Accurate High-Performance Route Planning

Transcription

1 Sanders/Schultes: Route Planning 1 Accurate High-Performance Route Planning Peter Sanders Dominik Schultes Institut für Theoretische Informatik Algorithmik II Universität Karlsruhe (TH) Gouda, July 11, 2006

2 Sanders/Schultes: Route Planning 2 How do I get there from here? Applications route planning systems in the internet (e.g. car navigation systems

3 Sanders/Schultes: Route Planning 3 Goals exact shortest (i.e. fastest) paths in large road networks fast queries fast preprocessing low space consumption? scale-invariant, i.e., optimised not only for long paths

4 Sanders/Schultes: Route Planning 4 Related Work method query prepr. space scale source basic A [Hart et al. 68] bidirected [Pohl 71] heuristic hwy hier [commercial] separator hierarchies o? [several groups 02] geometric containers [Wagner et al. 03] bitvectors ++ o [Lauther... 04] landmarks + ++ [Goldberg et al. 04] landmarks + reaches ++ o o o [Goldberg et al. 06] highway hierarchies here direct towards target exploit hierarchy

5 Sanders/Schultes: Route Planning 5 DIJKSTRA s Algorithm not practicable Dijkstra s t for large road networks (e.g. Western Europe: nodes) bidirectional Dijkstra s t improves the running time, but still too slow

6 Sanders/Schultes: Route Planning 6 Naive Route Planning 1. Look for the next reasonable motorway

7 Sanders/Schultes: Route Planning 7 Naive Route Planning 1. Look for the next reasonable motorway 2. Drive on motorways to a location close to the target

8 Sanders/Schultes: Route Planning 8 Naive Route Planning 1. Look for the next reasonable motorway 2. Drive on motorways to a location close to the target 3. Search the target starting from the motorway exit

9 Sanders/Schultes: Route Planning 9 Commercial Approach Heuristic Highway Hierarchy s t complete search in local area search in (sparser) highway network iterate highway hierarchy Defining the highway network: s t use road category (highway, federal highway, motorway,... ) + manual rectifications delicate compromise speed accuracy

10 Sanders/Schultes: Route Planning 10 Our Approach Exact Highway Hierarchy s t complete search in local area search in (sparser) highway network iterate highway hierarchy Defining the highway network: minimal network that preserves all shortest paths s t fully automatic (just fix neighborhood size) uncompromisingly fast

11 Sanders/Schultes: Route Planning 11 Our Approach Exact Highway Hierarchy s t complete search in local area search in (sparser) highway network contract network, e.g., iterate highway hierarchy s t

12 Sanders/Schultes: Route Planning 12 A Meaning of Local choose neighbourhood radius r(s) e.g. distance to the H-closest node for a fixed parameter H define neighbourhood of s: N (s) := {v V d(s,v) r(s)} example for H = rank s = s 3 N(s) 5 2 6

13 Sanders/Schultes: Route Planning 13 Highway Network N (s) N (t) s t Highway Edge (u,v) belongs to highway network iff there are nodes s and t s.t. (u,v) is on the canonical shortest path from s to t and v N (s) and u N (t)

14 Sanders/Schultes: Route Planning 14 Contraction highway nodes and edges

15 Sanders/Schultes: Route Planning 15 Contraction bypass node

16 Sanders/Schultes: Route Planning 16 Contraction shortcuts

17 Sanders/Schultes: Route Planning 17 Contraction bypass node

18 Sanders/Schultes: Route Planning 18 Contraction

19 Sanders/Schultes: Route Planning 19 Contraction enters component component (of bypassed nodes) leaves component

20 Sanders/Schultes: Route Planning 20 Contraction core

21 Sanders/Schultes: Route Planning 21 Contraction Which nodes should be bypassed? Use some heuristic taking into account the number of shortcuts that would be created and the degree of the node.

22 Sanders/Schultes: Route Planning 22 Construction Example: Western Europe, bounding box around Karlsruhe

23 Sanders/Schultes: Route Planning 23 Complete Road Network

24 Sanders/Schultes: Route Planning 24 Highway Network

25 Sanders/Schultes: Route Planning 25 Contracted Network (Core)gy

26 Sanders/Schultes: Route Planning 26 Highway Hierarchy: Level 1 and Level 2

27 Sanders/Schultes: Route Planning 27 Fast Construction Phase 1: Construction of Partial Shortest Path Trees For each node s 0, perform an SSSP search from s 0. A node s state is either active or passive. active s 0 is active. passive A node inherits the state of its parent in the shortest path tree. s 0 If the abort condition is fulfilled for a node p, p s state is set to passive. The search is aborted when all queued nodes are passive.

28 Sanders/Schultes: Route Planning 28 Fast Construction Abort Condition: Path P : N (s 1 ) s 0 s 1 p N (p) p is set to passive iff s 1 p p N (s 1 ) s 0 N (p) P N (s 1 ) N (p) 1

29 Sanders/Schultes: Route Planning 29 Fast Construction, Phase 2 Theorem: The tree roots and leaves encountered in Phase 1 witness all highway edges. The highway edges can be found in time linear in the tree sizes. N (s 0 ) N (t 0 ) leaf of partial s 0 t 0 shortest path tree Highway

30 Sanders/Schultes: Route Planning 30 Fast Construction Problem: very long edges, e.g. ferries

31 Sanders/Schultes: Route Planning 31 Faster Construction Solution: An active node v is declared to be a maverick if d(s 0,v) > f r(s 0 ). When all active nodes are mavericks, the search from passive nodes is no longer continued. superset of the highway network mavericks active s 0 f r(s 0 ) passive

32 Sanders/Schultes: Route Planning 32 Space Consumption Choose neighborhood sizes such that levels shrink geometrically linear space consumption Arbitrarily Small Constant Factor (not implemented): Large H 0 large level-0 radius small higher levels No r( ) needed for level-0 search (under certain assumptions) Mapping to next level by hash table

33 Sanders/Schultes: Route Planning 33 Query Bidirectional version of Dijkstra s Algorithm Restrictions: level 1 N (v) Do not leave the neighbourhood of the entrance point to the current level. s v level 0 N (s) Instead: switch to the next level. Do not enter a component of bypassed nodes. entrance point to level 0 entrance point to level 1 entrance point to level 2

34 Sanders/Schultes: Route Planning 34 Query (bidir. Dijkstra I) Operations on two priority queues Q and Q : void insert(nodeid, key) void decreasekey(nodeid, key) nodeid deletemin() node u has key δ(u) (tentative) distance from the respective source node

35 Sanders/Schultes: Route Planning 35 Query (bidir. Dijkstra II) query(s,t) { Q.insert(s,0); Q.insert(t,0); while ( Q Q /0) do { {, }; //select direction u := Q.deleteMin(); relaxedges(, u); } }

36 Sanders/Schultes: Route Planning 36 Query (bidir. Dijkstra III) relaxedges(, u) { } } foreach e = (u,v) E do { k := δ(u) + w(e); if v Q then Q.decreaseKey(v,k); else Q.insert(v,k);

37 Sanders/Schultes: Route Planning 37 Query (Hwy I) Operations on two priority queues Q and Q : void insert(nodeid, key) void decreasekey(nodeid, key) nodeid deletemin() node u has key (δ(u), l(u), gap(u)) (tentative) distance from the respective source node search level gap to the next neighbourhood border lexicographical order: <, >, <

38 Sanders/Schultes: Route Planning 38 Query (Hwy II) query(s,t) { Q.insert(s, (0,0,r 0 (s))); Q.insert(t, (0,0,r 0 (t))); while ( Q Q /0) do { {, }; //select direction u := Q.deleteMin(); relaxedges(, u); } }

39 Sanders/Schultes: Route Planning 39 Query (Hwy III) relaxedges(, u) { foreach e = (u,v) E do { gap := gap(u); if gap = then gap := r l(u) (u); for (l := l(u); w(e) > gap; l++, gap := r l (u)); // leave component // go upwards if l(e) < l then continue; // Restriction 1 if e enters a component then continue; // Restriction 2 k := (δ(u)+w(e), l, gap w(e)); if v Q then Q.decreaseKey(v,k); else Q.insert(v,k); } }

40 Sanders/Schultes: Route Planning 40 Query N 0 (s) gap(s) shortcut s s 1 s 1 Restriction 2 v s 2 =s 2 Restriction 1 u N 1 (s 1 ) N 2 (s 2 )

41 Sanders/Schultes: Route Planning 41 Query Theorem: We still find the shortest path.

42 Sanders/Schultes: Route Planning 42 Query Example: from Karlsruhe, Am Fasanengarten 5 to Palma de Mallorca

43 Sanders/Schultes: Route Planning 43 Bounding Box: 20 km Level 0 Search Space

44 Sanders/Schultes: Route Planning 44 Bounding Box: 20 km Level 0 Search Space

45 Sanders/Schultes: Route Planning 45 Bounding Box: 20 km Level 1 Search Space

46 Sanders/Schultes: Route Planning 46 Bounding Box: 20 km Level 1 Search Space

47 Sanders/Schultes: Route Planning 47 Bounding Box: 20 km Level 2 Search Space

48 Sanders/Schultes: Route Planning 48 Bounding Box: 20 km Level 2 Search Space

49 Sanders/Schultes: Route Planning 49 Bounding Box: 20 km Level 3 Search Space

50 Sanders/Schultes: Route Planning 50 Bounding Box: 80 km Level 4 Search Space

51 Sanders/Schultes: Route Planning 51 Bounding Box: 400 km Level 6 Search Space

52 Sanders/Schultes: Route Planning 52 Level 8 Search Space

53 Sanders/Schultes: Route Planning 53 Level 10 Search Space

54 Sanders/Schultes: Route Planning 54 Optimisation: Distance Table Construction: Construct fewer levels. e.g. 4 instead of 9 Compute an all-pairs distance table for the topmost level L entries

55 Sanders/Schultes: Route Planning 55 Distance Table Query: s t Abort the search when all entrance points in the core of level L have been encountered. Use the distance table to bridge the gap. 55 for each direction entries

56 Sanders/Schultes: Route Planning 56 Implementation C++ heavy use of templates reused binary heap priority queue reused DIJKSTRA lines of code incl. comments, excl. infrastructure (I/O, logging,...)

57 Sanders/Schultes: Route Planning 57 Implementation adjacency array graph representation forward + backward directed edges separate array for level specific data nodes r 0 r 0 level nodes r 1 r 2 r 3 r 4 r 1 r 2 r 3 edges

58 Sanders/Schultes: Route Planning 58 Experiments W. Europe (PTV) USA/CAN (PTV) #nodes #directed edges construction [min] search time [ms] speedup ( DIJKSTRA) 7 232

59 Sanders/Schultes: Route Planning 59 Shrinking of the Highway Networks Europe H = 40 H = 60 H = #edges level

60 Sanders/Schultes: Route Planning 60 Neighbourhood Size Preprocessing Time [min] Memory Overhead per Node [byte] Europe USA/CAN USA Query Time [ms]

61 Sanders/Schultes: Route Planning 61 Number of Levels Europe Memory Overhead per Node [byte] Memory Overhead Query Time Query Time [ms] Number of Levels

62 Sanders/Schultes: Route Planning 62 Contraction Rate Europe # Settled Nodes Settled Nodes Relaxed Edges # Relaxed Edges Contraction Rate

63 Sanders/Schultes: Route Planning 63 Queries Time Query Time [ms] Europe USA/CAN Dijkstra Rank

64 Sanders/Schultes: Route Planning 64 Queries Speedup Speedup (CPU Time) Europe USA/CAN Dijkstra Rank

65 Sanders/Schultes: Route Planning 65 Search Space Europe USA/CAN USA (Tiger) Frequency Search Space

66 Sanders/Schultes: Route Planning 66 Distance Table Accesses Europe USA/CAN USA (Tiger) Frequency #Accessed Entries

67 Sanders/Schultes: Route Planning 67 Summary exact routes in large street networks e.g. 18 million nodes quality advantage, advertisement argument fast search < 1 ms cheap, energy efficient processors in mobile devices low server load lots of room for additional functionality fast preprocessing low space consumption 20 min data base no manual postprocessing of data less dependence on data sources organic enhancement of existing commercial solutions

68 Sanders/Schultes: Route Planning 68 Work in Progress computation of M N distance tables joint work with [Knopp, Schulz] 1,2 combination with a goal directed approach (landmarks) joint work with [Delling, Holzer] 1 1 Universität Karlsruhe, Algorithmics I Group 2 PTV AG

69 Sanders/Schultes: Route Planning 69 Future Work fast, local updates on the highway network (e.g. for traffic jams) implementation for mobile devices (flash access... ) multi-criteria shortest paths joint work with [Müller-Hannemann, Schnee] 3 flexible objective functions 3 Technische Universität Darmstadt, Algorithmics Group

70 Sanders/Schultes: Route Planning 70 Industrial Cooperations We help transforming technology into products: consulting... Joint projects for further features...

71 Sanders/Schultes: Route Planning 71 Road Network of Europe

72 Sanders/Schultes: Route Planning 72 Road Network USA/Can

73 Sanders/Schultes: Route Planning 73 Canonical Shortest Paths SP : Set of shortest paths SP canonical P = s,...,s,...,t,...,t SP : s t SP

74 Sanders/Schultes: Route Planning 74 Canonical Shortest Paths N (s 1 ) N (t 1 ) 0 N (s 0 ) 0 1 N (v ) u 1 2 v s 2 0 s 1 t 2 1 t u v (a) Construction, started from s 0. N (s 1 ) u 1 v N (t 0 ) 2 2 N (u ) s 2 0 s 1 t 2 1 t u v 0 N (v) (b) Construction, started from s 1.

75 Sanders/Schultes: Route Planning u 1 2 v s 2 0 s t t u v (c) Result of the construction.

76 Sanders/Schultes: Route Planning 76 Abort Condition: Efficient Testing. Fast Construction Equivalent Formulation: N (s 1 ) s 0 s 1 u v w p N (p) p is set to passive iff d(s 1,v) r(s 1 ) < d(s 1,w) v p d(u, p) > r(p)