Algorithme bidirectionnel pour le plus court chemin multimodal bi-objectif contraint par un langage régulier Christian Artigues 1 Marie-José Huguet 1 Sandra Ulrich Ngueveu 1 Dominik Kirchler 2 , 3 , 4 Roberto Wolfler Calvo 3 1 LAAS - CNRS & Université de Toulouse, France 2 LIX, Ecole Polytechnique 3 LIPN, Université Paris 13 4 Mediamobile, Ivry sur Seine {artigues,huguet,ngueveu}@laas.fr, kirchler@lix.polytechnique.fr, roberto.wolfler@lipn.univ-paris13.fr ROADEF 2012 - Angers 1/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 2/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 3/38
Our Goal ◮ Find exact shortest path (two objectives: time and mode changes) ... ◮ ... by using car, bike, rental bike, public transport, walking ◮ ... respecting constraints/preferences (viability, e.g. private bike) ◮ ... and considering traffic conditions, time-tables. ◮ Speed-up techniques 4/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 5/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 6/38
Multimodal transportation network ◮ Multi-Layer Graph ◮ G = ( V , A ) ◮ transfer arcs 7/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 8/38
Bi-objective Regular Language Constrained Shortest Path Problem BI-RegL-CSPP ◮ G Σ = ( V , A ) , nodes v ∈ V , arcs ( i , j , l ) ∈ A , i , j ∈ V ◮ labels l ∈ Σ = { l 1 , l 2 , . . . } assigned to arcs (representing modes) ◮ restrictions/preferences modelled by regular language L 0 : concatenated labels on SP ∈ L 0 ◮ objective: min total cost (travel time) AND number of mode changes (transfers) ◮ Single obj. variant effic. solvable by a generalisation of Dijkstra’s algorithm D RegLC f f foot s 2 f t t transfer s 0 s 3 b bike t t t p public transportation s 1 s 4 p v rental bike (velib) b ftv ◮ Barrett et al. Formal-language-constrained path problems , 2000 ◮ Barrett et al. Engineering Label-Constrained SP Algorithms , 2008 ◮ Lozano, Storchi Shortest viable path algorithm in multimodal networks , 2001 9/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 10/38
BI-RegL-CSPP Example s x 2 x 3 s 1 5 w, b s 1 1 1 1 x 1 x 4 x 5 4 4 s 0 s 2 1 1 1 1 x 6 x 7 3 w, b w, b (a) multi-modal network (b) NFA for mode scenario M 2 = { w , b , s } Figure: a B I -R EG L-CSPP with 7 nodes, 12 arcs, 3 modes and a 3 state-NFA path ( x 1 , x 4 , x 5 ): 0 transfer, travel time 8, path ( x 1 , x 6 , x 7 , x 5 ): 2 transfers, travel time 5 path ( x 1 , x 2 , x 4 , x 3 , x 5 ): 4 transfers, travel time 4 path ( x 1 , x 6 , x 4 , x 7 , x 5 ) : infeasible 11/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 12/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 13/38
Extending Dijkstra algorithm: Labels and dominance rules Labels ◮ Label ( i , s , k ) ◮ t k is arrival time on i in state s with k transfers ◮ p k is predecessor of ( i , s , k ) on the shortest path from O to i . ◮ Only one label ( i , s , k ) must be kept (Bellman condition) BI-RegL-CSPP is polynomial Pareto dominance ◮ Discard any label ( i , s , k ) such that there exists an already visited label ( i , s , k ′ ) with k ′ ≤ k and t k ′ js ≤ t k js State-based dominance rule ◮ Lozano and Storchi 2001 proposed the concept of preference between states aiming at discarding a label ( i , s , k ) with a label ( i , s ′ , k ′ ) with s � = s ′ but some issued were detected ◮ The application causes implementation issues (need of knowing the set of modes taken in a partial path represented by a label) ◮ Dominated labels may be not discarded (a label is discarded only if it is dominated by all its preferred labels) ◮ The theorem only applies to the automaton used in Lozano and Storchi 2001 14/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 15/38
State-based dominance: example A preorder � on the automaton states S ◮ s � s ′ if δ ( m , m ′ , s ′ ) ⊆ δ ( m , m ′ , s ) for any transition ( m , m ′ ) in the automaton. State-based dominance rule ◮ A label ( i , s ′ , k ′ ) such that there is another label ( i , s , k ) with t isk ≤ t i , s ′ k ′ , k ≤ k ′ and s � s ′ can be discarded 16/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 17/38
State-based dominance: example s x 2 x 3 s 1 5 w, b s 1 1 1 1 x 1 x 4 x 5 4 4 s 0 s 2 1 1 1 1 x 6 x 7 3 w, b w, b (a) Simple multimodal nework (b) 3-state NFA s 0 s 1 s 2 w , w id ∅ id w , b id ∅ id w , s s 1 ∅ ∅ b , b id ∅ id b , w id ∅ id s , s ∅ id ∅ s , w ∅ s 2 ∅ (c) Function δ f ◮ s 0 � s 2 but s 2 �� s 0 . ◮ ( x 1 , x 6 , x 4 ) ( k ′ =2, t ′ =2, s ′ =2) is dominated by ( x 1 , x 2 , x 4 ) ( k =2, t =2, s =0) 18/38
Outline Introduction Problem definition Multi-Modal Transportation Network Bi-objective regular Language Constrained SPP Example State-based bidirectionnal algorithm for the time-independent BI-RegL-CSPP Extending dijkstra algorithm: Labels and pareto dominance A new state-based dominance rule Example of state-based dominance Bidirectional algorithm Computational experiments SDALT for the time-dependent BI-RegL-CSPP uniALT algorithm Our Contribution: SDALT Experiments 19/38
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