Mikael Hammar

Spanning spiders and light-splitting switches

Motivated by a problem in the design of optical networks, we ask when a graph has a spanning spider (subdivision of a star), or, more generally, a spanning tree with a bounded number of branch vertices. We investigate the existence of these spanning subgraphs in analogy to classical studies of Hamiltonicity.

F-Chord: Improved Uniform Routing on Chord

We propose a family of novel schemes based on Chord retain- ing all positive aspects that made Chord a popular topology for routing in P2P networks. The schemes, based on the Fibonacci number system, allow to improve on the maximum/average number of hops for lookups and the routing table size per node.

A System for Virtual Directories Using Euler Diagrams

In this paper, we describe how to use Euler Diagrams to represent virtual directories. i.e. collection of files that are computed on demand and satisfy a number of constraints. We, then, briefly describe the state of VennFS project that is currently modified to include this new capability. In particular, we show a data structure designed to answer queries about a given Euler Diagram and its sets. The data structure EulerTree described here is based on the R-Tree (see [Pankaj K. Agarwal, Mark de Berg, Joachim Gudmundsson, Mikael Hammar and Herman J. Haverkort, Box-trees and R-trees with near-optimal query time, in: Symposium on Computational Geometry, 2001, pp. 124-133]), a data structure designed for answering range queries over a family of shapes in the 2-dimensional space.

Box-trees and R-trees with near-optimal query time

A box-tree is a \ifasci so-called \emph{bounding-volume hierarchy} \else bounding-volume hierarchy \fi that uses axis-aligned boxes as bounding volumes. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. Finally, we present algorithms to convert box-trees to R-trees, resulting in R-trees with (almost) optimal query complexity.

Approximation Results for Kinetic Variants of TSP

We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem (TSP) where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results:• If the points all move with the same velocity, then there is a polynomial time approximation scheme for the Kinetic TSP. • The Kinetic TSP cannot be approximated better than by a factor of 2 by a polynomial time algorithm unless P = NP, even if there are only two moving points in the instance. • The Kinetic TSP cannot be approximated better than by a factor of

On R-trees with Low Stabbing Number

2^{\Omega(\sqrt{n})}

There are spanning spiders in dense graphs (and we know how to find them)

by a polynomial time algorithm unless P = NP, even if the maximum velocity is bounded. n denotes the size of the input instance.The last result is especially surprising in the light of existing polynomial time approximation schemes for the static version of the problem.

Concerning the Time Bounds of Existing Shortest Watchman Route Algorithms

The R-tree is a well-known bounding-volume hierarchy that is suitable for storing geometric data on secondary memory. Unfortunately, no good analysis of its query time exists. We describe a new algorithm to construct an R-tree for a set of planar objects that has provably good query complexity for point location queries and range queries with ranges of small width. For certain important special cases, our bounds are optimal. We also show how to update the structure dynamically, and we generalize our results to higher-dimensional spaces.

On R-trees with low query complexity

A spanning spider for a graph G is a spanning tree T of G with at most one vertex having degree three or more in T. In this paper we give density criteria for the existence of spanning spiders in graphs. We constructively prove the following result: Given a graph G with n vertices, if the degree sum of any independent triple of vertices is at least n - 1, then there exists a spanning spider in G. We also study the case of bipartite graphs and give density conditions for the existence of a spanning spider in a bipartite graph. All our proofs are constructive and imply the existence of polynomial time algorithms to construct the spanning spiders. The interest in the existence of spanning spiders originally arises in the realm of multicasting in optical networks. However, the graph theoretical problems discussed here are interesting in their own right.

Non-uniform deterministic routing on F-Chord(/spl alpha/)

A watchman route in a polygon P is a route inside P such that each point in the interior of P is visible from at least one point along the route. The objective of the shortest watchman route problem is to minimize the length of the watchman route for a given polygon. In 1991 Chin and Ntafos claimed an O(n4) algorithm, solving the shortest watchman route problem for simple polygons, given a starting point of the route. Later, improvements of this result were presented by Tan, Hirata and Inagaki, decreasing the time-bound to O(n2). We prove that the time bound analyses of these algorithms are erroneous and that their true time bound is Ω(2n). Furthermore, a modification to the latest algorithm is given, restoring its time bound to O(n2).