A temporally abstracted Viterbi algorithm (TAV) Shaunak
Chatterjee and Stuart Russell University of California, Berkeley
July 17, 2011
Slide 2
Earths history A timescale view Widely varying timescales are
pervasive in data Planning, simulation & state estimation More
efficient if timescale information is cleverly exploited
4.5Ga1Ma10000 yrs 600 yrs 1 yr2 days1 min
Slide 3
Where is Shaunak?
MondayTuesdayWednesdayThursdayFridaySaturdaySunday Berkeley
Barcelona Philadelphia Barcelona Burger Cheese steak
PaellaGazpachoTapasGazpacho Images: berkeley.edu, wikipedia,
food.com
Slide 4
State time trellis t=1t=2t=3t=4t=5 Berkeley Philadelphia
Montreal Toronto Barcelona Madrid Paris Marseille t=6
Slide 5
The Viterbi algorithm Viterbi, 1967 1 2 3 3 4 4 4 4 Berkeley
Philadelphia Montreal Toronto Barcelona Madrid Paris Marseille
t=1t=2t=3t=4t=5 t=6
Slide 6
1 2 3 3 4 4 4 4 3 4 4 4 6 6 8 8 t=1t=2t=3t=4t=5 t=6 Berkeley
Philadelphia Montreal Toronto Barcelona Madrid Paris Marseille The
Viterbi algorithm Viterbi, 1967
Slide 7
1 2 3 3 4 4 4 4 3 4 4 4 6 6 8 10 7 15 17 18 15 18 8
t=1t=2t=3t=4t=5 t=6 Berkeley Philadelphia Montreal Toronto
Barcelona Madrid Paris Marseille The Viterbi algorithm Viterbi,
1967
Slide 8
1 2 3 3 4 4 4 4 3 4 4 4 6 6 8 10 7 15 17 18 15 18 8
t=1t=2t=3t=4t=5 t=6 Berkeley Philadelphia Montreal Toronto
Barcelona Madrid Paris Marseille 13 15 10 11 13 15 The Viterbi
algorithm Viterbi, 1967
Slide 9
1 2 3 3 4 4 4 4 3 4 4 4 6 6 8 10 7 15 17 18 15 18 8
t=1t=2t=3t=4t=5 t=6 Berkeley Philadelphia Montreal Toronto
Barcelona Madrid Paris Marseille 13 15 10 11 13 15 14 15 16 17 11
12 14 15 16 17 18 12 13 15 16 The Viterbi algorithm Viterbi,
1967
Slide 10
1 2 3 3 4 4 4 4 3 4 4 4 6 6 8 10 7 15 17 18 15 18 8
t=1t=2t=3t=4t=5 t=6 Berkeley Philadelphia Montreal Toronto
Barcelona Madrid Paris Marseille 13 15 10 11 13 15 14 15 16 17 11
12 14 15 16 17 18 12 13 15 16 10 The Viterbi algorithm Viterbi,
1967
Slide 11
O(N 2 T) by using dynamic programming N T possible state
sequences Used in signal decoding, speech recognition, parsing and
many other applications For large N and T, this cost could be quite
prohibitive Every possible transition is considered In some cases,
many of these transitions are very unlikely to feature in the
optimal path
Slide 12
Abstraction U.S.A. Canada Spain France Europe North America
Berkeley Philly Montreal Toronto Barcelona Madrid Paris Marseille
t=1t=2t=3t=4t=5 t=6
Slide 13
Abstraction tree Abstractness Europe Spain Barcelona France
Madrid Paris Marseille
Slide 14
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 15
CFDP Step 1: Find the most likely sequence in the current
state-time trellis
Slide 16
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 17
CFDP Step 1: Find the most likely sequence in the current
state-time trellis Step 2: Refine along the most likely
sequence
Slide 18
CFDP Refinement Node-based refinement N.America Europe
N.America Spain France Node Refinement
Slide 19
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 20
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 21
CFDP Step 1: Find the most likely sequence in the current
state-time trellis Step 2: Refine along the most likely sequence
Step 3: Go to step 1 if step 2 performed any refinement; else
terminate
Slide 22
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 23
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 24
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 25
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 26
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 27
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 28
Cost bounds for abstract links Cost of an abstract link should
be a lower bound of the link refinements it encapsulates Standard
heuristic admissibility argument Correctness
Slide 29
Coarse-to-fine dynamic programming (CFDP) Raphael, 2001 t = 1t
= 2t = 3t = 4t = 5t = 6 Berkeley Philly Montreal Toronto Barcelona
Madrid Paris Marseille
Slide 30
Analyzing CFDP Great when large portions of the state-time
trellis are very unlikely Leading to fewer refinements An
appropriate abstraction hierarchy is required
Slide 31
Analyzing CFDP Great when large portions of the state-time
trellis are very unlikely Leading to fewer refinements An
appropriate abstraction hierarchy is required Computation
complexity Best case O(B 2 T (log N) 3 ) B is the branching factor
of the hierarchy Worst case O(N 4 T 2 )
Slide 32
An actual state trajectory JanDec Berkeley San Francisco
Stanford Sardinia Venice Milan Interlaken Los Angeles road trip
Yosemite road trip India trip Europe trip MaySep
Slide 33
Persistence a.k.a. Timescales JanDec Berkeley San Francisco
Stanford Sardinia Venice Milan Interlaken Los Angeles road trip
Yosemite road trip India trip Europe trip MaySep
Slide 34
Persistence and timescales People stay within the same
metropolitan area for weeks Change countries in months Continent
changes are more rare We do not need to consider transitions from
the Bay Area to other continents and countries at every time
step
Slide 35
A set of really good paths Set 1: All paths within California
for the entire month of April Set 2: All paths that visit
California and at least one other state in April Cost(Paths
April-in-California ) < Cost(Paths April-in-1+-states ) | Paths
April-in-California |
