Furthermore, if the worst case lookup time is restricted to k, then the lower bound becomes Ω(k·.n1/k).

Selected Topics from Computational Geometry, Data Structures and Motion Planning

The Voronoi diagram of a set of sites in the plane partitions the plane into regions, called Voronoi regions, one to a site. The Voronoi region of a site s is the set of points in the plane for which 8 is the closest site among all the sites. The Voronoi diagram has many applications in diverse fields, cf. Leven and Sharir [LS86] or Aurenhammer [Aur90] for a list of applications and a history of Voronoi diagrams. Different types of diagrams result from considering different notions of distance, e.g. Euclidean or Lp-norm or convex distance functions, and different sorts of sites, e.g. points, line segments, or circles. For many types of diagrams efficient construction algorithms have been found; these are either based on the divide-and-conquer technique due to Shamos and Hoey [SH75], the sweepline technique due to Fortune [For87], or geometric transforms due to Brown [Bro79] and Edelsbrunner and Seidel [ES86]. A unifying approach to Voronoi diagrams was proposed by Klein [Kle88a, Kle88b, Kle89a, KleS9b], cf. [ES86] for a related approach. Klein does not use the concept of distance as the basic notion but rather the concept of bisecting curve, i.e. he assumes for each pair {p, q} of sites the existence of a bisector J(p, q) which is homeomorphic to a line and divides the plane into a p-region and a q-region. The intersection of all p-regions for different qs is then the Voronoi-region of site p. He also postulates that Voronoi-regions are simply-connected and partition the plane. He shows that these so-called abstract Voronoi diagrams have already many of the properties of concrete Voronoi diagrams. In [MMO91] and the refinement [KMM91] we present a randomized incremental algorithm that can handle abstract Voronoi diagrams in (almost) their full generality. When n denotes the number of sites, the algorithm runs in O(nlog n) expected time, the average being taken over all permutations of the input. The algorithm is simple enough to be of great practical importance. It is uniform in the sense that only a single operation, namely the construction of a Voronoi diagram for 5 sites, depends on the specific type of Voronoi diagram and has to be newly programmed in order to adapt the algorithm to the type of the diagram. Moreover, this operation is the only geometric operation in our algorithm, and using this operation, abstract Voronoi diagrams can be constructed in a