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## Algorithms for Planar Graphs and Graphs in Metric Spaces

by
Christian Wulff-nilsen

Citations: | 1 - 0 self |

by
Christian Wulff-nilsen

Citations: | 1 - 0 self |

@MISC{Wulff-nilsen_algorithmsfor,

author = {Christian Wulff-nilsen},

title = {Algorithms for Planar Graphs and Graphs in Metric Spaces},

year = {}

}

Algorithms for network problems play an increasingly important and very useful representation that allows classical graph algorithms, such as Dijkstra and Bellman-Ford, to be applied. Real-life networks often have additional structural properties that can be exploited. For instance, a road network or a wire layout on a microchip is typically (near-)planar and distances in the network are often defined w.r.t. the Euclidean or the rectilinear metric. Specialized algorithms that take advantage of such properties are often orders of magnitude faster than the corresponding algorithms for general graphs. The first and main part of this thesis focuses on the development of efficient planar graph algorithms. The most important contributions include a faster single-source shortest path algorithm, a distance oracle with subquadratic preprocessing time, an O(n log n) time algorithm for the replacement paths problem, and a min st-cut oracle with nearlinear preprocessing time. We also give improved time bounds for computing various graph invariants such as diameter and girth. In the second part, we consider stretch factor problems for geometric graphs and graphs embedded in metric spaces. Roughly speaking, the stretch factor is a real value expressing how well a (geo-)metric graph approximates the underlying complete graph w.r.t. distances. We give improved algorithms for computing the stretch factor of a given graph and for augmenting a graph with new edges while minimizing stretch factor. The third and final part of the thesis deals with the Steiner tree problem in the plane equipped with a weighted fixed orientation metric. Here, we give an improved theoretical analysis of the strength of pruning techniques applied by many Steiner tree algorithms. We also present an algorithm that computes a so called Steiner hull, a structure that may help in the computation of a Steiner minimal tree.

metric space planar graph stretch factor replacement path problem second part corresponding algorithm geometric graph improved algorithm final part subquadratic preprocessing time steiner minimal tree new edge additional structural property stretch factor problem various graph invariant real value general graph efficient planar graph algorithm classical graph algorithm distance oracle called steiner hull improved time bound nearlinear preprocessing time many steiner tree algorithm improved theoretical analysis important contribution underlying complete graph steiner tree problem road network time algorithm main part thesis deal real-life network single-source shortest path algorithm specialized algorithm metric graph weighted fixed orientation useful representation network problem wire layout min st-cut oracle

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