Algorithms for Route Planning in Transportation Networks Dorothea Wagner j FACULTY FOR INFORMATICS INSTITUTE FOR THEORETICAL INFORMATICS · KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Motivation important application, e. g., navigation systems for cars, Google Maps, Bing Maps, . , timetable information. many commercial systems use heuristic methods, consider “reasonable” part of the network, have no quality guarantees. Find methods for route planning in transportation networks with provably optimal solutions regarding the quality of the routes. Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Problem request: find the best connection in a transportation network idea: network as graph G = (V ; E) edge weights are travel times shortest paths in G equal quickest connections classic problem (Dijkstra) problems: transport networks are huge Dijkstra too slow( > 1 second) Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Speed-Up Techniques observations: Dijkstra visits all nodes closer than the target unnecessary computations many requests in a hardly changing network idea: two-phase algorithm: offline: compute additional data during preprocessing online: speed-up query with this data 3 criteria: preprocessing time and space, speed-up over Dijkstra Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Showpiece of Algorithm Engineering Design E x p e e r z i m y l Algorithmics e a n t n A t Implemen Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Showpiece of Algorithm Engineering Realistic Real-world Data machine models Design E x p e e r z i Falsifiable m y l e a Hypotheses n t n A t Implemen Performance guarantees & algorithm dependability Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Shortcuts [SWW99, SS05, GSSD08] observation: nodes with low degree are not important contract graph iteratively remove such nodes add shortcuts to preserve distances between non-removed nodes query: bidirectional prune edges heading less important nodes Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Contraction Hierarchies [GSSD08] idea: solely use contraction approach: heuristically order nodes by “importance” contract nodes in that order node v contracted by 1 forall edges (u; v) and (v; w) do 2 if (u; v; w) unique shortest path then 3 add shortcut (u; w) with weight len(u; v) + len(v; w); query only looks at edges to more important nodes Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Example: CH Preprocessing 2 3 2 1 2 2 6 1 4 3 5 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Example: CH Preprocessing 2 5 1 2 2 6 4 3 5 3 2 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Example: CH Preprocessing 5 1 2 6 4 3 5 2 3 2 2 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Example: CH Preprocessing 5 3 6 4 5 1 2 2 3 2 3 2 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Example: CH Preprocessing 8 6 5 5 3 2 2 3 4 1 3 2 2 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics Example: CH Preprocessing 6 8 5 3 5 2 3 4 1 2 3 2 2 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics CH Query modified bidirectional Dijkstra upward graph G" := (V ; E") with E" := (u; v) E : u < v f 2 g downward graph G# := (V ; E#) with E# := (u; v) E : u > v f 2 g forward search in G" and backward search in G# 6 8 5 3 5 2 3 4 1 2 3 node order 2 2 t s 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics CH Query modified bidirectional Dijkstra upward graph G" := (V ; E") with E" := (u; v) E : u < v f 2 g downward graph G# := (V ; E#) with E# := (u; v) E : u > v f 2 g forward search in G" and backward search in G# 6 8 5 3 5 2 3 4 1 2 3 node order 2 2 t s 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics CH Query modified bidirectional Dijkstra upward graph G" := (V ; E") with E" := (u; v) E : u < v f 2 g downward graph G# := (V ; E#) with E# := (u; v) E : u > v f 2 g forward search in G" and backward search in G# 6 8 5 3 5 2 3 4 1 2 3 node order 2 2 t s 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics CH Query modified bidirectional Dijkstra upward graph G" := (V ; E") with E" := (u; v) E : u < v f 2 g downward graph G# := (V ; E#) with E# := (u; v) E : u > v f 2 g forward search in G" and backward search in G# 6 8 5 3 5 2 3 4 1 2 3 node order 2 2 t s 1 Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Practice to Theory: WeakCH Question: What is a good contraction order? up to now: solely heuristical [GSSD08] no guarantees WeakCH [BCRW13] balanced separator nodes are important resulting CH is called weak ! O(nα) separators O(nα) nodes in the search space ! order is independent of metric Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Practice to Theory: WeakCH Question: What is a good contraction order? up to now: solely heuristical [GSSD08] no guarantees WeakCH [BCRW13] balanced separator nodes are important resulting CH is called weak ! O(nα) separators O(nα) nodes in the search space ! order is independent of metric Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics New Challenges realistic customizable routes: user customizable metrics e.g., height restrictions, avoid freeways, eco-friendliness, . fast customization time per metric very small space overhead timetable information: consider public transportation networks develop new techniques robustness towards the input? multi-modal routes: change the type of transportation during the journey allow only “reasonable” transfers several constraints to the shortest path Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric ∞ some arc in the CH Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric ∞ 4 4 establish lower triangle inequality Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric 8 4 4 establish lower triangle inequality Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric 8 do this for all lower triangles Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric node order process arcs increasing by order Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric node order process arcs increasing by order Dorothea Wagner – Algorithms for Route Planning in Transportation Networks Institute for Theoretical Informatics Chair Algorithmics From Theory to Practice: Customizable Contraction Hierachies [DSW14] idea: CH topology is the same regardless of metric quickly introduce new metric node order process arcs increasing by order Dorothea Wagner – Algorithms for Route Planning in Transportation