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A Probabilistic Model for Road Selection in Mobile Maps Thomas C. van Dijk Jan-Henrik Haunert W2GIS, 5 April 2013

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Overview: a little bit of modelling, interpretation, algorithms, evaluation, outlook,

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Road selection focus -and- context map route map Depends on user task

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Road selection destination map Kopf et al. focus maps Yamamoto et al.

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Motivating example Pedestrian with a smartphone Current location available Wayfinding to a destination Interest in surroundings Freedom of movement Contrast: in-car sat nav

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From scoring to selection Maximise score using given number of road segments Minimise number of road segments selecting at least a given amount of score Additional constraints?

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Betweenness centrality Consider all shortest paths between all pairs of nodes; count how often each edge is used Folklore interpretation: People are likely to travel shortest paths Heavily traveled edges are important

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Betweenness centrality Consider 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 path Probability that you visit a certain edge

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PageRank primer Origin in web page ranking (Page, Brinn, Motwani, Winograd Google) Previous application in predicting human movement (Jiang) Usually stated as eigenvector calculation Also: interpretation in terms of random walks

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Random Surfer model Hypothetical user is at a web page Transitions along a uniformly random link All transitions take equal time Sometimes jumps to a uniformly random page on road segment turn Traveler road segment

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Non-uniform transition probability Walking along a road segment Coming up to a crossing Where to go? Shortest path? Straight on? Anywhere? Whats the network? Web pages, hyperlinks?

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Whats the network? Web pages? Links? Modelling user behaviour: routing decisions Road network -> bidirectional line graph linear dual graph (Winter)

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Bidirected line graph

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Non-uniform transition probability At a node of the line graph Follow an arc Where to go? Straight on? Constant given road network Shortest path? Constant given road network and destination

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Non-uniform damping PageRank jumps to uniformly random node Instead, jump to the current user location! Probabilistic interpretation: Consider random walk starting at user location Exponential distribution of walk length Score of node = probability of going to node

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Data set OpenStreetMap City of Wrzburg and surroundings Lower Franconia, Germany Approx 10 5 nodes Examples here: crop to the city itself 3786 nodes, 4987 road segments

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Demo

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Shortest-path bonus

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Non-turning bonus

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Runtime of simple implementation Calculate PageRank Loop over nodes Distribute score to neighbours Repeat Select nodes with high score Runtime: ~100 ms

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Direct selection Select all nodes with high rank Select no nodes with tiny rank O( log n ) time Correctness holds almost surely Almost immediately from Borgs et als algorithm for Significant PageRanks.

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Really big maps? 1.Just calculate, then select 2.Approximation algorithm: Runtime O( log(n) log( -1 ) -1 -2 ) # nodes Calculate + Select Direct selection (reasonable , ) 3786~100ms~200ms 10 5 ~10.000 ms~300ms

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Gaps Selection is not necessarily connected Simplified example:

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Gaps

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Strokes

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Overview: a little bit of modelling, interpretation, algorithms, evaluation, outlook,

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Q?

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A!