Route Planning

Route PlanningVehicle navigation systems,

Dijkstra’s algorithm, bidirectional search, transit-node routing

Vehicle navigation systems• Main tasks:

– positioning: locating the vehicle using GPS and/or dead reckoning with distance and heading sensors

– routing: determining a good route from a source to a destination

– guidance: providing visual and audio feedback on the route

Positioning• GPS: works well, except in “urban

canyons”• Urban canyons can give gross

errors in position due to reflections from buildings

• Urban canyons can give loss of signal

• Especially problematic when driving out of parking garages

Positioning• Dead reckoning: determine position from last known

position using distance and heading sensors (relative position)

• Use map matching: the shape of the route taken and where it matches on the map, to correct dead reckoning

Guiding• Top view, perspective view, overview• Schematic information on exit lanes• Spoken directions

Largely an HCI issue

Routing• Based on Dijkstra’s shortest path algorithm• Many improvements to deal with huge networks• Improvements use preprocessing

Routing• Based on Dijkstra’s shortest path algorithm• Many improvements to deal with huge networks• Improvements use preprocessing

Routing

Bidirectional search (from Bayreuth and from Erlangen)

• Based on Dijkstra’s shortest path algorithm• Many improvements to deal with huge networks• Improvements use preprocessing

Routing

Bidirectional search (from Bayreuth and from Erlangen)

• Based on Dijkstra’s shortest path algorithm• Many improvements to deal with huge networks• Improvements use preprocessing

Routing• Dijkstra’s algorithm takes O(n + m log m) time for a

graph with n nodes and m edges

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shortest path tree

Routing• Dijkstra’s algorithm takes O(n + m log m) time for a graph

with n nodes and m edges– Every node is handled only once– Its outgoing edges are considered only then– Considering an edge may lower the cost of its destination node– Nodes are stored by distance in a Fibonacci heap (it allows for a

very efficient decrease-value operation)

• Road networks have m = O(n), so it takes O(n log n) time

Routing• Very fast shortest path queries are needed in vehicle

navigation systems and by Google Maps• Idea: pre-compute the shortest path for every pair of

nodes and store it in a table

X much too much storage needed• Need other ways to answer shortest path queries faster

– Highway hierarchies– Transit-node routing

Transit-Node Routing• For a real road network, there exists a relatively small

set of nodes, such that any shortest path of sufficient length will pass at least one of them

• For the Netherlands, every shortest path of at least 100 km will use a highway, so we can take all highway exits

Highways and other major roads; every shortest path in the original road network of at least 60 km will use a highway or other major road

60 km

Transit-Node Routing• This relatively small set of nodes is called the set of

transit nodes

• Furthermore, for every node, there are (typically) only few nodes that are the first transit nodes encountered when going far enough (access nodes)

• For the USA, the road network has 24 million nodes and 58 million edges

• Transit-node routing uses 10,000 transit nodes and for each node there are ~10 access nodes

Transit-Node Routing• Store all distances between two transit nodes in a table• For every node, store the distance to its ~10 access

nodes in a table

• Use table look-up to determine shortest paths, if the distance between source and target is large enough– for the source and target, get the access nodes and distances– for every pair [access node of source, access node of target],

determine a candidate path length by 3 table look-ups

• If the distance is small, just run Dijkstra bidirectional

V : nodes of the input graphT : transit nodes chosen

transit nodes

transit nodes

nodes of Vi access nodes of i

transit nodetable

access nodetable/list

Transit-Node Routing• Trade-off:

many transit nodes: fast query time, large storage requirements, high preprocessing time

few transit nodes: slower query time, smaller storage requirements, lower preprocessing time

• Reported (road network USA):query time: 5 – 63 sstorage: 21 – 244 bytes/nodepreprocessing: 59 – 1200 minutes

Transit-Node Routing• Need (for preprocessing):

– a way to choose transit-nodes– a way to determine access nodes– a way to compute the transit node table and access node table

• Need (at query time):– a way to decide if a query is local ( use Dijkstra)

or not ( use table look-up)– a way to retrieve the shortest path itself

Choosing transit nodes• Use grid-based

partition and use intersections of the network and the grid

Choosing transit nodes• Use grid-based

partition and use intersections of the network and the grid

• Select nodes from Vinner based on whether they lie on some shortest path from a node in C to Vouter

C

inner

outer

Choosing transit nodes• Consider every center square C• Use 5 x 5 squares to define Vinner

• Use 9 x 9 squares to define Vouter

• Consider all paths from some node in C to some node in Vouter

• All nodes of Vinner on such a path will go in the set of transit nodes (eventually united over all C)

• Run Dijkstra from every node in C until all nodes in Vouter are settled

Choosing transit nodes• Given s and t, if they are more than 4 grid cells apart

(horizontally or vertically), their shortest path must contain a transit node

• This provides an easy test for locality of any query (later, during query time)

Computing access nodes• For each transit node u, run Dijkstra until all shortest

paths from u pass another transit node• Every node v in the shortest path tree from u before

another transit node is reached gets u as one of its access nodes

u

vshortest path tree from u

Computing the tables• The access node table is automatically computed when

the access nodes are determined• Distances between “near” transit nodes are also

computed transit node graph• Dijkstra on the transit node graph gives the transit node

table

Summary• Vehicle navigation systems rely on positioning, routing,

and guidance• Route planning relies on Dijkstra’s algorithm and

techniques to speed up queries, like preprocessing using the transit nodes idea

Documents

Route Planning