algorithms of multi-modal route planning based on the ... · is a novel approach.his ideais...

Photogrammetrie • Fernerkundung • Geoinformation 5/2009, S. 431–444

DOI: 10.1127/1432-8364/2009/0031 1432-8364/09/0031 $ 3.50© 2009 DGPF / E. Schweizerbart'sche Verlagsbuchhandlung, D-70176 Stuttgart

Algorithms of Multi-Modal Route Planning Based onthe Concept of Switch Point

Lu Liu & Liqiu Meng, München

Keywords: Multi-Modal Navigation, Shortest Path Algorithm, Network Analysis

street to the Chinese Tower of English Gardenfor a tour with a car. In this case, it does notmake sense to give the result of directly driv-ing the car from the origin to the entrance ofChinese Tower even if some of the streets inthe garden are allowed for cars. Since the useris not allowed to park his car at the door of thetower, he has to find an appropriate place alongthe route where parking is possible. As a re-sult, the route is divided by the parking placeinto two segments with different transporta-tion modes – by car and by foot. This is a sim-ple case of multi-modal route planning prob-lem with two involved travel modes.

1 Introduction

The purpose of multi-modal route planningis to provide the user with an optimal routebetween the source and the target of a trip.The route may utilize several transportationmodes including car driving, public transpor-tation, cycling, walking, etc. (Hochmair 2008).Taking several accessible transportationmodes into account for the travel plan is a verycommon practice in everyday life.

A typical application scenario of multi-modal route planning, for example, is to iden-tify the best route from the garage exit ofTechnische Universität München in Hess

Summary: The paper addresses the task of generi-cally finding the shortest path in multi-modal net-works with the multi-modal route planning problemin transportation field as a special case. The multi-modal networks can be modelled by a data struc-ture based on the core concept of Switch Pointwhich abstracts the places where it is allowed forchanging from one mode to another. Two routingalgorithms Multi-Modal Bellman-Ford (MMBF)and Multi-Modal Dijkstra (MMD) were elicitedwhich are respectively rooted in the classical label-correcting and label-setting methods. Both MMBFand MMD are capable of finding in multi-modalnetworks the shortest paths in spite of differentcomputing complexity. The feasibility of the ap-proach was verified in our prototype system. Theresults of our experiments conducted on real trans-portation networks showed the differences betweenthe proposed algorithms in terms of computing per-formance.

Zusammenfassung: Algorithmen zur multi-moda-len Routenplanung auf Grundlage des Schaltpunk-tes. Der Beitrag befasst sich mit der Aufgabe zumAuffinden der kürzesten Route in einem multi-mo-dalen Netzwerk, wobei die multi-modale Routen-planung aus dem Bereich der Verkehrstechnik alseine spezielle Anwendung betrachtet wird. DieModellierung eines multi-modalen Netzwerks ba-siert auf dem Kernkonzept Schaltpunkt. Bei einemSchaltpunkt handelt es sich um eine Stelle, wo eineModalität auf eine andere umgeschaltet werdendarf. Zwei Routenalgorithmen Multi-Modal Bell-man-Ford (MMBF) und Multi-Modal Dijkstra(MMD) wurden dargestellt. Als Grundlage dazudienen zwei klassische Methoden – „Label-Korrek-tur“ und „Label-Einstellung“. Beide MMBF undMMD sind in der Lage, die kürzeste Route in ei-nem beliebigen multi-modalen Netzwerk aufzufin-den. Die Machbarkeit des Ansatzes zur multimoda-len Routenplanung wurde in einem Prototypensys-tem bestätigt. Ergebnisse aus unseren Untersu-chungen der realen Verkehrsnetzwerke zeigen je-doch, dass sich die beiden Algorithmen hinsichtlichdes Rechenaufwands voneinander unterscheiden.

432 Photogrammetrie • Fernerkundung • Geoinformation 5/2009

Modesti & Sciomachen 1998, Ziliaskopoulos

& Wardell 2000, Lozano & Storchi 2001,Lozano & Storchi 2002, Boussedjra et al.2004, Bielli et al. 2006, Hochmair 2008,Zografos & Androutsopoulos 2008). Theyproposed effective approaches for finding thebest route in static or even dynamic multi-modal transportation networks. Nevertheless,most of the proposed routing algorithms arecoupled with the specific transportation modecombinations, which may limit their accep-tance in a variety of applications, although it istrue that the path should respect a set of con-straints on the sequence of the used modes,i. e., the path must be viable (lozano & storchi

2001). The method proposed by frank (2008)is a novel approach. His idea is to combine thenavigation graph and business graph together,and then apply the traditional shortest path al-gorithms in the product graph. In addition, theresearchers from Universität Karlsruhe havealso made substantial contribution to this top-ic. They proposed a multi-modal path findingmethod based on (regular-) language-con-strained shortest path algorithm (Barrett etal. 2008, Pajor 2009) generalized from (for-mal-) language-constrained algorithm appliedin transportation field by Barrett et al. (1998).Unfortunately, as they remarked in (Delling

et al. 2009), using a fast routing algorithm insuch a label-constrained scenario is very com-plicated and hence, another challenging task.In our previous work (Liu & Meng 2008), wegave a preliminary investigation of the multi-modal extension of Bellman-Ford algorithm,but lacked any theoretical analysis of the pro-posed algorithm. At that time, we didn’t real-ize that our method for generalizing mono-modal Bellman-Ford into multi-modal situa-tion is generic enough that it can also be ap-plied in the label-correcting algorithms (e. g.,Dijkstra’s algorithm).

Our concern with the multi-modal routeplanning is more general. The main purposeof our work is to propose a solution which canfind the optimal route in a static N-modal net-work for an arbitrary mode combination,where N indicates the number of modes. Moreprecisely, we treat the mode combination as apart of the input. To approach such a routingproblem, we have scrutinized the input of themulti-modal route planning and developed a

Another typical scenario which is a little bitmore complicated would be to find an optimalroute in Munich from a suburban place such asSeefeld to the Chinese Tower of English Gar-den for a tour with a car after work. In thiscase, directly driving to a parking place andthen walking to the destination may not be abest choice because there would be trafficjams in the downtown area of Munich. To stopthe car at a suburban train station, and thentravel to the destination by inner-city publictransportation system may be a better solu-tion. The task is not trivial because only anexperienced inhabitant who is familiar withthe traffic context in Munich would tend tomake such a decision. This is a routing prob-lem involving multiple networks with multipleobjectives, constraints and some dynamic andfuzzy information.

A number of mature mono-modal naviga-tion services such as car navigation, pedestri-an navigation, public transportation informa-tion systems which can do route planning for aspecific network are already available on themarket. Some public transportation informa-tion systems can be envisaged as an embry-onic form of multi-modal route planning sys-tems as they can serve the user with an esti-mated travel time and an overview map thatsketches the walking routes from the originallocations to various stations. couckuyt et al.(2006) from Microsoft have patented the basicconcepts of multi-modal navigation and somebasic functions of such a system. And rehrl

et al. (2007) described the requirements of amultimodal transportation routing system inmore detail. hoel et al. (2005) from ESRI pro-posed their approach for efficient modeling ofthe multi-modal network, which was imple-mented as a tool in ArcGIS Network Analysistoolbox. The modeling issue of a multi-modalfreight transportation network was discussedin (southworth & peterson 2000). Their pro-posed data models can be regarded as multi-layer graphs connected by transfer nodes orarcs, while some other researchers (Modesti

& Sciomachen 1998, Boussedjra et al. 2004)built a single graph containing all the informa-tion of different transportation modes. Theproblem of finding the optimal path in a multi-modal network was also being investigated inthe past two decades (Boardman et al. 1997,

L. Liu & L. Meng, Algorithms of Multi-Modal Route Planning 433

ferent modes, mi and mj as two distinctive ele-ments of M, only the points satisfying somespecial conditions are eligible to be the SwitchPoints from mi to mj . All the special condi-tions can be expressed by a N × N Switch PointMatrix SPM (see Fig. 1). The matrix elementsare denoted by λSP

mi mj, λSPmi mj = nil when mi = mj.

That means all the values on the diagonal ofSPM are nil because it is meaningless toswitch from a mode to itself. λSP

mi mj, mi ≠ mj in-dicates the condition that should be satisfiedfor the switching from mi to mj. A practical ex-ample of λSP

mi mj is that if mi and mj indicate themodes of car driving and walking respective-ly, then λSP

mi mj, should be “the place is allowed topark a car”.

Fig. 1: Switch Point Matrix (SPM).

According to the graph theory, a graph is com-posed of a vertex set V and an edge set E,which is denoted by G = {V, E}. λSP

mi mj can beexpressed by some special attributes of verti-ces in G. Therefore, a vertex υ is the SwitchPoint between two different modes mi and mj,if and only if

1) υ is accessible by both Gmi and Gmj;2) λ (υ) = λSP

mi mj.

where Gmi and Gmj denote the graphs of modemi and modemj respectively, λ is a functionand can return the attribute of a vertex.

The υSPmi mj between mode mi and mode mj is

not unique. All the υSPmi mj constitute a set of

Switch Points. The fact that | VSPmi mj | > 1 in most

cases indicates the main difficulty of multi-modal route planning. If the Switch Points be-tween any two modes can be uniquely deter-mined, i. e., | VSP

mi mj | ≡ 1, mi, mj M, i ≠ j, theproblem can be reduced to finding the shortestpaths that visit specified intermediate nodes ina multi-modal network. This is a typical kind

corresponding mathematic model. Based onthis model, two algorithms which can solve ageneral problem of finding the multi-modalshortest path are implemented. The main con-tributions of this paper consist of: 1) a datamodel based on the concept of Switch Pointfor modeling the multi-modal networks; 2)two multi-modal shortest path algorithms gen-eralized from the classical mono-modal short-est path algorithms, and the generalizationsare based on the same basic principle; 3) a pro-totype system in which the proposed datamodel and algorithms are implemented.

The paper is organized as follows. In Sec-tion 2 we present the definition of Switch Pointwhich is the core concept of our data modeland its matrix expression, propose the multi-modal graph set containing the Switch Points,describe the multi-modal shortest path prob-lem and present its mathematical formulation.A general multi-modal shortest path problemcan be solved by Multi-Modal Bellman-Ford(MMBF) and Multi-Modal Dijkstra (MMD).The two proposed algorithms are described,demonstrated and analyzed in detail in Sec-tion 3. The experiment results related to thecity of Munich in our prototype system areshown in Section 4. Finally, Section 5 givesthe conclusions and an outlook.

2 Data Model

2.1 Switch Point

In modern urban transportation systems,many inter-modal facilities such as parkingplaces, park and ride lots, transit hubs, trail-heads, etc. are provided besides the basictransportation networks. These facilities makeit easy to transfer between different transpor-tation modes. With the collection and digitali-zation of inter-modal facilities from the realworld and the integration of their informationwith navigational databases, it becomes pos-sible to conduct automatic multi-modal routeplanning.

In our multi-modal data model, the inter-modal facilities are abstracted as points wherea travel mode can switch from one to another.Therefore, we call them Switch Points. Gener-ally speaking, given a set M containing N dif-


parking place connecting the motorized net-work and the pedestrian network.Switch Point is significant to the multi-mo-

dal route planning problem. A network datasetcan not support multi-modal routing applica-tion without the information of Switch Point.The definition of a multi-modal shortest pathproblem and its solutions are all based on theconcept of Switch Point.

From the transportation point of view,Switch Point is an abstraction of the place inthe real world where people can change fromone transportation mode to another. We canenumerate different Switch Points of differenttraffic mode-pairs in a table such as Tab. 1which we call SPM-T (SPM in Transporta-tion). Given an ordered transportation modetuple (mi, mj ), we can get the values ofSPM-T (mi, mj ) by looking up the matrix.

With Tab. 1 we do not attempt to enumerateall possible traffic modes which may for ex-ample include airplane, ship, inter-city train,bicycle, roller skating, etc. Instead, we focuson the most usual ones in our everyday life.The SPM-T has two properties:

SPM-T is asymmetrical.Taking the car driving and walking modesfor example, SPM-T (D, W) is not equal toSPM-T (W, D). The reason is obvious: a drivercan park his car at any available parking placeand transfer to pedestrian mode, however inthe reverse situation, the driver must walk to aplace (e. g., the parking place where his/hercar is parked, or a car rental company) wherethere is a car he/she can use.

SPM-T is scale-dependent.Tab. 1 just shows the case of inner-city mapscale. At a more detailed level, we may findthat there are no switch points between car

of constrained shortest path problem, which isconsidered by (Bajaj 1971) according to thetaxonomy of shortest path algorithms (deo &pang 1984).

For the purpose of route planning in multi-modal networks and their various combina-tions, we have to pay special attention to theSwitch Point. Conceptually, Switch Points aresimilar to point features which can reside inone or more connectivity groups of the multi-modal network model in ArcGIS NetworkAnalysis toolbox (Hoel et al. 2005). However,in our Switch Point-based data model, the va-lidity of a vertex as a Switch Point depends onthe aforementioned two conditions. A υSP

mi mj

connecting two different modes may not begeometrically identical in the two correspond-ing networks. Taking urban transportationnetwork for example, two nodes with distinctgeographic coordinates are eligible to be theSwitch Point connecting the two networks ifthey share some attribute values (e. g., nodeID). Fig. 2 demonstrates a Switch Point whichis a car entrance and a pedestrian exit of a

NodeID: 10002Attribute: parking place(car entrance)

NodeID: 10002Attribute: parking place(pedestrian exit)

Motorized road

Pedestrian way

Fig. 2: Example of a Switch Point expressed bytwo geographically different nodes in two net-works.

Tab. 1: An example of SPM-T. D, P and W denote car driving, public transportation and walkingrespectively.

D P W

D nil P+R lots parking lots for cars

P P+R lots where a car isavailable

nil public transportation stations

W some place where a car isavailable or can be rented

public transportation stations nil


denoting MMGS. λ : V → Λ, V = kM Vkis the vertex label function, and CM ={ck : Ek→ R+, kM} is the set of cost functionsthat map the edges of different modes to posi-tive real-valued costs. The source node andtarget node are denoted by S, SVm1 andT, TVmN respectively. With the list M as in-put, we can get the switch point value λSP

mi mi + 1 =SPM (mi, mi + 1), i [1, N− 1] by looking up inthe SPM.At first, we give the formalized description

of double-modal shortest path problem as asimplified multi-modal shortest path problem.With the inputs listed above, in a double-mo-dal shortest path problem, we are given a se-quential modes list M = m1, m2, m1 ≠ m2 com-posed by two different modes, a set of verticesattributes Λ and a double-graph GM ={Gm1 = {Vm1, Em1}, Gm2 = {Vm1, Em2}, with vertexlabel function λ : V → Λ, V = Vm1 Vm2, andcost functions cm1 : Em1→R+ and cm2 : Em2→R+.CM = {cm1, cm2} is the cost function set com-posed of the two functions. With theM = m1, m2 as a part of the input, we canget the λSP

m1 m2. The path cost of mode kpk = υ0, υ1,…, υt is the sum of the costs of itsconstituent edges (cf. (1)).We define the double-modal shortest path costfrom S to T with tow modes m1, m2 (2).

A double-modal shortest path from vertex Sto vertex T with two modes m1, m2 involvedin is then defined as any path p

m1, m2with cost

c (pm1, m2

) = δ (S, T, m1, m2).Based on the formalized description of dou-

ble-modal shortest path problem, we can gen-eralize it to the multi-modal situation. Thepath cost of mode m pk = υ0, υ1,…, υt is thesame as defined in double-modal shortest pathproblem by Eq. (1).

driving and public transportation because it isimpossible to drive a car directly into a busjust like taking a bicycle into a suburban train.In other words, no Switch Point can exist thatconnects car driving mode and public trans-portation mode. There must be a pedestrianmode in between. On the other hand, whenzooming out to a much smaller map scale,some traffic modes such as pedestrian, bi-cycle, inner-city public transportation, etc.may become meaningless and therefore shouldbe ignored.

In our real life, a route planning problemwith multiple travel modes can be very com-plicated. Still, it can be reduced to a shortestpath problem or some of its variations. In ourapproach, we describe the original routingproblem as a multi-modal shortest path prob-lem.

2.2 Formalized Description

The input of a multi-modal shortest path prob-lem contains five parts: 1)Multi-Modal GraphSet (MMGS); 2) attributes of vertices in theMMGS; 3) Switch Point Matrix (SPM); 4) asequential list of modes which will be con-tained in the final route; 5) a source node anda target node.

The notational conventions used in this pa-per are identical with the second edition of thetextbook Introduction to Algorithms (Cormen

et al. 2001). Consequently, we are given a se-quential list M = m1, m2,…, mN, N ≥ 2,mi ≠mi+1, i [1, N− 1] composed by N modes, aset of vertices attributes Λ and a vertex-la-beled, non-negative weighted, acyclic, direct-ed multi-graph GM = {Gk = {Vk, Ek} | kM}

(1)

δ (S, T, m1, m2) = (2)

δ (S, T, M) = (3)


3 Multi-modal Shortest PathAlgorithms

Our investigation starts with the double-mod-al shortest path problem which can be thengeneralized to cope with the multi-modal situ-ation. We implemented two double-modalshortest path algorithms based on the classicallabel-correcting and label-setting algorithms,i. e., Bellman-Ford algorithm (Bellman 1958,Cormen et al. 2001) and Dijkstra’s algorithm(Dijkstra 1959, Cormen et al. 2001) respec-tively, and extended them to Multi-ModalBellman-Ford (MMBF) and Multi-Modal Di-jkstra (MMD). A key step was to make a fewmodifications with respect to Switch Pointduring the initialization process.

3.1 Multi-modal Bellman-Ford

Bellman-Ford search is a well-tested label-correcting algorithm used to find the shortestpath from a single source in a graph withoutnegative-weighted cycles. Generally speak-ing, Bellman-Ford search procedure consistsof two main steps: the initialization and thetraversal on the whole graph (i. e., iterativelyrelaxation of all the edges). After the traversalstep, the distance value on each vertex indi-cates the shortest distance value from thesource to this vertex.

To solve the double-modal shortest pathproblem, we execute Bellman-Ford searchtwice, the first time in the Gm1 with the sourceS and the second time in the Gm2. The initiali-zation step in the second time is different fromthe first time. Instead of setting all the distancevalues on the vertices to infinity, we kept thevalues calculated on the Switch Points in Gm1.After this modified initialization, we did Bell-man-Ford traversal in Gm2 and got the finalshortest paths from SVm1 to any vertexυVm1 with mode m1 and any vertex υVm2with modes m1 and m2 via Switch Points.Fig. 3 demonstrates the work flow of the

double-modal Bellman-Ford algorithm on twographs with a single source-target pair. Thefirst graph is Gm1 = {Vm1, Em1} with 6 verticesand 9 edges of mode m1, and the second one isGm2 = {Vm2, Em2} with 7 vertices and 18 edgesof mode m2. The source is vertex S in Gm1. The

We define the multi-modal shortest pathcost from S to T with the mode list M by (3).

An N-modal shortest path from vertex S tovertex T is then defined as any path pM withcost c (pM) = δ (S, T, M).

If we restrict the discussion in urban trans-portation field, the shortest path with N travelmodes reveals some other interesting proper-ties besides the lowest cost.

The number of switch points in the multi-modal shortest path is limited.There is a set of alternative transportationmodes available for users to make their travelplan. Although people may travel using morethan one mode, they are not disposed to under-take too many modal switches. This propertycan be seen as a constraint on the number ofmodal transfers on a multi-modal shortestpath.

The sequence of switch points in the multi-modal shortest path has some regularity.The regularity of the switch points sequencecan be set equal to the transportation mode se-quence. A multi-modal path may become lesslogical if the involved transportation modecombination seldom or never appears in atravel plan. This property can be seen as aconstraint on the mode list.

By taking the constraints into account thepath will become “viable” as defined in (lo-zano & storchi 2001). However, finding themulti-modal shortest path and determiningthe relative logic of mode combination shouldbe treated as two separate problems. The at-tempt to couple the algorithms with the con-crete mode constraints may limit the accep-tance level of routing algorithms themselvesin applications other than transportation routeplanning, e. g., computer network routing, cir-cuit designing, etc.

Our concern with the multi-modal routeplanning is more general in the sense thatthere are N modes involved and the modecombination can be arbitrarily set. The solu-tion is provided by two multi-modal shortestpath algorithms described in the next section.


where the routines multimodalinitialize andBellmanfordsearch work as follows:

ThecomputingcomplexityisO (∑kM | Vk | | Ek |)for the MMGS Gm and the mode list M,since the multimodalinitialize takes Θ (Vk )time and the Bellmanfordsearch takesΘ ((|Vk | − 1) |Ek |) time in each of the Npasses.

Theorem 3-1 (Effectiveness of the MMBF)Let MMBF be run on a vertex-labeled,non-negative weighted, acyclic, directedMMGS GM = {Gk = {Vk, Ek} | kM} whereM = m1, m2,…, mN , N ≥ 2, mi ≠ mi + 1,i [1, N− 1] with source S and cost functionset CM = {ck : Ek → R+, kM}. We can havedistance [mi] [υ] = δ (S, υ, m1,…, mi ) for allvertices υVmi.

target is vertex T in Gm2. Fig. 3 (a) and (d) arethe original graphs. The vertices in dark greyC and D indicate Switch Points from mode m1to m2. The distance values after initializationand searching processes are shown within thevertices. The thick edges in Fig. 3 (c) indicatethe shortest paths from S to the Switch Points.The second initialization is illustrated in Fig. 3(e). The thick edge in Fig. 3 ( f ) indicates theshortest path from Switch Point to the target.In this example, the double-modal shortestpath from S to T is S→ B→ D (Switch Point)→ T with the final distance value of 29.

In a multi-modal situation with an orderedmode list M, we can also get the shortest pathsby applying the modified initialization in thegraphs from Gm2 and traverse graphs of fur-ther modes. Thus, the double-modal Bellman-Ford algorithm is generalized into multi

modalBellmanford (MMBF). The wholeprocess can be described as follows:

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Fig. 3: Work flow of the double-modal Bellman-Ford algorithm.


3.2 Multi-modal Dijkstra

In MMBF algorithm, the key action is the res-ervation of the distance values on SwitchPoints during the initialization step in eachround of Bellman-Ford searching from themode m2 to mn through which the multi-modalshortest path can be finally found. This ideacan also be applied in Dijkstra search which isa well-tested label-setting algorithm. Againstthe assumption made in our early study (liu &meng 2008) that only double-modal Bellman-Ford has the potential to be generalized to amulti-modal situation while the double-modalDijkstra search can only find a good but notnecessarily the optimal path. With our mostrecent experiments we have proved the possi-bility of finding the shortest path using dou-ble-modal Dijkstra or its further extensionmultimodaldijkstra (MMD). The MMD al-gorithm can be described as follows.

where the multimodalinitialize is the sameas in MMBF and dijkstrasearch works in thefollowing way:

ProofAfter the first round Bellmanfordsearch onGm1, distance [m1] [VSP

m1 m2] = δ (S, VSPm1 m2, m1). Af-

ter the second round multimodalinitialize

on Gm2, all the δ (S, VSPm1 m2, m1) are kept as the

initial values of the vertices υSPm1 m2 VSP

m1 m2, m1,i. e., distance [m2] [VSP

m1 m2] = δ (S, VSPm1 m2, m1). To

prove distance [m2] [υ] = δ (S, υ, m1, m2) afterthe second round Bellmanfordsearch, weadd a virtual source vertex Sm2 directly con-necting to the vertices υSP

m1 m2 VSPm1 m2 in Gm2

which becomes G'm2 (Fig. 4(a)). The costs ofthe edges from Sm2 to VSP

m1 m2 are exactlyδ (S, VSP

m1 m2, m1). As a result, there are | Vm2 | + 1vertices in G'm2, and a normal Bellman-Ford’salgorithm should take | Vm2 | iterations of therelaxes of all | Em2 | + | VSP

m1 m2 | edges. In fact, thestatus of Gm2 after multimodalinitialize isexactly the same as that after the initializationand the first iteration of relaxes all the edges inG'm2. That means the algorithm just needs to dothe remaining | Vm2 | − 1 iterations inG'm2, whichis exactly the work of Bellmanfordsearch inGm2. After that, we can get the shortest dis-tance values on all the vertices in Gm2, i. e.,distance [m2] [υ] = δ (S, υ, m1, m2). In general,we can add virtual source vertex Smi in thegraph of the ith mode i [2, N], and the edgeswhose costs are δ (S, VSP

mi− 1mi, m1,…, mi ) fromSmi to the Switch Points from the last graph toGmi. After the multimodalinitialize andBellmanfordsearch in Gmi, which is equiva-lent to the normal Bellman-Ford search inG'mi = {Vmi {Smi}, Emi {Smi→VSP

mi− 1mi}}, we canhave distance [mi] [υ] = δ (S, υ, m1,…, mi ). ■

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Fig. 4: The effectiveness proof of MMBF by adding virtual sources and the corresponding edges.


when the target is reached. This improvementcan also be used in MMD with a single source-target pair. If we are only interested in theshortest path between source and target viamode sequence M, the search process can beterminated and continued into the next graphwhen it has reached all the υSP

mi mi + 1 VSPmi mi + 1 from

Gm1 to Gm N− 1, and be finally terminated when it

has reached the target in GmN. In this case, the

original multimodaldijkstra becomes mul-timodaldijkstra-target depicted as follows.

where the dijkstrasearch should be changedinto dijkstrasearch-target:

This improvement by terminating the search-ing process ahead of time can not be applied inMMBF because it is based on label-correctingmethod which can get the shortest distancevalue on all the vertices only after the very lastiteration.

According to our complexity analysis,MMD with an ordinary array as its min-prior-ity queue is faster than MMBF. MMD-Tshould be faster than MMD practically al-though in the worst case they have the samerunning time. The performance differencesbetween MMBF and MMD were verified inour experiments described in section 4. Theaverage improvement of MMD-T can also beseen from the results.

4 Experiments

We implemented the two proposed algorithmsin C# and developed a prototype system by

When the min-priority queue Q is implement-ed as an ordinary array, its computing timecomplexity is O (∑kM | Vk |2) for the MMGSGM and the mode list M, since the multi

modalinitialize takes Θ (Vk) time and thedijkstrasearch with an ordinary array as itsmin-priority queue takes O (| Vk |2) time in eachof the N passes.

The demonstration of MMD in a double-modal situation with a single source-targetpair can also be expressed by Fig. 3. The sta-tuses of the graphs after the two roundsmultimodalinitialize are exactly the same asthat in MMBF. This means, the statuses of thegraphs after dijkstrasearch are the same asthat shown in Fig. 3 (c) and ( f ).

Theorem 3-2 (Effectiveness of the MMD)MMD, run on a vertex-labeled, non-negativeweighted, acyclic, directed MMGSGM = {Gk = {Vk, Ek} | kM} whereM = m1, m2,…, mN , N ≥ 2, mi ≠ mi + 1,i [1, N− 1] with source S and cost functionset CM = {ck : Ek → R+, kM}, terminates withdistance [mi] [υ] = δ (S, υ, m1,…, mi ) for allvertices υVmi.

ProofThe basic method used in the effectivenessproof of MMBF can also be used here. In gen-eral, we can add a virtual source vertex Smi inthe graph of the ith mode i [2, N] and theedges whose costs are δ (S, VSP

mi− 1mi, m1,…, mi )from Smi to the switch points from Gmi− 1to Gmi. Gmi becomes G'mi = {Vmi {Smi},Emi {Smi→VSP

mi− 1mi}}. The multimodalini-tialize in Gmi is equivalent to the normal Dijk-stra’s initialization together with the firstround relaxes on the virtual edges emittedfrom Smi in G'mi . As a result, each time theexecution of multimodalinitialize anddijkstrasearch in Gmi is equivalent to thenormal Dijkstra’s algorithm in G'mi . Therefore,when the MMD terminates, we have dis-tance [mi] [υ] = δ (S, υ, m1,…, mi ). ■

For any vertex in any Gmi, the distance valueon it indicates the shortest distance from thesource inGm1 to this vertex via the mode se-quence (m1,…, mi) after MMD. For the classi-cal Dijkstra’s algorithm with a single source-target pair input, the performance can be con-siderably improved by terminating the search


of Munich from two different sources. One isOpenStreetMap (OSM) which has been en-riched with Switch Point information, theother is a navigation dataset provided by Unit-ed Maps Co., Ltd. (UM) which integrates the

Visual Studio 2005 Professional. The pro-grams run on a PC with a 2.2 GHz Intel Core2Duo CPU and 2 GB physical memory underWindows XP Professional SP3. Our tests wereconducted on the spatial datasets of a portion

Tab. 2: The size of test networks.

Data source Area of coverage(width, length)*

Mode | V | | E | | E | / | V |

UM(20.34, 26.18)

D 19471 44979 2.31

W 20516 57694 2.81

(9.829, 11.00) U 64 132 2.06

OSM (4.970, 4.663)D 4807 9125 1.90

W 9077 22482 2.48* The unit of width and length is km.

Tab. 3: Computing speed of MMBF and MMD.

Data source Mode list MMBF(s) MMD(s)

UM

1-modal

D 354.9 10.28

W 536.7 12.27

U 0.003125 0.001094

2-modal

D, W 862.7 24.81

W, U 467.3 12.21

U, W 467.7 13.13

3-modalD, W, U 817.1 25.11

W, U, W 936.5 25.57

4-modal D, W, U, W 1287 37.98

OSM1-modal

D 15.99 0.4941

W 76.42 2.239

2-modal D, W 90.78 3.013

Tab. 4: Computing speed of MMD and MMD-T on UM dataset.

Mode list MMD (s) MMD-T (s) Improvement. ratio (%)

2-modal

D, W 23.70 16.93 40.0

W, U 14.96 12.02 24.5

U, W 12.70 9.066 40.1

3-modalD, W, U 23.75 19.27 23.2

W, U, W 23.83 19.91 19.7

4-modal D, W, U, W 35.99 27.65 30.2

Avg: 29.6


ing the performances between MMBF andMMD on the two datasets; the other is the testof single source-target-pair MMD to investi-gate how much performance improvement canbe made by MMD-T. For a given combinationof dataset, mode list and multi-modal shortestpath algorithm, an average running time of100 s execution of MMD with randomly se-lected source (and target for MMD-T) and 10 sfor MMBF was recorded. The reported run-ning times do not include data input or log out-put. The SPM (or, more precisely, SPM-T)used for the experiment is shown in Tab. 1 inSection 2. It should be noticed that the Dijk-

information from Navteq and ATKIS (the of-ficial topographic cartographic informationsystem in Germany). The two datasets differin the areas they cover as well as the size oftheir corresponding networks.

The basic information of the two datasets islisted in Tab. 2. There are three transportationmodes of network in UM dataset: motorizedways denoted by D, pedestrian ways denotedby W and underground lines denoted by U. InOSM dataset, there are D and W.

The experiments consist of two groups: oneis the test of single-source multi-modal short-est path algorithms for the purpose of compar-

Fig. 5: Comparison between the routing results of our prototype system and Google Maps.

Fig. 6: A multi-modal routing result with three modes.


Two algorithms MMBF and MMD were de-veloped on the basis of label-correcting andlabel-setting algorithms respectively. They re-veal the complexities of O (∑kM | Vk | | Ek |) andO (∑kM | Vk |2) respectively. The effectivenessof the algorithms has been proved, which canensure that the found paths are the shortest interms of distances or weights. Our experi-ments were conducted on real transportationnetworks from two different data sources.MMBF is much slower than MMD, and MMD-T runs 29.6% faster than MMD on average incase of the single source-target pair. From thepractical point of view, the MMBF is ineffi-cient for real-time applications although it isable to find the multi-modal shortest path.

The conducted experiments have shown theconvincing process of solving a multi-modalroute planning problem. However, multi-mod-al routing in our real life can take various farmore complicated forms which will challengethe computing power of routing algorithms.Therefore, our work for the next steps will fo-cus on two points:

1) To explore the further potential of com-puting performance of the routing algo-rithmsThe multi-modal routing planning algo-rithms introduced in this paper are based onthe two earliest and well-tested shortestpath algorithms. In fact, some researchershave proposed improved versions of mono-modal shortest path algorithms (Gallo &Pallottino 1988, Cherkassky et al. 1996).According to the evaluation done by (Zhan

& Noon 1998) using real road networks andthe comparison work between label-settingand label-correcting algorithms for comput-ing single source-target pair shortest pathsby (Zhan & Noon 2000), TWO-Q is themost efficient routing algorithm for shortestpath finding in road networks (Pallottino

1984). We attempt to implement a newmulti-modal shortest path algorithm basedon TWO-Q and test its performance in thenear future.

2) To take the knowledge from specific fieldinto consideration.There is a lot of context knowledge from thetransportation field which should be embed-ded in the route planning. For example, the

stra search can reveal considerably differentcomputing performance if it is applied in adifferent data structure. For this reason, weused an ordinary array as the min-priorityqueue without using any built-in data struc-tures or methods provided by .net frameworkclass library. In this way, our test results areindependent of the internal optimization madeby Microsoft.The experimental results of the first group

are listed in Tab. 3 which shows that theMMBF is much slower than MMD, which hasconfirmed our analysis of theoretical com-plexities.

The experimental results of the secondgroup are listed in Tab. 4 which shows that theaverage improvement made by MMD-T overMMD amounts to 29.6%.

Fig. 5 shows the routing result of our algo-rithm in comparison with Google’s route froma crossing of motorized roads to a pedestrianjunction in the English Garden. The destina-tion is shown by a checkered flag. We can seethat the car segments (blue solid line) of thetwo routes are exactly the same. However, thepedestrian segment (red dashed line) in ourroute is missing in Google’s route. The reasonfor that is in Google Maps (and almost all thecurrently available routing systems) only onetravel mode can be selected, e. g., either by caror by foot, but not both. Our routing algorithmallows the car-pedestrian combination with anappropriate parking place as the Switch Point.

Fig. 6 shows another multi-modal routingresult calculated by our algorithms. Threemodes are involved in the route: car driving(blue solid line), walking (red dashed line), un-derground train (black solid line) and walkingagain.

5 Conclusions and Outlook

The multi-modal route planning problem ad-dressed in this paper originates from the trans-portation field. However, we developed a mul-ti-modal routing strategy in a general sense,thus opened up its applicability for the fieldsbeyond transportation. We proposed a datamodel with Switch Point as the core conceptand gave the formal description of the multi-modal shortest path based on Switch Point.


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Acknowledgements

The research work is sponsored by UnitedMaps Co., Ltd. The high-quality and multi-modal navigation dataset is jointly created byUnited Maps Co., Ltd. and Department ofCartography at Technische UniversitätMünchen in the frame of a project on data in-tegration. The authors would like to thank Mr.Hongchao Fan who gave useful suggestionsabout improving this paper.

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Address of the Authors:

M.Eng. lu liu, Prof. Dr.-Ing. liqiu meng, Tech-nische Universität München, Department ofCartography, D-80333 Munich, Germany, Tel.:+49-89-2892-2586, -2825, Fax: +49-89-2809573,e-mail: [email protected], [email protected]

Manuskript eingereicht: Mai 2009Angenommen: Juli 2009

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algorithms of multi-modal route planning based on the ... · is a novel approach.his ideais...

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