PowerPoint Presentation

A Probabilistic Modelfor Road Selectionin Mobile MapsThomas C. van DijkJan-Henrik Haunert

W2GIS, 5 April 2013

Overview: a little bit of modelling, interpretation, algorithms, evaluation, outlook, Road selectionfocus-and-context maproute map

Depends on user taskRoad selection

destination mapKopf et al.focus mapsYamamoto et al.

Motivating examplePedestrian with a smartphoneCurrent location availableWayfinding to a destination

Interest in surroundingsFreedom of movementContrast: in-car sat navFrom scoring to selectionMaximise scoreusing given number of road segmentsMinimise number of road segmentsselecting at least a given amount of score

Additional constraints?

Betweenness centralityConsider all shortest paths between all pairs of nodes; count how often each edge is used

Folklore interpretation:People are likely to travel shortest pathsHeavily traveled edges are importantBetweenness centralityConsider all shortest paths between all pairs of nodes; count how often each edge is used

Probabilistic interpretation:Start at a uniformly random node,uniformly random destination,take a uniformly random shortest pathProbability that you visit a certain edgePageRank primerOrigin in web page ranking(Page, Brinn, Motwani, Winograd Google)Previous application in predicting human movement (Jiang)

Usually stated as eigenvector calculationAlso: interpretation in terms of random walks

Random Surfer modelHypothetical user is at a web page

Transitions along a uniformly random link

All transitions take equal time

Sometimes jumps to a uniformly random pageon road segmentturnTravelerroad segmentNon-uniform transition probabilityWalking along a road segmentComing up to a crossingWhere to go?

Shortest path? Straight on? Anywhere?

Whats the network?Web pages, hyperlinks?Whats the network?Web pages? Links?

Modelling user behaviour: routing decisions

Road network -> bidirectional line graphlinear dual graph (Winter)Bidirected line graphBidirected line graphBidirected line graphBidirected line graphBidirected line graphNon-uniform transition probabilityAt a node of the line graphFollow an arcWhere to go?

Straight on?Constant given road networkShortest path?Constant given road network and destinationNon-uniform dampingPageRank jumps to uniformly random nodeInstead, jump to the current user location!

Probabilistic interpretation:Consider random walk starting at user locationExponential distribution of walk lengthScore of node = probability of going to nodeData setOpenStreetMapCity of Wrzburg and surroundingsLower Franconia, GermanyApprox 105 nodesExamples here: crop to the city itself3786 nodes, 4987 road segments

Demo

Shortest-path bonus

Non-turning bonus

Runtime of simple implementationCalculate PageRankLoop over nodesDistribute score to neighboursRepeatSelect nodes with high score

Runtime: ~100 msDirect selectionSelect all nodes with high rankSelect no nodes with tiny rank

O( log n ) timeCorrectness holds almost surelyAlmost immediately from Borgs et als algorithm for Significant PageRanks.Really big maps?Just calculate, then selectApproximation algorithm:Runtime O( log(n) log(-1) -1 -2 )# nodesCalculate + SelectDirect selection (reasonable , )3786~100ms~200ms105~10.000 ms~300msGapsSelection is not necessarily connectedSimplified example:

Gaps

Strokes

Overview: a little bit of modelling, interpretation, algorithms, evaluation, outlook, Q?A!