Carpooling: the 2 Synchronization Points Shortest Paths Problem - - PowerPoint PPT Presentation

Carpooling: the 2 Synchronization Points Shortest Paths Problem

Arthur Bit-Monnot Christian Artigues Marie-Jos´ e Huguet Marc-Olivier Killijian September 5, 2013

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Carpooling

Two users:

• a passenger
• a driver

With their own origin and destination

Goal

Find paths minimizing the travel time Including:

• pick-up point
• drop-off point

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Outline

Shortest Paths: pre-requisites Solving our carpooling problem Experiments Conclusion

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Multi-modal graph

• Mixed car, foot and public transportation edges
• FIFO
• Public transportation → time-dependent

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Shortest Path Problem

Dijkstra Shortest Path from one node to all (One-to-All)

Properties

• Nodes are settled only once
• Nodes are settled with increasing cost
• Time-independent: backward search (All-to-One)

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Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival

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Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 2, t = 2) (c = 0, t = 0)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 2, t = 2) (c = 0, t = 0) (c = 2, t = 2)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 2, t = 2) (c = 0, t = 0) (c = 2, t = 2) (c = 4, t = 4)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 2, t = 2) (c = 0, t = 0) (c = 2, t = 2)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 2, t = 2) (c = 0, t = 0) (c = 2, t = 2) (c = 12, t = 12)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 5, t = 2) (c = 6, t = 0)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 5, t = 2) (c = 6, t = 0) (c = 7, t = 4)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 5, t = 2) (c = 6, t = 0) (c = 7, t = 4) (c = 21, t = 18)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 5, t = 2) (c = 6, t = 0) (c = 7, t = 4) (c = 21, t = 18) (c = 8, t = 2)

Best Origin Problem (BOP)

Given several origins with initial cost and arrival times, select the

• rigin minimizing the cost at the arrival
• 1
• 2

x d

t = 2 → ∆ = 2 t = 0 → ∆ = 2 t = 4 → ∆ = 14 t = 2 → ∆ = 10

(c = 5, t = 2) (c = 6, t = 0) (c = 7, t = 4) (c = 21, t = 18) (c = 8, t = 2) (c = 18, t = 12)

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Best Origin Problem (BOP)

Consistency between cost and arrival time

Given 2 labels (c, t) et (c′, t′), Consistency if : c < c′ ⇔ t < t′

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Best Origin Problem (BOP)

Consistency between cost and arrival time

Given 2 labels (c, t) et (c′, t′), Consistency if : c < c′ ⇔ t < t′ Need to take cost and arrival time into account Mono-objective variant of Martins’ algorithm

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Best Origin Problem (BOP)

Consistency between cost and arrival time

Given 2 labels (c, t) et (c′, t′), Consistency if : c < c′ ⇔ t < t′ Need to take cost and arrival time into account Mono-objective variant of Martins’ algorithm

Properties

• Finds best origin for all nodes
• Labels settled by increasing cost
• Node’s best cost → first settled label

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Best Origin Problem (BOP)

Dominance rules (cx, tx) dominates (c′

x, t′ x) if and only if: .

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Best Origin Problem (BOP)

Dominance rules (cx, tx) dominates (c′

x, t′ x) if and only if: .

Exact tx ≤ t′

x and cx − c′ x ≤ tx − t′ x

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Best Origin Problem (BOP)

Dominance rules (cx, tx) dominates (c′

x, t′ x) if and only if: .

Exact tx ≤ t′

x and cx − c′ x ≤ tx − t′ x

Heuristic tx ≤ t′

x and cx ≤ c′ x

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Best Origin Problem (BOP)

Dominance rules (cx, tx) dominates (c′

x, t′ x) if and only if: .

Exact tx ≤ t′

x and cx − c′ x ≤ tx − t′ x

Heuristic tx ≤ t′

x and cx ≤ c′ x

Complexity O(|E| · |V |2)

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Outline

Shortest Paths: pre-requisites Solving our carpooling problem Experiments Conclusion

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Problem definition

Oc r s Dc Op Dp

P

1

P

2

P3 P

4

P

5

Cost to be minimized

arrival(pedestrian) - departure(pedestrian) + arrival(driver) - departure(driver) cost = c(P1) + c(P2) + waiting time + 2 × c(P3) + c(P4) + c(P5)

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Solving principle

cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5) Oc Dc Op Dp

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

r2 r3 cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

r2 r3 s1 s2 s3

P3

cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

r2 r3 s1 s2 s3

P3

s4

P4

cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

r2 r3 s1 s2 s3

P3

s4

P4

cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5) s2 s3

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

r2 r3 s1 s2 s3

P3

s4

P4

s2 s3 cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

P5

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Solving principle

Oc Dc Op Dp r1 r2 r3

P1

r4

P2

r2 r3 s1 s2 s3

P3

s4

P4

s2 s3 cost = c(P1) + c(P2) + wait + 2 × c(P3) + c(P4) + c(P5)

P5

s3 r3

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Solving principle

Algo Source Dest. Settled Nodes Problem A1

• p

All N1 SPP A2

• c

All N2 SPP A3 Xin = N1 ∩ N2 All N3 Best Orig. (Dijkstra) A4 dc All N4 SPP (backward) A5 Xoff = N3 ∩ N4 dp N5 Best Orig. (Martins)

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Solving principle

Algo Source Dest. Settled Nodes Problem A1

• p

All N1 SPP A2

• c

All N2 SPP A3 Xin = N1 ∩ N2 All N3 Best Orig. (Dijkstra) A4 dc All N4 SPP (backward) A5 Xoff = N3 ∩ N4 dp N5 Best Orig. (Martins) Integrated approach: select the algorithm with lowest cost in heap.

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Restrictions on pick-up and drop-off points

Integrating knowledge of the problem Goal: restrain the considered pick-up and drop-off point Oc Dc Op Dp r2 r3 r4 s2 s3 s4

A1 A2 A3 A4 A5

Zup Zoff Stop conditions:

• Zup explored → stop A1 and A2
• Zoff explored → stop A3 and A4

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A*: guided search Guiding towards a node d

Principle

Explore nodes close to the destination first Heuristic hd(n) : lower bound of the distance n → d

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A*: guided search Guiding towards a node d

Principle

Explore nodes close to the destination first Heuristic hd(n) : lower bound of the distance n → d

Guiding towards an area Z

HZ(n) = minz∈Zhz(n) Guiding towards the closest node in the area

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A*: guided search Guiding towards a node d

Principle

Explore nodes close to the destination first Heuristic hd(n) : lower bound of the distance n → d

Guiding towards an area Z

HZ(n) = minz∈Zhz(n) Guiding towards the closest node in the area Landmarks: HZ(n) computed only once.

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Outline

Shortest Paths: pre-requisites Solving our carpooling problem Experiments Conclusion

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Experiments : Data

Instances Carpooling Bordeaux-Toulouse-Albi Graph South West of France

• 639 765 nodes
• 21 439 public transportation nodes
• 5 millions edges

Restrictions Pick-up/Drop-off areas

• Entire cities
• Nodes accessible in 10 minutes

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Experimental Results

Settled Labels/node Restrictions Runtime (ms) labels in A5 Cost (s)

• 48 377

5 610 354 21.52 24 607

• 4 316

1 793 205 1.17 24 621

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Experimental Results

Settled Labels/node Restrictions Runtime (ms) labels in A5 Cost (s)

• 48 377

5 610 354 21.52 24 607

• 4 316

1 793 205 1.17 24 621 cities-guided 5 910 928 487 13.99 24 610 cities-guided 853 378 404 1.26 24 623

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Experimental Results

Settled Labels/node Restrictions Runtime (ms) labels in A5 Cost (s)

• 48 377

5 610 354 21.52 24 607

• 4 316

1 793 205 1.17 24 621 cities-guided 5 910 928 487 13.99 24 610 cities-guided 853 378 404 1.26 24 623 10-min-guided 220 122 706 4.54 24 881 10-min-guided 195 120 126 1.15 24 881

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Outline

Shortest Paths: pre-requisites Solving our carpooling problem Experiments Conclusion

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Conclusion

• Study of the Best Origin Problem
• Efficient method to solve the carpooling problem
• Integration of user preferences to provide a further speed up

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Questions?

?

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