寻路算法 Transit Node Routing based on Highway Hierarchies

Transit Node Routing based on Highway Hierarchies Peter Sanders Dominik SchultesRoute PlanningGoals:exact shortest(i.e.fastest)paths in large road networks fast queriesfast preprocessinglow space consumptionApplications:route planning systems in the internetcar navigation systems···Motivation‘Problem’:existing solutions are already‘too fast’.Example:perform a query(using hwy.hierarchies):≈1msvisualise the path(using our Java application):≈400msCounter-Argument:applications that require a lot of queries(and only a few paths)massive traffic simulationsoptimisations in logistics systemsKarlsruhe→CopenhagenKarlsruhe→BerlinKarlsruhe→ViennaKarlsruhe→MunichKarlsruhe→RomeKarlsruhe→ParisKarlsruhe→LondonKarlsruhe→BrusselsKarlsruhe→CopenhagenKarlsruhe→BerlinKarlsruhe→ViennaKarlsruhe→MunichKarlsruhe→RomeKarlsruhe→ParisKarlsruhe→LondonKarlsruhe→BrusselsFirst ObservationFor long-distance travel:leave current locationvia one of only a few‘important’traffic junctions,called access points( we can afford to store all access points for each node) [in Europe:about10access points per node on average]Karlsruhe→BerlinKarlsruhe→BerlinKarlsruhe→BerlinSecond ObservationEach access point is relevant for several nodes.union of the access points of all nodes is small,called transit node set( we can afford to store the distances between all transit node pairs) [in Europe:about10000transit nodes]Transit Node RoutingPreprocessing:identify transit node set T⊆Vcompute complete|T|×|T|distance tablefor each node:identify its access points(mapping A:V→2T),store the distancesQuery(source s and target t given):computed top(s,t):=min{d(s,u)+d(u,v)+d(v,t):u∈A(s),v∈A(t)}Transit Node RoutingLocality Filter:local cases must befiltered( special treatment) L:V×V→{true,false}¬L(s,t)implies d(s,t)=d top(s,t)ExampleRelated Workseparator-based implementation[Müller et al.2006]–determine separator nodes(=transit nodes)–partition the graph into small components–access points of node u:border nodes of u’s component–localityfilter:“same component?”grid-based implementation[Bast,Funke,Matijevic2006]–compute geometric subdivision of the network into cells–access points:border nodes needed for‘long-distance’travel–transit nodes:union of all access points–localityfilter:“less than a certain number of grid cells apart?”Our Approach:Highway Hierarchies1complete search within a local areasearch in a (thinner )highway network =minimal graph that preserves all shortest pathscontract network,e.g.,iterate highway hierarchyExample:Karlsruhe123Local Areachoose neighbourhood radius r(s)(by a heuristic)define neighbourhood of sN(s):={v∈V|d(s,v)≤r(s)}Highway NetworkEdge(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 tand(u,v)is not entirely within N(s)or N(t)ContractionDistance Table:Search Space ExampleDistance TableCompute an all-pairs distance tablefor the core of the topmost levelℓ.13465×13465entriesAbort the search when all entrance points in thecore of levelℓhave been encountered.≈55for each direction Use the distance table to bridge the gap.≈55×55entriesHH-based Transit Node RoutingCompute an all-pairs distance tablefor the core of the topmost levelℓ.13465×13465entries transit node set TAbort the search when all entrance points in thecore of levelℓhave been encountered.≈55for each direction do not‘search’,just perform look-upsUse the distance table to bridge the gap.≈55×55entriesMissing Pieces1.localityd(s,t)=d u(s,t)<d top(s,t) L(s,t):=“disks of s and t overlap”Missing Pieces2.too many‘entrance points’(55)solution:fall back on comparatively few‘access points’(10) (motivated by the observations from[Bast,Funke,Matijevic2006])Missing Piecespute top distance table in the original graphsolution:[Knopp,S,S,Schulz,Wagner2007]“Computing Many-to-Many Shortest PathsUsing Highway Hierarchies”(e.g.10000×10000table in one minute)for each t∈T,perform backward search up to top levelℓ,store search space entries(t,u,d(u,t))arrange search spaces:group entries by ufor each s∈T,perform forward search,at each node u,scan all entries(t,u,d(u,t))andcompute d(s,u)+d(u,t)Second Layer(to deal with medium range queries)secondary transit node set T2⊃Tsecondary access mapping A2:V→2T2d(u,v):u,v∈T2∧d(u,v)=d top(u,v) secondary dist.tablesecondary localityfilter L2¬L2(s,t)impliesd(s,t)=d top(s,t)ORd(s,t)=min{d(s,u)+d(u,v)+d(v,t):u∈A2(s),v∈A2(t)}Two Concrete VariantsPreprocessingfor each secondary transit node t∈T2:–perform backward highway search–stop at primary transit nodes( set of backward access points)–store search space entries(t,u,d(u,t))arrange search spacesPreprocessingfor each secondary transit node s∈T2:–perform forward highway search–stop at primary transit nodes( set of forward access points)–at each node u and for each search space entry(t,u,d(u,t)):∗compute distance d u(s,t):=d(s,u)+d(u,t)via u∗compute distance d top(s,t)via top layer∗if d u(s,t)<d top(s,t)then·add entry d u(s,t)to the secondary distance table·ensure that the disks around s and t contain u(use similar procedure for lower layers)ExperimentsW.Europe(PTV)USA(TIGER/Line)#nodes#directed edges#road categoriesspeed range[km/h]Preprocessinglayer2|T||A|space timeUSA timeeco184379 4.910674 5.72443:25 dist eco10235210.9EUR timeeco118356 5.5112939.92512:44 dist eco6977513.1Querylayer2[%]wrong cont’d time0.14 1.1311.5µs gen0.00140.01381.578.1087.5µs0.54 2.8713.4µs gen0.00150.01904.6818.32107.4µsDijkstra RankQ u e r y T i m e [µs ]2526272829210211212213214215216217218219220221222223224510204010030010005102040100300100Local Queries:USA,travel time metricDetermine Shortest Pathsfrom source s,look iteratively for the next adjacent node s′thatleads to fwd acc pnt ud(s,s′)+d(s′,u)=d(s,u)analogously,from target to bwd acc pntfrom fwd acc pnt u,look iteratively for the next adjacent node u′in the topmost level that leads to bwd acc pnt vd(u,u′)+d(u′,v)=d(u,v)(if necessary,use a hidden path instead)unpack shortcutsif path not through top layer,fall back on highway query。