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

In this paper, we consider Dijkstra’s algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. We assume that a layout of the graph is given. From this layout, in the preprocessing, we determine for each edge a geometric object containing all nodes that can be reached on a shortest path starting with that edge. Based on these geometric objects, the search space for online computation can be reduced significantly. We present an extensive experimental study comparing the impact of different types of objects. The test data we use are traffic networks, the typical field of application for this scenario.

[1]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[2]  Geert-Jan Giezeman,et al.  On the design of CGAL a computational geometry algorithms library , 2000 .

[3]  Laurent Siklóssy,et al.  TRAINS, An Active Time-Table Searcher , 1988, ECAI.

[4]  Karsten Weihe,et al.  Dijkstra's algorithm on-line: an empirical case study from public railroad transport , 2000, JEAL.

[5]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[6]  Ulrik Brandes,et al.  GraphML Progress Report , 2001, GD.

[7]  Kurt Mehlhorn,et al.  LEDA: a platform for combinatorial and geometric computing , 1997, CACM.

[8]  Mikkel Thorup,et al.  Undirected single source shortest paths in linear time , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[9]  Thomas Preuss,et al.  An Integrated Traffic Information System , 1997 .

[10]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[11]  G. Toussaint.  Solving geometric problems with the rotating calipers , 1983 .

[12]  Jeffrey Scott Vitter,et al.  Shortest paths in euclidean graphs , 2005, Algorithmica.

[13]  K. Nachtigall.  Time depending shortest-path problems with applications to railway networks , 1995 .

[14]  Sakti Pramanik,et al.  HiTi graph model of topographical road maps in navigation systems , 1996, Proceedings of the Twelfth International Conference on Data Engineering.

[15]  Kurt Mehlhorn,et al.  A platform for combinatorial and geometric computing , 1995 .

[16]  Ulrich Meyer,et al.  Single-source shortest-paths on arbitrary directed graphs in linear average-case time , 2001, SODA '01.

[17]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[18]  Hermann A. Maurer,et al.  New Results and New Trends in Computer Science , 1991, Lecture Notes in Computer Science.

[19]  Jürgen Teich,et al.  Minimal enclosing parallelogram with application , 1995, SCG '95.

[20]  Madhav V. Marathe,et al.  A computational study of routing algorithms for realistic transportation networks , 1999, JEAL.

[21]  Shashi Shekhar,et al.  Path computation algorithms for advanced traveller information system (ATIS) , 1993, Proceedings of IEEE 9th International Conference on Data Engineering.

[22]  Andrew V. Goldberg,et al.  A Simple Shortest Path Algorithm with Linear Average Time , 2001, ESA.

[23]  Christos D. Zaroliagis,et al.  Using Multi-level Graphs for Timetable Information in Railway Systems , 2002, ALENEX.

[24]  Madhav V. Marathe,et al.  Classical and Contemporary Shortest Path Problems in Road Networks: Implementation and Experimental Analysis of the TRANSIMS Router , 2002, ESA.

[25]  F. Benjamin Zhan,et al.  Shortest Path Algorithms: An Evaluation Using Real Road Networks , 1998, Transp. Sci..

[26]  Karsten Weihe,et al.  Pareto Shortest Paths is Often Feasible in Practice , 2001, WAE.

[27]  Frank Schulz,et al.  Using Multi-Level Graphs for Timetable Information , 2001 .

[28]  Seth Pettie,et al.  Experimental Evaluation of a New Shortest Path Algorithm , 2002, ALENEX.

[29]  F. Benjamin Zhan,et al.  A Comparison Between Label-Setting and Label-Correcting Algorithms for Computing One-to-One Shortest Paths ∗ , 2000 .