Partitioning of Public Transit Networks

Transcription

1 Partitioning of Public Transit Networks [Bachelor s thesis] Matthias Hertel Albert-Ludwigs-Universität Freiburg

2 Contents Introduction Motivation Goal 2 Data model 3 Algorithms K-means Merging algorithm METIS PUNCH 4 Evaluation

3 Transfer Patterns [] with partitioning Transfer Patterns = sequences of transfers on optimal routes Freiburg Zürich: {[Freiburg, Zürich], [Freiburg, Basel, Zürich]}

4 Transfer Patterns [] with partitioning Transfer Patterns = sequences of transfers on optimal routes Freiburg Zürich: {[Freiburg, Zürich], [Freiburg, Basel, Zürich]} Compute Transfer Patterns between stations of the same partition border stations b(c x ) and b(c y ) reduced runtime reduced space

5 Transfer Patterns [] with partitioning Transfer Patterns = sequences of transfers on optimal routes Freiburg Zürich: {[Freiburg, Zürich], [Freiburg, Basel, Zürich]} Compute Transfer Patterns between stations of the same partition border stations b(c x ) and b(c y ) reduced runtime reduced space Query A B : A b(c A ) b(c B ) B little slower query times

6 Goal Partition the stations of a public transit network, such that partitions are small most traffic lies inside the partitions

7 Dataset schedule of Deutsche Bahn (205) only local traffic (no ICEs and ICs) modelled as undirected weighted graph stations nodes connections edges frequencies edge weights heuristical footpaths (distance 400 m; weight 200,000)

8 K-means-clustering [2] uses only geographic data Algorithm k-means-clustering initialize while assignments change do update assignments update means end while

9 Merging algorithm [3] hierarchical merges neighboured partitions hyperparameter k = number of partitions hyperparameter U = upper bound partition size order distinguished by a utility function

10 Merging algorithm [3] hierarchical merges neighboured partitions hyperparameter k = number of partitions hyperparameter U = upper bound partition size order distinguished by a utility function f (u, v) = s(u) s(v) ( w(u,v) + w(u,v) ) s(u) s(v) s(u) = size of u s(v) = size of v w(u, v) = sum of edge weights between u and v

11 METIS [4] graph partitioning framework state of the art can be downloaded hyperparameter k = number of partitions three phases (next slide)

12 METIS Figure : The three phases of METIS (Source: [4])

13 PUNCH [5] partitioning using natural cut heuristics hyperparameter U = upper bound partition size two phases filtering phase assembly phase

14 PUNCH Filtering phase: contract regions that are separated by small cuts main graph bridge main graph U < U

15 PUNCH Assembly phase initial solution: run merging algorithm on filtered graph local optimization: uncontract small regions rerun merging algorithm take better solution

16 Comparison: cut size 0 0 k-means merging PUNCH METIS 0 9 cut size maximum partition size Figure : Cut size over maximum partition size.

17 Comparison: cut edges k-means merging PUNCH METIS number of cut edges maximum partition size Figure : Cut edges over maximum partition size.

18 PUNCH - unweighted graph main graph 0

19 PUNCH - unweighted graph main graph 0 main graph 3 minimum cut preserved

20 PUNCH - weighted graph main graph 200,000 0

21 PUNCH - weighted graph main graph 200,000 0 main graph 3 200,000 minimum cut not preserved

22 The gain of footpaths merging algorithm with U=4,000 Figure : no footpaths

23 The gain of footpaths merging algorithm with U=4,000 Figure : no footpaths Figure : with footpaths

24 Conclusions K-means better than expected traffic geographically clustered

25 Conclusions K-means better than expected traffic geographically clustered merging algorithm and METIS produce good results

26 Conclusions K-means better than expected traffic geographically clustered merging algorithm and METIS produce good results arbitrary utility functions can be used with the merging algorithm

27 Conclusions K-means better than expected traffic geographically clustered merging algorithm and METIS produce good results arbitrary utility functions can be used with the merging algorithm PUNCH: filtering phase must use edge weights

28 Conclusions K-means better than expected traffic geographically clustered merging algorithm and METIS produce good results arbitrary utility functions can be used with the merging algorithm PUNCH: filtering phase must use edge weights footpaths prohibit geographically overlapping partitions

29 Questions? Thank you for your attention!

30 Bibliography H. Bast, E. Carlsson, A. Eigenwillig, R. Geisberger, C. Harrelson, V. Raychev, and F. Viger, Fast routing in very large public transportation networks using transfer patterns, in Algorithms ESA 200. Springer, 200, pp J. MacQueen, Some methods for classification and analysis of multivariate observations, in Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol., no. 4. Oakland, CA, USA., 967, pp M. G. van der Horst, Optimal route planning for car navigation systems, Master s thesis, Technische Universitaet Eindhoven, G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM Journal on scientific Computing, vol. 20, no., pp , 998. D. Delling, A. V. Goldberg, I. Razenshteyn, and R. F. Werneck, Graph partitioning with natural cuts, in Parallel & Distributed Processing Symposium (IPDPS), 20 IEEE International. IEEE, 20, pp

31 Cut size with k-means k=00 k=200 k=400 k= cut size round Figure : Cut size over maximum partition size.

32 METIS unbalancing ratio r s(p) r N k metis, r=.5 metis, r=2.0 metis, r= maximum partition size number of partitions Figure : Maximum partition size over number of partitions.

33 METIS metis, r=.5 metis, r=2.0 metis, r= cut size number of partitions Figure : Cut size over number of partitions.

34 PUNCH Filtering phase, pass : contract bridge-separated regions main graph bridge main graph 3 0

35 PUNCH Filtering phase, pass 2: contract simple paths main graph main graph 5

36 PUNCH Filtering phase, pass 3: contract two-cut-separated regions main graph main graph 0 2

37 PUNCH Filtering phase, pass 4: contract natural cut -separated regions Figure : Finding a natural cut (Source: [5])

38 k-means merging PUNCH METIS partitions max. part. size 4,05,873,975 3,32 cut size cut edges 2,273 9,497 3,97 8,562 cut edges (%) border nodes 5,564 2,954 7,669 2,00 border nodes (%) runtime (s) Table : Results of the four algorithms with about 8 partitions.