Objectives for Lecture 6: Pathfinding

3. Objectives for Lecture 6: Pathfinding Depth of understanding • After the lecture you are able to…

4. 1. Chapter: Introduction 1.1 Navigation 1.2 ModellingStreetswith Graphs 2. Chapter: Algorithms 2.1 British Museum Algorithm 2.2 Breadth First, Depth First, Best First 2.3 Dijkstra, A* 3. Chapter: Application 4. Chapter: Summary

5. Introduction – NavigationThe human approach https://mapstyle.withgoogle.com/

6. Introduction – NavigationThe human approach https://mapstyle.withgoogle.com/

7. Introduction – NavigationThe human approach https://mapstyle.withgoogle.com/

8. The human approachtofindingroutes on a streetmapis a veryvisualone. Feasiblepathsaredevelopedbyvisuallytracing different streetlines in thegeneraldirectionofthetarget. Obstacles, deadendsandrestrictionsareintuitivelytakenintoaccount, although – in typicalroutingsituations – no intensive thinkingorreflectioniscarried out. The human actoffinding a pathbetween a startingpointand a destinationcanbeconsidered „intelligent“. As a matter offact, we do not understandthe human approachoffindingroutes on maps in detail. Hence, wecan not modelitdirectly. Instead, searchalgorithmshaveevolved, solvingthe same problemsbysupposedly different means. Somepopularonesoftheseshallbeintroduced in thislecture. The questionthatremainsiswhetheror not thepresentedalgorithmscanthemselvesbeconsidered „intelligent“ solelybecausetheysolve a problemthathumanssolvebymeansof human intelligence.

9. Introduction – NavigationDefinition „The process or activity of accurately ascertaining one's position and planning and following a route.”https://en.oxforddictionaries.com/definition/navigation

10. Introduction – NavigationDefinition 1. Navigation HourstoMinutes 2. Path Following MinutestoSeconds 3. Control/ Stabilization SecondstoMiliseconds

11. Introduction –NavigationHistorical Approaches Paper Map Direct Transmission „Play“ SpecificPaths 1930 IterAvto Paper Map Gyroscope ScrollsMap 1981 electroGyrocator https://ndrive.com/brief-history-gps-car-navigation/

12. Introduction –NavigationHistorical Approaches Digital Map Dead Reckoning Cassettes 1985 Etak Navigator Digital Color Map Dead Reckoning CD-ROM 1987 Toyota CD-ROM Navigation System https://ndrive.com/brief-history-gps-car-navigation/

13. Introduction –NavigationHistorical Approaches Digital Color Map GPS 1990 Mazda Eunos Cosmo GPS Dead Reckoning MapMatching Speech Recognition Dynamic Routing Points of Interest (POI) Modern Navigation Systems https://ndrive.com/brief-history-gps-car-navigation/

14. Introduction –NavigationWhatisneededforcarnavigation?

15. Introduction – ModellingStreetswithGraphsWhatneedstobemodelled? Road GeometryPosition Length Start - End Road ConnectivityIntersections ConnectionsFly-Over Road AttributesTurn Restrictions Speed Limits Type Segment Segment 2 Segment 1

16. Introduction – ModellingStreetswithGraphsData Sources GPS Tracks SatelliteImagery https://www.maps.google.de https://buy.garmin.com/de-DE/DE/p/550460

17. Introduction – ModellingStreetswithGraphsOSM Open SteetMap (OSM) • Worldwide Map Data • Free • Started 2004 in London • 1 000 000 ContributingUsers • DefinesData Modelfor Mapping https://upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Openstreetmap_logo.svg/1200px-Openstreetmap_logo.svg.png

18. Introduction – ModellingStreetswithGraphsOSM Nodes Definedbylatitude, longitudeandnode id. Representsarbitrarylocations.Attributes specifiedbytags. Node_ID = 1 Lat = 48.1 Lon = 11.5 Node_ID = 2 Lat = 48.4 Lon = 12.5 k : highway v : traffic light k : highway v : bus_stop A tag consistsof a keyand a value. Tags describefeaturesofmapelements. Keys andvaluesarefreetext but conventionsexist.

19. Introduction – ModellingStreetswithGraphsOSM Ways Definedby an orderedsetofnodes. Groups nodesintolines. Attributes specifiedbytags. Way_ID = 1 Node_ID = 1 Lat = 48.1 Lon = 11.50 Node_ID = 2 Lat = 48.1 Lon = 11.51 Node_ID = 3 Lat = 48.1 Lon = 11.52 k : highway v : primary A tag consistsof a keyand a value. Tags describefeaturesofmapelements. Keys andvaluesarefreetext but conventionsexist.

20. Introduction – ModellingStreetswithGraphsOSM Relations Definedby an orderedlistofnodes, waysand/orrelations. Defineslogicalorgeographicrelationshipsbetweenotherelements. Relation_ID = 1 A tag consistsof a keyand a value. Tags describefeaturesofmapelements. Keys andvaluesarefreetext but conventionsexist.

21. Introduction – ModellingStreetswith GraphsGraphs – Basic Elements Node Directed Edge Undirected Edge

22. Introduction – ModellingStreetswith GraphsGraphs– Basic Elements Weighted Edge 3 Unweighted Edge Edge weightsareinterpretableascost(e.g. time, distance …)

23. Introduction – ModellingStreetswithGraphsGraphs– Formal Description

24. Introduction – ModellingStreetswithGraphsFrom OSM toRoutable Graphs Initial Graph OSM Data providesgeometryandspeedlimits 50 km/h 50 km/h 50 km/h 50 km/h 30 km/h 50 km/h 30 km/h 30 km/h

25. Introduction – ModellingStreetswithGraphsFrom OSM toRoutable Graphs Augmented Graph Edge weights like distancesandtimespansarecalculated 100 m 7.2 s 100 m 7.2 s 50 m 3.6 s 80 m 5.8 s 30 m 3.6 s 100 m 7.2 s 40 m 4.8 s 100 m 12 s

26. Introduction – ModellingStreetswithGraphsFrom OSM toRoutable Graphs Compressed Graph Graph compressionreducescomplexity: Reductionofcomputationaleffort 100 m 7.2 s 150 m 10.8 s 180 m 13 s 30 m 3.6 s 100 m 7.2 s 40 m 4.8 s 100 m 12 s

27. Introduction – ModellingStreetswithGraphsFrom OSM toRoutable Graphs Compressed Graph withturn restrictionsandturn penalties Turn restrictionsarederivedfrom OSM tags, turn penaltiesmaybeintroducedfor time optimal routing 10.8 s 13 s 100 m 12 s

28. Introduction – ModellingStreetswith GraphsLectureAssumptions

29. Introduction –NavigationWhatisneededforcarnavigation?

30. AlgorithmsOverview Routing GenerateRoutable Graph Determine Origin / Destination B B B A A A D D D Find Path C C C

31. AlgorithmsSearch Trees A B C D D B Search Tree Graph A D C

32. AlgorithmsSearch Trees {A} A B C D D {A, B} {A, B, D} {A, B, C} {A, B, C, D} A searchtreecanbemadefrom a graph Eachchildnodedenotes a paththatis a one-stepextensionofthepathdenotedbyitsparent Convertinggraphsintosearchtrees:Tracing out all possiblepathsuntilnofurtherextensionispossible

33. AlgorithmsPathfinding: Problem Formulations B B • A : Find a Path • Depth First • Breadth First • Best First A A D D C C • B : Find an optimal Path • Dijkstra • A*

34. AlgorithmsCompleteness A searchalgorithmiscompleteifitisguaranteedto find anexistingsolution in a finite amountof time. (Almost) all algorithms in thislecture

35. AlgorithmsPathfindingExample Graph 8 5 4 3 6.4 5

36. AlgorithmsGround Rules Example Graph Invalid Path PathsneverbitetheirowntailGenerally applicable

37. AlgorithmsGround Rules Example Graph 1 TiesareresolvedlexicallyApplicableforexamplegraph 2 3

38. Algorithms – British MuseumDescription • Algorithm: • Randomlyappend Nodes topathwhile not stuck. • Ifdestination was reached: Success • If stuck, startover

39. AlgorithmsMunich Demo Graph

40. AlgorithmsMunich Demo Graph Properties:1. BidirectionalEdges 2. StaticWeights 3. Weights = Distances 4. Main Roads 5. No Turn Restrictions 6. Basic Road Geometry

41. Algorithms – British MuseumMunich Demo

42. Algorithms – Depth FirstDescription • Algorithm:From „ArtificialIntelligence“ by Patrick H. Winston • Form a one-element queueconsistingof a zero-lengthpaththatcontainsonlytherootnode • Untilthefirstpath in thequeueterminates at thegoalnodeorthequeueisempty, • Remove thefirstpathfromthequeue; createnewpathsbyextendingthefirstpathto all theneighborsofthe terminal node • Reject all newpathswithloops • Add thenewpaths, ifany, tothefrontofthequeue • Ifthegoalnodeisfound, announcesuccess; otherwiseannouncefailure

43. Algorithms – Depth FirstDescription • Algorithm:From „ArtificialIntelligence“ by Patrick H. Winston • Form a one-element queueconsistingof a zero-lengthpaththatcontainsonlytherootnode • Untilthefirstpath in thequeueterminates at thegoalnodeorthequeueisempty, • Remove thefirstpathfromthequeue; createnewpathsbyextendingthefirstpathto all theneighborsofthe terminal node • Reject all newpathswithloops • Add thenewpaths, ifany, tothefrontofthequeue • Ifthegoalnodeisfound, announcesuccess; otherwiseannouncefailure Divingintothe Search Tree

44. Algorithms – DepthFirstCharacteristic Search Tree S G narrowanddeep

45. Algorithms – DepthFirstPathfindingExample Graph A

46. Algorithms – DepthFirstPathfindingExample Graph A B

47. Algorithms – DepthFirstPathfindingExample Graph A B C

48. Algorithms – DepthFirstPathfindingExample Graph A B Backtrack Backtrace C

49. Algorithms – DepthFirstPathfindingExample Graph A B C E

50. Algorithms – DepthFirstPathfindingExample Graph A B C C E