Colpaert, Pieter

Delva, Harm

Rojas Melendez, Julian Andres

Verborgh, Ruben

English

Ghent University Academic Bibliography

Abstract.Decentralized Publication and Consumption of TransferFootpathsHarm Delva, Julián Andrés Rojas Meléndez, Pieter Colpaert, Ruben VerborghIDLab, Department of Electronics and Information Systems, Ghent University – imecUsers expect route planners that combine all modes of transportationto propose good journeys to their destination. These route planners use data fromseveral sources such as road networks and schedule-based public transit. Wefocus on the link between the two; specifically, the walking distances betweenstops. Research in this field so far has found that computing these pathsdynamically is too slow, but that computing all of them results in a quadraticallyscaling number of edges which is prohibitively expensive in practice. Thecommon solution is to cluster the stops into small unconnected graphs, but thisrestricts the amount of walking and has a significant impact on the travel times.Moreover, clustering operates on a closed-world assumption, which makes itimpractical to add additional public transit services. A decentralized publishingstrategy that fixes these issues should thus (i) scale gracefully with the number ofstops; (ii) support unrestricted walking; (iii) make it easy to add new services and(iv) support splitting the work among several actors. We introduce a publishingstrategy that is based on the Delaunay triangulation of public transit stops, whereevery triangle edge corresponds to a single footpath that is precomputed. Thisguarantees that all stops are reachable from one another, while the number ofprecomputed paths increases linearly with the number of stops. Each publictransit service can be processed separately, and combining several operators’ canbe done with a minimal amount of work. Approximating the walking distancewith a path along the triangle edges overestimates the actual distance by 20% onaverage. Our results show that our approach is a middle-ground betweencompleteness and practicality. It consistently overestimates the walking distances,but this seems workable since overestimating the time needed to catch aconnection is arguably better than recommending an impossible journey. Theestimates could still be improved by combining the great-circle distance with ourapproximations. Alternatively, different triangulations could be combined tocreate a more complete graph.1. IntroductionFueled by the need for more sustainable transportation, users can pick between moreforms of transport than ever before – and they expect that route planners can combinethese to find the ideal journey. Those route planners integrate various kinds of dataCopyright © 2019 for this paper by its authors. Use permitted under Creative CommonsLicense Attribution 4.0 International (CC BY 4.0).such as road networks and schedule-based public transit. In this article, we focus onthe link between the two, specifically on how to walk between stops. The walkingtime from one stop to another is commonly referred to as a footpath [1, 2].Computing these paths dynamically is often too slow in practice [3, 4], yet computingthem beforehand quickly becomes prohibitively expensive in practice [5, 1, 2]. Asmall country such as Belgium already contains 100,000 public transit stops, yielding10 billion pairs of stops. These scaling issues are often avoided by only evaluatingalgorithms on small unconnected footpath graphs. However, this restricts the amountof walking in a journey and has a significant impact on travel times [4]. On top ofthat, determining which stops belong to the same graph ultimately relies on a closed-world assumption; which makes it impractical to add additional public transitservices.Our goal is to find a method to compute and publish these footpaths in a scalable anddecentralized manner. This implies that we need not publish all the paths, but ideallythe ones that are published should allow for unrestricted walking. In practice thismeans that every stop in the footpath graph must be reachable from every other stop ifthere is a path between those stops in the real world. Various actors should be able toadd information to this footpath graph, and adding a new service should not requirechanges in existing information. These actors would need a common frame ofreference to combine their efforts, which can be provided by Semantic Webtechnologies.2. Related workThe publication of transfer footpaths is closely related to the publication of roadnetworks. The Routable Tiles [6] dataset was recently proposed as a scalable and cost-efficient way to republish the OpenStreetMap roads using Linked Data Fragments [7].The data is fragmented into hierarchical tiles, which are interconnected withhypermedia controls for automatic data discovery. This allows clients to find anddownload the data they need without having to ingest en entire data dump, whilebeing significantly cheaper than a full-fledged querying API.Determining which stops have to be connected with each other can be seen as aproblem in the field of computational geometry. One of the classical problems in thisfield is the creation of a point set triangulation. A point set triangulation of a set ofpoints  can be defined as a maximal set of non-crossing edges between the points of . Any triangulation of  points has exactly  edges where  is the numberof points on the convex hull. Triangulations are not unique however, and certaintriangulations are NP-hard to compute [8].One particularly interesting triangulation, which can be computed in just time, is the Delaunay triangulation [9]. This triangulation tries to maximize theminimum angle of all the triangles, effectively avoiding long and narrow triangles.PP n 3n − h − 3 hO(n logn)The Delaunay graph, which is simply the graph representation of the triangulation,contains several other interesting graphs such as the nearest neighbor graph and theminimal spanning tree of  [10, 11]. They are also known to be good approximationsof complete graphs: the length of the shortest path between two points alongDelaunay edges is at most  times the Euclidean distance betweenthem [12].3. MethodP4π/(3 ) ≈3 2.418  Fig. 1: Illustration of how two footpath graphs are merged. The subfigures from top tobottom, left to right: (i) an existing triangulation of a larger network; (ii) theWe use a Delaunay triangulation of the public transit stop locations to approximatethe full footpath graph, based on the insights from the related work section. Thetriangle edges represent which stops will be directly connected through a footpath.Our representation of a footpath differs slightly from the conventional one; we usewalking distances instead of walking times to avoid making assumptions aboutwalking speeds. A road network route planner is used to compute the walkingdistances between the stops.Delaunay triangulations can be generalized to any metric space, but the walkingdistance is not a mathematical metric because it could, in theory, lack the requiredsymmetry property. Even on foot, you cannot always retrace your steps – either due tolegal restrictions or due to physical limitations. And, equally importantly, this wouldbring us back to the case where an impractical number of path lengths have to beprecomputed. This is why we the triangulation uses the Euclidean distance betweenthe stops’ WGS84 coordinates instead. Other metrics such as the great-circle distancecan be used, but none of them have any relation to the walking distance so we optedfor the most simple metric to implement. This also means that the theoreticalguarantees of the Delaunay triangulation only hold for the Euclidean distance becausethat’s the metric was used for its construction.The paths between stops of a single operator can be calculated independently from therest. Combining the graphs of two operators is done by comparing their individualtriangulations to the triangulation of their combined stops – and computing themissing footpaths. This process is illustrated in Fig. 1 where the graph of a city’spublic transit network is merged with the one from the region around it. Thecombined graph is the union of each operators’ graph with the additional edges thatconnect the two graphs. Operators typically have their own service area, which meansthat the vast majority of paths will be between stops of the same operator, thusmaking it possible for anyone to combine several operators’ graphs with minimalwork. The results can be shared and reused, which stands in stark contrast to theconventional approaches. The paths can be published similar to the Routable Tilesdataset so that sharing them does not come at a significant extra cost.4. Resultstriangulation of an additional regional operator; (iii) the edges that are needed toattach the new triangulation to the existing one and (iv) the end result.We aim to verify two claims about our method: (i) the graph we construct is almost asgood as the complete graph for route planning and (ii) footpath graphs can be mergedwith minimal effort. To verify the first claim we have used a graph of paths betweenbus stops to approximate the distances between railway stations. Edges between trainstations and bus stops have also been added to make the railway stations reachable.Fig 2 shows that the approximation overestimates the actual distance by about 20% onaverage. This seems reasonable for route planners – it is better to underestimate thetime needed to catch a connection than to recommend an impossible route.Table 1: The number of paths that need to be computed to connect two operators.Case Paths of firstoperatorPaths of secondoperatorAdditional requiredpathsFlanders + Brussels 107,171 7,969 2,508Flanders +Wallonia107,171 94,730 4,0202b: The same linear regression plot butwith 8 bins. The mean and standarddeviation is shown for each bin, alongwith the 95% CI in the grey shadedarea. The approximation error does notexceed 25% in 95% of cases.2a: Linear regression plot comparingthe actual distances to theapproximations. The relativeapproximation error decreases as thedistance increases. Note that theapproximations are strictlyoverestimations due to the triangleinequality.Fig. 2: The relative difference between the actual and the approximated distance has ahigher variance for shorter paths. Every data point in these plots is a path in betweenrailway stations where the distance is approximated using a graph of bus stops and thefootpaths between bus stops and railway stations.To verify that combining the graphs from two operators does not require excessivework, we consider both the usual and the worst case scenario, respectively beingmostly disjoint service areas and entirely overlapping service areas. We have takenthe public transit operators from the Belgian regions of Flanders and Wallonia for theusual case and we revisit the case from Fig. 1 for the worst case, where the Brusselspublic transit network is added to the one from Flanders. The results in Table 1 showthat in the usual scenario merging two operators is relatively straight-forward as mostwork has already been done. The worst case scenario shows that it is possible thatmerging two operators can still require a significant amount of work, although thevast majority has already been done. Our implementation contains some additionalexperiments and is available on https:/ / github.com/hdelva/planner.js/tree/footpaths asa frozen branch of an experimental serverless route planner.5. Conclusion and future workThis paper introduced a decentralized publishing method for transit footpaths basedon Delaunay triangulations. We have shown that different actors can add informationto the footpath graph, splitting the work between them. The amount of paths scaleslinearly with the amount of public transit stops, and every stop is reachable fromevery other. This last property could prove useful in its own right, as it allows forunrestricted walking unlike conventional methods. However, this comes at the cost ofprecision; the walking distance between stops is consistently slightly overestimatedwhich could cause route planners to discard the most optimal journeys. Future workcould focus on comparing the results from a route planner with a complete footpathgraph to the one as we describe in this paper. If the difference in quality is negligibleour methods could prove useful even for centralized services.The approximations may still be improved with different kinds of point settriangulations and several triangulations could even be combined to create a morecomplete graph. Alternatively, combining the haversine distance and the currentlyapproximated distance seems promising as well – although this could lead tounderestimations. Another open question is how scalable this approach is when moremodes of transportation are required. It seems undesirable to create separate graphsfor bicycles, electric bikes, wheel chairs, etc. but they might be able to shareinformation.References1. Delling, D., Pajor, T., Werneck, R.F.: Round-based public transit routing.Transportation Science. 49, 591–604 (2014).2. Dibbelt, J., Pajor, T., Strasser, B., Wagner, D.: Connection scan algorithm. Journalof Experimental Algorithmics (JEA). 23, 1–7 (2018).3. Delling, D., Dibbelt, J., Pajor, T., Wagner, D., Werneck, R.F.: Computingmultimodal journeys in practice. In: International Symposium on ExperimentalAlgorithms. pp. 260–271. Springer (2013).4. Wagner, D., Zündorf, T.: Public transit routing with unrestricted walking. In: 17thWorkshop on Algorithmic Approaches for Transportation Modelling,Optimization, and Systems (ATMOS 2017). Schloss Dagstuhl-Leibniz-Zentrumfuer Informatik (2017).5. Bast, H., Carlsson, E., Eigenwillig, A., Geisberger, R., Harrelson, C., Raychev, V.,Viger, F.: Fast Routing in Very Large Public Transportation Networks UsingTransfer Patterns. In: Algorithms - ESA 2010, 18th Annual EuropeanSymposium. Proceedings, Part I. pp. 290–301 (2010).6. Colpaert, P., Abelshausen, B., Rojas Meléndez, J., Delva, H., Verborgh, R.:Republishing Open Street Map’s roads as Linked Routable Tiles. In: Proceedingsof the 16th ESWC: Posters and Demos (2019).7. Verborgh, R., Vander Sande, M., Hartig, O., Van Herwegen, J., De Vocht, L., DeMeester, B., Haesendonck, G., Colpaert, P.: Triple Pattern Fragments: a Low-costKnowledge Graph Interface for the Web. Journal of Web Semantics. 37–38, 184–206 (2016).8. Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. Journal of theACM (JACM). 55, 11 (2008).9. Leach, G.: Improving worst-case optimal Delaunay triangulation algorithms. In:4th Canadian Conference on Computational Geometry. p. 15. Citeseer (1992).10. Jaromczyk, J., Toussaint, G.T.: Relative neighborhood graphs and their relatives.Proceedings of the IEEE. 80, 1502–1517 (1992).11. Matula, D.W., Sokal, R.R.: Properties of Gabriel graphs relevant to geographicvariation research and the clustering of points in the plane. Geographical analysis.12, 205–222 (1980).12. Keil, J.M., Gutwin, C.A.: Classes of graphs which approximate the completeEuclidean graph. Discrete & Computational Geometry. 7, 13–28 (1992).CEUR-WS.org/Vol-2447 - Semantics for Transport and Aproaches for Making Data Interoperable 2019file:///C|/Users/Administrator/Documents/dstevens/DSpubds/DS261_i_i_i.html[9/25/2019 10:20:52 AM]CEUR-WS.org/Vol-2447 urn:nbn:de:0074-2447-4Copyright © 2019 for the individual papers by the papers' authors.Copyright © 2019 for the volume as a collection by its editors. Thisvolume and its papers are published under the Creative CommonsLicense Attribution 4.0 International (CC BY 4.0).SEM4TRA-AMAR 2019Semantics for Transport and Aproachesfor Making Data InteroperableJoint Proceedings of the 1st International Workshop On Semantics ForTransport and the 1st International Workshop on Approaches forMaking Data Interoperableco-located with 15th Semantics Conference (SEMANTiCS 2019)Karlsruhe, Germany, September 9, 2019.Edited byLucie-Aimee Kaffee *,**Kemele M. Endris **,***Maria-Esther Vidal **Marco Comerio ****Mersedeh Sadeghi *****David Chaves-Fraga ******Pieter Colpaert ******** University of Southampton, United Kingdom** TIB Leibniz Information Centre for Science and Technology, Germany *** L3S Research Center, Germany **** CEFRIEL, Italy ***** Politecnico di Milano, Italy ****** Universidad Politécnica de Madrid, Spain ******* Ghent University, Belgium CEUR-WS.org/Vol-2447 - Semantics for Transport and Aproaches for Making Data Interoperable 2019file:///C|/Users/Administrator/Documents/dstevens/DSpubds/DS261_i_i_i.html[9/25/2019 10:20:52 AM]Table of ContentsPreface1st International Workshop on Approaches for Making DataInteroperableAnalysis of Current RDM Applications for the Interdisciplinary Publication of ResearchDataAndré Langer, Ellen Bilz, Martin GaedkeAssessing the Quality of R2RML Mappings Ademar Crotti Junior, Jeremy Debattista, Declan O'SullivanFoundations to Query Labeled Property Graphs using SPARQL*Olaf Hartig1st International Workshop On Semantics For TransportDecentralized Publication and Consumption of Transfer FootpathsHarm Delva, Julián Andrés Rojas Meléndez, Pieter Colpaert, Ruben VerborghPredicting Phase Durations of Traffic Lights Using Live Open Traffic Lights DataBrecht Van de Vyvere, Karel D’haene, Kurt D’haene, Pieter Colpaert, Ruben VerborghTurn Transportation Data into EU Compliance through Semantic Web-based SolutionsMarco Comerio, Alessio Carenini, Mario Scrocca, Irene CelinoAutomated Mapping for Semantic-based Conversion of Transportation Data FormatsMarjan Hosseini, Safia Kalwar, Matteo Rossi, Mersedeh SadeghiONETT: Systematic Knowledge Graph Generation for National Access PointsDavid Chaves-Fraga, Adolfo Antón, Jhon Toledo, Oscar Corcho2019-09-20: submitted by David Chaves-Fraga, metadata incl. bibliographic data published under Creative Commons CC02019-09-20: published on CEUR-WS.org |valid HTML5|

Decentralized publication and consumption of transfer footpaths

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Decentralized publication and consumption of transfer footpaths

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