Jens Gustedt

Two linear time Union-Find strategies for image processing

Abstract We consider Union-Find as an appropriate data structure to obtain two linear time algorithms for the segmentation of images. The linearity is obtained by restricting the order in which Unions are performed. For one algorithm the complexity bound is proven by amortizing the Find operations. For the other we use periodic updates to keep the relevant part of our Union-Find-tree of constant height. Both algorithms are generalized and lead to new linear strategies for Union-Find that are neither covered by the algorithm of Gabow and Tarjan (1984) nor by the one of Dillencourt et al. (1992).

On the pathwidth of chordal graphs

Abstract In this paper we first show that the pathwidth problem for chordal graphs is NP-hard. Then we give polynomial algorithms for subclasses. One of those classes are the k -starlike graphs – a generalization of split graphs. The other class are the primitive starlike graphs – a class of graphs where the intersection behavior of maximal cliques is strongly restricted.

PRO:a model for parallel resource-optimal computation

We present a new parallel computation model that enables the design of resource-optimal scalable parallel algorithms and simplifies their analysis. The model rests on the novel idea of incorporating relative optimality as an integral part and measuring the quality of a parallel algorithm in terms of granularity.

Efficient union-find for planar graphs and other sparse graph classes

We solve the Union-Find problem (UF) efficiently for the case the input is restricted to several graph classes, namely partial k-trees for any fixed k, d-dimensional grids for any fixed dimension d and for planar graphs. For the later we develop a technique of decomposing such a graph into small subgraphs, patching, that might be useful for other algorithmic problems on planar graphs, too.

Finiteness Theorems for Graphs and Posets Obtained by Compositions

We investigate classes of graphs and posets that admit decompositions to obtain or disprove finiteness results for obstruction sets. To do so we develop a theory of minimal infinite antichains that allows us to characterize such antichains by means of the set of elements below it.In particular we show that the following classes have infinite antichains with respect to the induced subgraph/poset relation: interval graphs and orders, N-free orders, orders with bounded decomposition width. On the other hand for orders with bounded decomposition diameter finiteness of all antichains is shown. As a consequence those classes with infinite antichains have undecidable hereditary properties whereas those with finite antichains have fast algorithms for all such properties.

Memory Management for Union-Find Algorithms

We provide a general tool to improve the real time performance of a broad class of Union-Find algorithms. This is done by minimizing the random access memory that is used and thus to avoid the well-known von Neumann bottleneck of synchronizing CPU and memory. A main application to image segmentation algorithms is demonstrated where the real time performance is drastically improved.

Minimum Spanning Trees for Minor-Closed Graph Classes in Parallel

For each minor-closed graph class we show that a simple variant of Borůvkas algorithm computes a MST for any input graph belonging to that class with linear costs. Among minor-closed graph classes are e.g planar graphs, graphs of bounded genus, partial k-trees for fixed k, and linkless or knotless embedable graphs. The algorithm can be implemented on a CRCW PRAM to run in logarithmic time with a work load that is linear in the size of the graph. We develop a new technique to find multiple edges in such a graph that might have applications in other parallel reduction algorithms as well.

A Compact Data Structure and Parallel Algorithms for Permutation Graphs

Starting from a permutation of {0, ..., n−1} we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 efficiently. The parallel algorithms obtained have a workload of O(m+n log n) where m is the number of edges of the permutation graph. They run in time O(log2n) on a CREW PRAM.

Testing Hereditary Properties Efficiently on Average

We use the quasi-ordering of substructure relations such as induced and weak subgraph, induced suborder, graph minor or subformula of a CNF formula to obtain recognition algorithms for hereditary properties that are fast on average. The ingredients needed besides inheritance are independence of the occurrence of small substructures in a random input and the existence of algorithms for recognition that are at most exponential.

Weak-Order Extensions of an Order

In this paper, at first we describe a graph representing all the weak-order extensions of a partially ordered set and an algorithm generating them. Then we present a graph representing all of the minimal weak-order extensions of a partially ordered set, and implying a generation algorithm. Finally, we prove that the number of weak-order extensions of a partially ordered set is a comparability invariant, whereas the number of minimal weak-order extensions of a partially ordered set is not a comparability invariant.