1. Route Planning Tabulation Dijkstra Bidirectional A* Landmarks Reach
ArcFlags
Transit Nodes
Contraction Hierarchies
Hub-based labelling
Good overview in JEA10 paper Bauer,Delling,Sanders,Schieferdecker,Schultes,Wagner. Combining Hierarchical and Goal-Directed Speed-Up Techniques for Dijkstra’s algorithm.
Slides with more detailed proofs:
[ADGW11] Ittai Abraham, Daniel Delling, Andrew V. Goldberg, Renato Fonseca F. Werneck. A Hub-Based Labeling Algorithm for Shortest Paths in Road Networks. Proc. 10th International Symposium on Experimental Algorithms (SEA), LNCS 6630, 2011,
[BFMSS07] Holger Bast, Stefan Funke, Domagoj Matijevic, Peter Sanders, and Dominik Schultes. In Transit to Constant Time Shortest-Path Queries in Road Networks. Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments (ALENEX), 2007.
[GSSD08] Robert Geisberger, Peter Sanders, Dominik Schultes, and Daniel Delling.
Contraction Hierarchies: Faster and Simpler Hierarchical Routing in Road Networks.
Proc. 7th International Workshop on Experimental Algorithms (WEA), LNCS 5038, 2008,

2. Route Planning
Input: Directed weighted graph G Query(s,t) – find shortest route in G from s to t Lot of algorithm engineering work for road networks Example: US Tigerline, 58 M edges & 24 M vertices
No preprocessing
With preprocessing
Fast query time
Variations of Dijksta’s algorithm
Query Time ↔ Space trade-off
Trivial: Distance table O(1) time & O(n2) space
Practice: Try to exploit graph properties

3. Route Planning – no preprocessing (non-negative edge weights)
Dijkstra Build shortest path tree T Visit vertices in increasing distance to s Bidirectional Dijkstra Grow s.p. tree Tf from s and Tb to t Maintain best so far s→t distance μ Termination condition: nextf + nextb  μ
Dijkstra – uses efficient priority queue to find next vertex to include into T; maintain for each vertex tentative distance
Bidirectional Dijkstra:
μ initially , update when relaxing eges
nextf + nextb are the distance to the next vertices to be included in Tf and Tb
Correctness of termination: The shortest path p has lengthμ and contains an edge (v,w), where v is in Tf, but w has distance nextf. But then distance w to t is μ-nextfnextb and w must be in Tb, i.e. the edge (v,w) has been relaxed to dist(p) when vTf and wTb Tf s t G Tb u

4. Dijkstra vs Bidirectional Dijkstra
Bidirectional Dijkstra (example area Washington state)

5. A*  Goal directed Input: Weighted graph G with non-negative edges
Query(s,t): Shortest route queries
Idea Let h(v) be ”heights” & define w’(u,v) = w(u,v) + h(v) - h(u)
Fact w’(s→v1vk→t) = w(s→v1vk→t) + h(t) - h(s)  G and G’ have identical shortest paths
Fact If w’0  we can use Dijkstra’s algorithm If w’0 and h(t)=0  h(v) lower bound on distance v→t
Ex. 1 Planar graphs with L2 distance, let h(v) = |t-v|2  triangle inequality ensures w’ non-negative
Ex. 2 h(v) = dG(v,t)  w’(s,t) =  Dijkstra’s algorithm would only explore the shortest path
Note Bidirectional A*  Bidirectional Dijkstra and A* combined
Intuitive idea: We would like to run ”downhill”, i.e. t should have low height and s high height
Height can also be thought of as a potential
Ex. 2 – not practical example: Knowing dG(v,t) would require computing the shortest paths. But it motivates landmarks.
Bidirectional A* can also use two different height functions, but then definition and termination condition more envolved, and the height functions must be consistent t v u
h(u)
h(v)
w(u,v)

6. A* Washington DC -> Los Angeles, h = spherical distance

7. Landmarks Select a small number of vertices L (Landmarks)
For all nodes v store distance vector d(v,l) to all landmarks lL
Idea In A* algorithm fix one landmark lL, and use h(v) = d(v,l) (valid by triangle inequality)
Practice: Use more than one landmark to find lower bounds on d(v,t) Dynamicly increase landmark set during search Bidirectional A* s l t v
d(t,l)
d(v,l)
d(v,t)
w’=0
Space increases with number of landmarks !

8. Bidirectional A* with Landmarks
Werneck & Goldberg were at MicroSoft
Yellow & Red = landmarks, Yellow = used landmarks during the bidirectional search

9. Reach For all nodes v store
Reach(v) = max(s,t) : v on shortest path s→t min{ d(s,v), d(v,t) }
Idea Reach(v) defines ball around v. If both s and t outside ball, v is not on shortest path
Query Prune edges (u,v) in Dijkstra, when relaxing (u,v) and
Reach(v) < min{ d(s,u)+w(u,v) , LowerBound(v,t) }
Practice: Approximate Reach for fast preprocessing
d(s,v) v
d(v,t) s t

10. Reach(v)
Previous examples: Colors did not illustrate visited edges, but vertices visited were very close

11. Shortcuts
A directed path u→v can be shortcut by a new edge (u,v) Idea: Shortcuts reduce Reach(x) of vertices x along the shortcut path (s→t distances are unchanged) u x s v t
shortcut

12. Reach(v) + Shortcuts

13. Reach(v) + Shortcuts + Landmarks

14. Experiments – Northwest US

15. Arc Flags Queries Prune edges e where Afe[i] = false and tCi
Partition vertices into k components C1,...,Ck.
For all edges e = (u,v) store a bitvector Afe[1..k], where
Afe[i] = true  Exist shortest path u→t where e is first edge and tCi
Queries Prune edges e where Afe[i] = false and tCi
Preprocessing Expensive ! C4 C2 C3 Ci G C1 u v t e Ck

16. Practice: Combine recursively with Highway Hierarchies
Transit Node Routing
recurse u v
Idea All shortest paths s→t, where s and t are far away, must cross few possible transit nodes
Identify few transit nodes in graph ~ n
Compute All-Pair-Shortest-Path matrix for transit nodes
For each vertex s find very few transit node distances (US ~10)
Query(s,t) far away queries For all (u,v), transit nodes u and v for s and t respectively, find d(s,t)=d(s,u)+d(u,v)+d(v,t) using table lookup
Locality filter = table over when to switch to other algorithm
Practice: Combine recursively with Highway Hierarchies s t

17. Transit Node Routing
Holger Bast, Stefan Funke, Domagoj Matijevic, Peter Sanders, and Dominik Schultes. In Transit to Constant Time Shortest-Path Queries in Road Networks. Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments (ALENEX), 2007.

18. Highway Hierachies For each node find H closest nodes (Neighborhood)
Highway edge (u,v)  exist some shortest path s→t containing (u,v), where sH and tH
Contract & Recurse  Hierarchy
Queries
Heuristics similar to Reach
Bidrectional Dijkstra (skipping lower level edges)

19. Contraction Hierarchies
Order nodes v1,...,vn in increasing order of importance
Repeatedly contract unimportant nodes by adding shortcuts required by shortest paths
Many heuristics in construction phase
Query: Bidirectional – only go to more important nodes

20. Hub Labelling
For all nodes v store two lists Lf(v) and Lb(v), such that for all (s,t) pairs, the shortest path s→t contains a node u, where uLf(s)Lb(t) Trivially exist; hard part is to limit space usage u s t
Lf(s) u
d(s,u)
Lb(t) u
d(u,t)
small
small

21. Hub Labelling comparison
Contraction hierarchies
Arc flags
Transit node
Table from page 239
Conclusion: Best query time for Hub labelling approach ~ time for 5 memory access (cache lines); space still reasonable
HL = Hub Labelling (prefix = space optimized; global/local = optimized for global/local queries)
CH = Contraction Hierarchies
CHASE = Contraction Hierarchies + Arc flags
HPML = High Performance Multi Level Routing
TNR = Transit Node Routing (TNR+AF = Transit Node Routing + ArcFlags + Contraction Hierarchies)
Hub labelling
Ittai Abraham, Daniel Delling, Andrew V. Goldberg, Renato Fonseca F. Werneck. A Hub-Based Labeling Algorithm for Shortest Paths in Road Networks. Symposium on Experimental Algorithms (SEA), Lecture Notes in Computer Science, Volume 6630, 2011, pp