Using Multi-Level Graphs for Timetable Information Frank Schulz, - - PowerPoint PPT Presentation

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Using Multi-Level Graphs for Timetable Information Frank Schulz, - - PowerPoint PPT Presentation

Sep 28, 2022 654 likes •1.3k views

Using Multi-Level Graphs for Timetable Information Frank Schulz, Dorothea Wagner, and Christos Zaroliagis 1 Overview 1. Introduction Timetable Information - A Shortest Path Problem 2. Multi-Level Graphs Speed-Up Technique for Shortest

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Using Multi-Level Graphs for Timetable Information

Frank Schulz, Dorothea Wagner, and Christos Zaroliagis

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1

Overview

  • 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
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2

Overview

  • 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
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3

Application Scenario

Timetable Information System

  • Large timetable

(e.g., 500.000 departures, 7.000 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
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4

Shortest Path Problem

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

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5

Contribution

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
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6

Overview

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

Multi-Level Graphs

Given

  • a weighted digraph G = (V, E)
  • a sequence of subsets of V

V ⊃ S1 ⊃ . . . ⊃ Sl 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
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7

Multi-Level Graphs

Given

  • a weighted digraph G = (V, E)
  • a sequence of subsets of V

V ⊃ S1 ⊃ . . . ⊃ Sl 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
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8

Level Construction

G a b c

G = (V, E), S1 = {a, b, c}, S2 = {a, c}

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8

Level Construction

G a b c Level 1 Level 0

Connected Components in G − S1

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Level Construction

G a b c Level 1 Level 0

No internal vertex of path belongs to S1

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8

Level Construction

G a b c Level 1 Level 0

⇒ Edge in level 1

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8

Level Construction

G a b c Level 1 Level 0

No internal vertex of path belongs to S1

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8

Level Construction

G a b c Level 1 Level 0

⇒ Edge in level 1

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8

Level Construction

G a b c Level 1 Level 0

No internal vertex of path belongs to S1

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8

Level Construction

G a b c Level 1 Level 0

⇒ Edge in level 1

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8

Level Construction

E G a b c

1

Level 1 Level 0

All edges in level 1

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8

Level Construction

E

1

Level 1

Consider now iteratively graph (S1, E1)

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8

Level Construction

E E

2 1

Level 2 Level 1

Connected components in G − S2

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8

Level Construction

E E

2 1

Level 2 Level 1

No internal vertex of path belongs to S2

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8

Level Construction

E E

2 1

Level 2 Level 1

⇒ Edge in level 2

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8

Level Construction

E E

2 1

Level 2 Level 1

All edges in level 2

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8

Level Construction

Level 2 Level 1 Level 0 E E G a b c

2 1

3-Level Graph M(G, S1, S2)

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Multi-Level Graphs

Given

  • a weighted digraph G = (V, E)
  • a sequence of subsets of V

V ⊃ S1 ⊃ . . . ⊃ Sl 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
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10

Component Tree

Level 2 Level 1 Level 0 E E G

2 1

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10

Component Tree

Level 2 Level 1 Level 0

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10

Component Tree

root Level 2 Level 1 Level 0

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10

Component Tree

root Level 2 Level 1 Level 0

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10

Component Tree

root Level 2 Level 1 Level 0

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10

Component Tree

root Level 2 Level 1 Level 0

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Multi-Level Graphs

Given

  • a weighted digraph G = (V, E)
  • a sequence of subsets of V

V ⊃ S1 ⊃ . . . ⊃ Sl 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
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12

Define Subgraph

root Level 2 Level 1

t s

Level 0

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12

Define Subgraph

root Level 2 Level 1

t s

Level 0

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12

Define Subgraph

G

t s

root E E

2 1

Level 2 Level 1 Level 0

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12

Define Subgraph

G

t s

root E E

2 1

Level 2 Level 1 Level 0

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12

Define Subgraph

G

t s

root E E

2 1

Level 2 Level 1 Level 0

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12

Define Subgraph

t s

root E E

2 1

Level 2 Level 1 G Level 0

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13

Multi-Level Graphs

Given

  • a weighted digraph G = (V, E)
  • a sequence of subsets of V

V ⊃ S1 ⊃ . . . ⊃ Sl 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
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14

Lemma

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

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15

A Different Pair s, t

root Level 2 Level 1 Level 0

s t

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15

A Different Pair s, t

E E G

2 1

Level 2 Level 1 Level 0

s t

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15

A Different Pair s, t

Level 2 Level 1 Level 0 E

1

t s

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16

Overview

  • 1. Introduction √
  • 2. Multi-Level Graphs √
  • 3. Timetable Information Graphs
  • 4. Experiments
  • 5. Conclusion & Outlook
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Station Graph

Patras Korinthos Athens

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Train Graph

Athens Korinthos Patras

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18

Train Graph

Athens Korinthos Patras

00:00 24:00

time

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18

Train Graph

Athens Korinthos Patras

00:00 24:00

time

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18

Train Graph

Athens Korinthos Patras

00:00 24:00

time

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19

Query

Single-source some-targets shortest path problem

Athens Korinthos Patras

00:00 24:00

time departure arrival

10:00

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20

Multi-Level Train Graph

  • Given l subsets of stations Σ1 ⊃ . . . ⊃ Σl
  • Component tree in station graph
  • Define subgraph for a pair of stations
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21

Overview

  • 1. Introduction √
  • 2. Multi-Level Graphs √
  • 3. Timetable Information Graphs √
  • 4. Experiments
  • 5. Conclusion & Outlook
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22

Experiments

Given one instance G of a traingraph

  • Timetable of German trains, winter 1996/97
  • ∼ 500000 vertices, ∼ 7000 stations

Evaluate multi-level train graph M(G, Σ1, . . . , Σl) Investigate dependence on

  • number of levels
  • input sequence Σ1 ⊃ . . . ⊃ Σl
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Sets of stations Σi

Three criteria to sort stations A Importance regarding train changes B Degree in station graph C Random Consider the first nj stations according to A, B, C

  • n1 = 1974, . . . , n10 = 50

❀ Sets of stations ΣA

j , ΣB j , ΣC j

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2-Level Graphs M(G, ΣX

j )

Number of Additional Edges

1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06 200 400 600 800 1000 1200 1400 1600 1800 2000 importance degree random

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Evaluation of M(G, Σ1, . . . , Σl)

Consider only criterion A in the following Average Speed-up over 100000 queries

  • r = Runtime for Dijkstra in G
  • q = Runtime for Dijkstra in subgraph of M
  • Speed-up = r/q
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2-Level Graphs M(G, ΣA

j )

Average Speed-up

1 1.5 2 2.5 3 3.5 4 200 400 600 800 1000 1200 1400 1600 1800 2000 speedup

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Problem with small ΣA

j

Big components ⇒ Level 1 rarely used

E G

1

s t

Level 1 Level 0

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2-Level Graphs M(G, ΣA

j )

Queries Using Level 1

2 4 6 8 10 12 14 200 400 600 800 1000 1200 1400 1600 1800 2000 40 50 60 70 80 90 100 110 speedup of queries using level 1 percentage of queries using level 1

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3-Level Graphs M(G, ΣA

j , ΣA k )

Average Speed-up

4 5 6 7 8 9 10 11 50 100 150 200 250 300 350 400 450 500 1 7 8 9

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l-Level Graphs M(G, Σ1, . . . , Σl−1)

Best Average Speed-up (CPU-time, #Edges hit)

2 4 6 8 10 12 14 16 18 1 2 3 4 5 6 CPU-speedup edge-speedup

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31

Overview

  • 1. Introduction √
  • 2. Multi-Level Graphs √
  • 3. Timetable Information Graphs √
  • 4. Experiments √
  • 5. Conclusion & Outlook
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Conclusion & Outlook

Experimental evaluation of multi-level graphs

  • Input graph from timetable information
  • Best number of levels: 4, 5
  • Sizes of Σi are crucial

Outlook

  • Other input graphs
  • Relation

(Σ1, . . . , Σl) ↔ speed-up

⊲ Here: Criteria A, B, C; Sizes of Σi

  • Theoretical analysis
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Overview

  • 1. Introduction √
  • 2. Multi-Level Graphs √
  • 3. Timetable Information Graphs √
  • 4. Experiments √
  • 5. Conclusion & Outlook √