Using Multi-Level Graphs for Timetable Information. Frank Schulz, Dorothea Wagner, and Christos Zaroliagis

Transcription

1 Using Multi-Level Graphs for Timetable Information Frank Schulz, Dorothea Wagner, and Christos Zaroliagis

2 Overview 1 1. Introduction Timetable Information - A Shortest Path Problem 2. Multi-Level Graphs Speed-Up Technique for Shortest Path Algorithms 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook

3 Overview 2 1. Introduction Timetable Information - A Shortest Path Problem 2. Multi-Level Graphs Speed-Up Technique for Shortest Path Algorithms 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook

4 Application Scenario 3 Timetable Information System Large timetable (e.g., departures, stations) Central server (e.g., 100 on-line queries per second) Fast Algorithm Simple Queries Input: departure and arrival station, departure time Output: train connection with earliest arrival time

5 Shortest Path Problem 4 Graph model Solving a query finding a shortest path Speed-up techniques needed Commercial products Heuristics that don t guarantee optimality Scientific work Geometric techniques Hierarchical graph decomposition

6 Contribution 5 Multi-Level Graph Approach Hierarchical graph decomposition technique For general digraphs Idea Preprocessing: Construct multiple levels of additional edges On-Line Phase: Compute shortest paths in small subgraphs Experimental Evaluation For the given application scenario With real data

7 Overview 6 1. Introduction 2. Multi-Level Graphs 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook

8 Multi-Level Graphs 7 Given a weighted digraph G = (V, E) a sequence of subsets of V V S 1... S l Outline Construct l levels of additional edges M(G) Component Tree Define subgraph of M(G) for a pair s, t V Use subgraph to compute s-t shortest path

9 Multi-Level Graphs 7 Given a weighted digraph G = (V, E) a sequence of subsets of V V S 1... S l Outline Construct l levels of additional edges M(G) Component Tree Define subgraph of M(G) for a pair s, t V Use subgraph to compute s-t shortest path

10 Level Construction 8 a b c G G = (V, E), S 1 = {a, b, c}, S 2 = {a, c}

11 Level Construction 8 Level 1 Level 0 a b c G Connected Components in G S 1

12 Level Construction 8 Level 1 Level 0 a b c G No internal vertex of path belongs to S 1

13 Level Construction 8 Level 1 Level 0 a b c G Edge in level 1

14 Level Construction 8 Level 1 Level 0 a b c G No internal vertex of path belongs to S 1

15 Level Construction 8 Level 1 Level 0 a b c G Edge in level 1

16 Level Construction 8 Level 1 Level 0 a b c G No internal vertex of path belongs to S 1

17 Level Construction 8 Level 1 Level 0 a b c G Edge in level 1

18 Level Construction 8 Level 1 E 1 Level 0 a b c G All edges in level 1

19 Level Construction 8 Level 1 E 1 Consider now iteratively graph (S 1, E 1 )

20 Level Construction 8 Level 2 E 2 Level 1 E 1 Connected components in G S 2

21 Level Construction 8 Level 2 E 2 Level 1 E 1 No internal vertex of path belongs to S 2

22 Level Construction 8 Level 2 E 2 Level 1 E 1 Edge in level 2

23 Level Construction 8 Level 2 E 2 Level 1 E 1 All edges in level 2

24 Level Construction 8 Level 2 E 2 Level 1 E 1 Level 0 a b c G 3-Level Graph M(G, S 1, S 2 )

25 Multi-Level Graphs 9 Given a weighted digraph G = (V, E) a sequence of subsets of V V S 1... S l Outline Construct l levels of additional edges M(G) Component Tree Define subgraph of M(G) for a pair s, t V Use subgraph to compute s-t shortest path

26 Component Tree 10 Level 2 E 2 Level 1 E 1 Level 0 G

27 Component Tree 10 Level 2 Level 1 Level 0

28 Component Tree 10 root Level 2 Level 1 Level 0

29 Component Tree 10 root Level 2 Level 1 Level 0

30 Component Tree 10 root Level 2 Level 1 Level 0

31 Component Tree 10 root Level 2 Level 1 Level 0

32 Multi-Level Graphs 11 Given a weighted digraph G = (V, E) a sequence of subsets of V V S 1... S l Outline Construct l levels of additional edges M(G) Component Tree Define subgraph of M(G) for a pair s, t V Use subgraph to compute s-t shortest path

33 Define Subgraph 12 root Level 2 Level 1 Level 0 s t

34 Define Subgraph 12 root Level 2 Level 1 Level 0 s t

35 Define Subgraph 12 root Level 2 E 2 Level 1 E 1 Level 0 s t G

36 Define Subgraph 12 root Level 2 E 2 Level 1 E 1 Level 0 s t G

37 Define Subgraph 12 root Level 2 E 2 Level 1 E 1 Level 0 s t G

38 Define Subgraph 12 root Level 2 E 2 Level 1 E 1 Level 0 s t G

39 Multi-Level Graphs 13 Given a weighted digraph G = (V, E) a sequence of subsets of V V S 1... S l Outline Construct l levels of additional edges M(G) Component Tree Define subgraph of M(G) for a pair s, t V Use subgraph to compute s-t shortest path

40 Lemma 14 The length of a shortest s-t path is the same in the s-t subgraph of M(G) and G. Subgraph of M(G): s t Original Graph G: G

41 A Different Pair s, t 15 root Level 2 Level 1 Level 0 s t

42 A Different Pair s, t 15 Level 2 E 2 Level 1 E 1 Level 0 s t G

43 A Different Pair s, t 15 Level 2 Level 1 E 1 Level 0 s t

44 Overview Introduction 2. Multi-Level Graphs 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook

45 Station Graph 17 Patras Korinthos Athens

46 Train Graph 18 Patras Korinthos Athens

47 Train Graph 18 00:00 Patras Korinthos Athens 24:00 time

48 Train Graph 18 00:00 Patras Korinthos Athens 24:00 time

49 Train Graph 18 00:00 Patras Korinthos Athens 24:00 time

50 Query 19 Single-source some-targets shortest path problem 00:00 Patras Korinthos Athens 10:00 departure 24:00 time arrival

51 Multi-Level Train Graph 20 Given l subsets of stations Σ 1... Σ l Component tree in station graph Define subgraph for a pair of stations

52 Overview Introduction 2. Multi-Level Graphs 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook

53 Experiments 22 Given one instance G of a traingraph Timetable of German trains, winter 1996/ vertices, 7000 stations Evaluate multi-level train graph M(G, Σ 1,..., Σ l ) Investigate dependence on number of levels input sequence Σ 1... Σ l

54 23 Sets of stations Σ i Three criteria to sort stations A Importance regarding train changes B Degree in station graph C Random Consider the first n j stations according to A, B, C n 1 = 1974,..., n 10 = 50 Sets of stations Σ A j, ΣB j, ΣC j

55 2-Level Graphs M(G, Σ X j ) 24 Number of Additional Edges 7e+06 6e+06 importance degree random 5e+06 4e+06 3e+06 2e+06 1e

56 Evaluation of M(G, Σ 1,..., Σ l ) 25 Consider only criterion A in the following Average Speed-up over queries r = Runtime for Dijkstra in G q = Runtime for Dijkstra in subgraph of M Speed-up = r/q

57 2-Level Graphs M(G, Σ A j ) 26 Average Speed-up 4 speedup

58 27 Problem with small Σ A j Big components Level 1 rarely used Level 1 E 1 Level 0 s t G

59 2-Level Graphs M(G, Σ A j ) Queries Using Level 1 speedup of queries using level 1 percentage of queries using level

60 3-Level Graphs M(G, Σ A j, ΣA k ) 29 Average Speed-up

61 l-level Graphs M(G, Σ 1,..., Σ l 1 ) 30 Best Average Speed-up (CPU-time, #Edges hit) CPU-speedup edge-speedup

62 Overview Introduction 2. Multi-Level Graphs 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook

63 Conclusion & Outlook 32 Experimental evaluation of multi-level graphs Input graph from timetable information Best number of levels: 4, 5 Sizes of Σ i are crucial Outlook Other input graphs Relation (Σ 1,..., Σ l ) speed-up Here: Criteria A, B, C; Sizes of Σ i Theoretical analysis

64 Overview Introduction 2. Multi-Level Graphs 3. Timetable Information Graphs 4. Experiments 5. Conclusion & Outlook