Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th...

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Hyperstar: A multi-path A* algorithm 7 th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell Dept of Civil & Environmental Engineering Imperial College London

Transcript of Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th...

Page 1: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Hyperstar: A multi-path A*

algorithm

7th Mathematics of Networks MeetingBristol University

27 June 2008

Professor Michael G H Bell

Dept of Civil & Environmental Engineering

Imperial College London

Page 2: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Rapid spread of Satnav

• Satnav devices have spread rapidly (4m in UK at present)

• Relatively accurate electronic maps (NavTeq, Teleatlas, etc.)

• However, link travel times are crude and seem to be based on free flow values

• TMC/TPEG congestion warning messages lead to rerouting

Page 3: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

ARIAdNE: Penalty A* algorithm

Page 4: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

ARIAdNE field trials

Page 5: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Garmin vs ARIAdNE arrival times

Page 6: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Garmin vs ARIAdNE routes

Page 7: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Garmin - ARIAdNE

ARIADNE betterARIADNE worse

Page 8: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Why ARIAdNE works better

© Crown copyright/database right 2006. An Ordnance Survey/EDINA supplied service

Page 9: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Comments

• The penalty A* algorithm works

• But, nice to be able to generate all routes of interest at once

• Spiess and Florian hyperpath algorithm does something similar

• Can it be adapted?

Page 10: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Dijkstra’s algorithm

1. Start at destination and set uj = ∞ for j ≠

destination and udest = 0

2. Put dest in OPEN

3. Search OPEN for smallest ui

4. For nodes j reached from i if uj > ui + cij

then uj = ui + cij

5. Put nodes j in OPEN and transfer i to CLOSED

6. Return to Step 3 until origin in CLOSED

Page 11: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

A* algorithm

1. Start at destination and set uj = ∞ for j ≠

destination and udest = 0

2. Put dest in OPEN

3. Search OPEN for smallest ui + hi,orig

4. For nodes j reached from i if uj > ui + cij

then uj = ui + cij

5. Put nodes j in OPEN and transfer i to CLOSED

6. Return to Step 3 until origin is CLOSED

Page 12: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Hyperpath algorithm

• Hyperpath is a bundle of potentially optimal paths

• Every link has both a cost and a service frequency

• Where there is choice within the hyperpath, allocation is proportional to service frequency (the strategy)

• Elemental path only added to hyperpath if the expected cost of travel is reduced

Page 13: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Hyperpath algorithm

1. Start at destination and set uj = ∞ for j ≠

destination, udest = 0 and Fi = 0

2. Put dest in OPEN

3. Search OPEN for smallest ui

4. For nodes j reached from i if uj > ui + cij then

uj = (Fi ui + fij cij ) / (Fi + fij), Fi = Fi + fij and add

link (i,j) to HYPERPATH

5. Put nodes j in OPEN and transfer i to CLOSED

6. Return to Step 3 until origin is CLOSED

Page 14: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Reinterpreting the hyperpath

algorithm

• Note: 1 / fij = link headway = max link delay = dij

• Allocation: Minmax exposure to delay

� pij dij = pik dik if links (i,j) and (i,k) attractive

� pij ∝ 1 / dij = fij

• Attractive: Add link to hyperpath if “expected” travel time reduced. Expected by whom? A risk averse traveller.

Page 15: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Singular hyperpath:

No delay

Page 16: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Hyperpath:

Medium max link delays

Page 17: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Hyperpath:

Large max link delays

Page 18: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Effect of A* speed-up

Page 19: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

1. Start at destination and set uj = ∞ for j ≠

destination, udest = 0 and Fi = 0

2. Put dest in OPEN

3. Search OPEN for smallest ui + hi,orig

4. For nodes j reached from i if uj > ui + cij then

uj = (Fi ui + fij cij ) / (Fi + fij), Fi = Fi + fij and add

link (i,j) to HYPERPATH

5. Put nodes j in OPEN and transfer i to CLOSED

6. Return to Step 3 until origin is CLOSED

H* algorithm

Page 20: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Discussion

• Approaches for handling uncertain delays in road networks examined in context of vehicle navigation � must be efficient �based on A*

• Approach 1: Avoid unreliable links �Penalty A* method

• Approach 2: Seek bundle of routes that may be optimal by adapting A*, with actual route determined by TMC/TPEGmessages

Page 21: Hyperstar: A multi-path A* algorithm - MoN) meetingsHyperstar: A multi-path A* algorithm 7th Mathematics of Networks Meeting Bristol University 27 June 2008 Professor Michael G H Bell

Danke für IhreAufmerksamkeit!

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