# Geometric Shortest Path Containers

@inproceedings{Wagner2004GeometricSP, title={Geometric Shortest Path Containers}, author={Dorothea Wagner and Thomas Willhalm and Christos D. Zaroliagis}, year={2004} }

In this paper, we consider Dijkstra’s algorithm for the single source single target shortest path problem in large sparse graphs. The goal is to reduce the response time for on-line queries by using precomputed information. Due to the size of the graph, preprocessing space requirements can be only linear in the number of nodes. We assume that a layout of the graph is given. In the preprocessing, we determine from this layout a geometric object for each edge containing all nodes that can be…

## 6 Citations

Partition-Based Speed-Up of Dijkstra ’ s Algorithm

- 2004

Determining the shortest path from one node to another in a graph is probably the most popular question in graph theory. If the graph is non-negatively weighted, Dijkstra’s algorithm is the classic…

Speed-Up Techniques for Shortest-Path Computations

- Computer ScienceSTACS
- 2007

A condensed overview of new developments and extensions of classic results for Dijkstra's algorithm is provided and how combinations of speed-up techniques can be realized to take advantage from different strategies are discussed.

Approximate distance oracles for geometric spanners

- Computer Science, MathematicsTALG
- 2008

The main result is a data structure that answers (1 + ϵ)-approximate shortest-path-length queries in constant time, and hence functions as an approximate distance oracle.

## References

SHOWING 1-10 OF 64 REFERENCES

Geometric Speed-Up Techniques for Finding Shortest Paths in Large Sparse Graphs

- Mathematics, Computer ScienceESA
- 2003

Dijkstra’s algorithm for the single source single target shortest paths problem in large sparse graphs is considered to reduce the response time for online queries by using precomputed information.

Semidynamic Algorithms for Maintaining Single-Source Shortest Path Trees

- Mathematics, Computer ScienceAlgorithmica
- 1998

The algorithms for the incremental problem (handling edge insertions and cost decrements) work for any graph; they have optimal space requirements and query time, but their performances depend on the class of the considered graph.

Incremental algorithms for minimal length paths

- Computer Science, MathematicsSODA '90
- 1990

The problem of maintaining on-line a solution to the All Pairs Shortest Paths Problem in a directed graph G = (V,E) where edges may be dynamically inserted or have their cost decreased is considered and a new data structure is introduced which is able to answer queries concerning the length of the shortest path between any two vertices in constant time.

Fully dynamic output bounded single source shortest path problem

- Mathematics, Computer ScienceSODA '96
- 1996

We consider the problem of maintaining the distances and the shortest paths from a single source in either a directed or an undirected graph with positive real edge weights, handling insertions,…

Undirected single source shortest paths in linear time

- Computer Science, MathematicsProceedings 38th Annual Symposium on Foundations of Computer Science
- 1997

A deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with integer weights, avoids the sorting bottle-neck by building a hierarchical bucketing structure, identifying vertex pairs that may be visited in any order.

Materialization Trade-Offs in Hierarchical Shortest Path Algorithms

- Computer ScienceSSD
- 1997

Experiments with the Twin Cities metropolitan road-map show that materializing the shortest-path-cost view for the boundary graph provides the best savings in computation time, for a given amount of storage and a small number of fragments.

Experimental Evaluation of a New Shortest Path Algorithm

- Computer ScienceALENEX
- 2002

Results are presented which show the new algorithm to run faster than Dijkstra's on a variety of sparse graphs when the number of vertices ranges from a few thousand to a few million, and when computing single-source shortest paths from as few as three different sources.

An Incremental Algorithm for a Generalization of the Shortest-Path Problem

- Mathematics, Computer ScienceJ. Algorithms
- 1996

An efficient incremental algorithm for the single-source shortest-path problem with positive edge lengths is obtained and is able to handle “multiple heterogeneous modifications”: between updates, the input graph is allowed to be restructured by an arbitrary mixture of edge insertions, edge deletions, and edge-length changes.

A new approach to dynamic all pairs shortest paths

- Mathematics, Computer ScienceJACM
- 2004

A fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in O(n2log3n) amortized time per update and unit worst-case time per distance query, where n is the number of vertices.

Efficient Algorithms for Shortest Paths in Sparse Networks

- Computer Science, MathematicsJ. ACM
- 1977

Algorithms for finding shortest paths are presented which are faster than algorithms previously known on networks which are relatively sparse in arcs, and a class of “arc set partition” algorithms is introduced.