Kurt Mehlhorn

Faster algorithms for the shortest path problem

Efficient implementations of Dijkstras shortest path algorithm are investigated. A new data structure, called the <italic>radix heap</italic>, is proposed for use in this algorithm. On a network with <italic>n</italic> vertices, <italic>m</italic> edges, and nonnegative integer arc costs bounded by <italic>C</italic>, a one-level form of radix heap gives a time bound for Dijkstras algorithm of <italic>O</italic>(<italic>m</italic> + <italic>n</italic> log <italic>C</italic>). A two-level form of radix heap gives a bound of <italic>O</italic>(<italic>m</italic> + <italic>n</italic> log <italic>C</italic>/log log <italic>C</italic>). A combination of a radix heap and a previously known data structure called a <italic>Fibonacci heap</italic> gives a bound of <italic>O</italic>(<italic>m</italic> + <italic>n</italic>a @@@@log <italic>C</italic>). The best previously known bounds are <italic>O</italic>(<italic>m</italic> + <italic>n</italic> log <italic>n</italic>) using Fibonacci heaps alone and <italic>O</italic>(<italic>m</italic> log log <italic>C</italic>) using the priority queue structure of Van Emde Boas et al. [ 17].

The LEDA Platform of Combinatorial and Geometric Computing

We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our web-pages for more information.

A faster approximation algorithm for the Steiner problem in graphs

Abstract We present a new implementation of the Kou, Markowsky and Berman algorithm for finding a Steiner tree for a connected, undirected distance graph with a specified subset S of the set of vertices V . The total distance of all edges of this Steiner tree is at most 2(1-1/ l ) times that of a Steiner minimal tree, where l is the minimum number of leaves in any Steiner minimal tree for the given graph. The algorithm runs in O(| E |+| V |log| V |) time in the worst case, where E is the set of all edges and V the set of all vertices in the graph.

Congruence, similarity, and symmetries of geometric objects

We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point setA exactly or approximately into a point setB. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).

Randomized and deterministic simulations of PRAMs by parallel machines with restricted granularity of parallel memories

SummaryThe present paper provides a comprehensive study of the following problem. Consider algorithms which are designed for shared memory models of parallel computation (PRAMs) in which processors are allowed to have fairly unrestricted access patterns to the shared memory. Consider also parallel machines in which the shared memory is organized in modules where only one cell of each module can be accessed at a time. Problem. Give general fast simulations of these algorithms by these parallel machines.Each of our solutions answers two basic questions. (1) How to initially distribute the logical memory addresses of the PRAM, to be simulated, among the physical locations of the simulating machine? (2) How to compute the physical location of a logical address during the simulation?We utilize two main ideas for the first question.(a)Randomization. The logical addresses are randomly distributed among the memory modules. This is done using universal hashing.(b)Copies. We keep copies of each logical address in several memory modules. In a typical time cycle of the PRAM some number of memory requests has to be satisfied. As a primary objective, our simulations minimize the maximum number of memory requests which are assigned to the same module. Our solutions also optimize the following computational resources. They minimize the size of the physical memory, the time for computing the mapping from logical to physical addresses and the space for storing this mapping.We discuss extensions of our solutions to various PRAMs and various shared memory parallel machines. Our solution is also applicable to synchronous distributed machines with no shared memory where the processors can communicate through a bounded degree network.

A new data structure for representing sorted lists

SummaryIn this paper we explore the use of weak B-trees to represent sorted lists. In weak B-trees each node has at least a and at most b sons where 2a≦b. We analyse the worst case cost of sequences of insertions and deletions in weak B-trees. This leads to a new data structure (level-linked weak B-trees) for representing sorted lists when the access pattern exhibits a (time-varying) locality of reference. Our structure is substantially simpler than the one proposed in [7], yet it has many of its properties. Our structure is as simple as the one proposed in [5], but our structure can treat arbitrary sequences of insertions and deletions whilst theirs can only treat non-interacting insertions and deletions. We also show that weak B-trees support concurrent operations in an efficient way.

Four results on randomized incremental constructions

We prove four results on randomized incremental constructions (RICs): an analysis of the expected behavior under insertion and deletions, a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, a tail estimate for the space complexity of RICs, a lower bound on the complexity of a game related to RICs.

Dynamic fractional cascading

The problem of searching for a key in many ordered lists arises frequently in computational geometry. Chazelle and Guibas recently introduced fractional cascading as a general technique for solving this type of problem. In this paper we show that fractional cascading also supports insertions into and deletions from the lists efficiently. More specifically, we show that a search for a key inn lists takes timeO(logN +n log logN) and an insertion or deletion takes timeO(log logN). HereN is the total size of all lists. If only insertions or deletions have to be supported theO(log logN) factor reduces toO(1). As an application we show that queries, insertions, and deletions into segment trees or range trees can be supported in timeO(logn log logn), whenn is the number of segments (points).

Computing a maximum cardinality matching in a bipartite graph in time O n 1.5 m/ log n

Abstract We show how to compute a maximum cardinality matching in a bipartite graph of n vertices in time O(n1.5 m log n ). For dense graphs this improves on the O( n m) algorithm of Hopcroft and Karp. The speed-up is obtained by an application of the fast adjacency matrix scanning technique of Cheriyan, Hagerup and Mehlhorn.

Randomized incremental construction of abstract Voronoi diagrams

Abstract Voronoi diagrams were introduced by R. Klein [14, 11, 12] as an axiomatic basis of Voronoi diagrams. We show how to construct abstract Voronoi diagrams in time O(n log n) by a randomized algorithm; the algorithm is based on Clarkson and Shors randomized incremental construction technique [6]. The new algorithm has the following advantages over previous algorithms: It can handle a much wider class of abstract Voronoi diagrams than the algorithms presented in [14, 17]. It can be adapted to a concrete kind of Voronoi diagram by providing a single basic operation, namely the construction of a Voronoi diagram of five sites. Moreover, all geometric decisions are confined to the basic operation, and using this operation, abstract Voronoi diagrams can be constructed in a purely combinatorial manner.