DD Lite: Efficient Incremental Search with State Dominance

1
DD Lite Efficient Incremental Search with State
Dominance

• Paper by G. Ayorkor Mills-Tettey, Anthony Stentz,
  and M. Bernardine Dias

Presented on 1 October 2007 in 16-735 Motion
Planning by Ross A. Knepper and Sean Hyde
2
D Lite

• As we saw in class last week, D Lite is an
  optimal, efficient algorithm for performing
  incremental search in a 2D grid.
• It optimizes an objective function, g(s).
• For example, D Lite can be used to find the
  lowest time-cost path in a grid
• total cost19

S
G
3
D Lite

• As we saw in class last week, D Lite is an
  optimal, efficient algorithm for performing
  incremental search in a 2D grid.
• It optimizes an objective function, g(s).
• For example, D Lite can be used to find the
  lowest time-cost path in a grid
• total cost20
• even when costs change.

S
G
4
Preview DD Lite

• DD Lite modifiesD Lite in order to reason
  about additional cost dimensions such as energy
  expenditure.

DD Lite path
D Lite path
5
Augmented States

• D Lite uses a state like (x, y).
• Some problems have other information that is
  important to find the best path.
• Augment the state with extra terms to indicate
  other factors.
• Example. Battery energy level s(x, y, e).
• What are the implications of that extra dimension?

6
Effect of State Augmentation

• In normal D Lite, two equal-cost paths to the
  same position constitute a tie, which is broken
  arbitrarily.
• With an augmented state, there are more states to
  search, but the answer which optimizes the whole
  state will be found.
• How to handle extra dimension(s) efficiently?

7
State Dominance

• Definition. State s1 dominates another state s2
  when no solution through s2 leads to a solution
  as good as the best solution that can be obtained
  through s1.
• Note that dominance defines only a partial
  ordering on the set of all states, S.
• In practical usage, selection of dominance
  neighbors is always problem-specific.

Example. Suppose there are two ways for a Mars
Rover to get to the same position in the same
amount of time, but one of them uses less battery
power. s1 (xi, yi, e1) dominates s2 (xi,
yi, e2) when e1 lt e2.
8
Dominance Relations

• Definition. A dominance relation exists between
  two states when one state dominates the other.
• Properties
• Non-reflexivity A state cannot dominate itself.
• Non-symmetry If a state u dominates a state v,
  then v does not dominate u.
• Transitivity If a state u dominates another
  state v, and v in turn dominates w, then u
  dominates w.
• In general, it can be hard to know whether two
  states have a dominance relation or not.

9
Why Use State Dominance?

• A dominated state can never lead to a better
  solution than the best solution that can be
  obtained from the dominating state.
• Can prune dominated states out of the search
  without loss of optimality.
• Consequently, the path well find from start to
  goal will be free of dominated states.
• Speeds up the search
• Breaks ties in the best cost path using a second
  meaningful metric.

10
State Dominance Example

• There are separate time- and energy- cost maps
  show the respective time and energy penalties for
  crossing each cell.
• The shade of a 2D cell in the energy cost map
  represents the derivative of energy the rate at
  which the battery level changes.
• The 3D state contains position and an absolute
  energy level, which is the result of traversing
  the cost map.

11
State Dominance Example
Time-cost map
Energy-cost map
S
S
G
G

• There are separate time- and energy- cost maps
  show the respective time and energy penalties for
  crossing each cell.
• The shade of a 2D cell in the energy cost map
  represents the derivative of energy the rate at
  which the battery level changes.
• The 3D state contains position and an absolute
  energy level, which is the result of traversing
  the cost map.

12
State Dominance Example
Time-cost map
S
G

• Goal is set to energy0.
• State (x,y,e)(2,2,15) dominates (2,2,33).
• Less energy expenditure leads to a better
  solution.

13
From D Lite to DD Lite

• Changes include tweaks to
• Objective function
• Algorithm
• Key change
• In addition to keeping track of one-step
  lookahead of the objective function, also keep
  track of one-step lookahead of whether or not a
  state is dominated.

14
DD Lite Extra Bookkeeping

• The g and rhs values of a node are augmented to
  track dominance of the state.
• The dominance component can take two values
  NOT_DOMINATED or DOMINATED.
• We define NOT_DOMINATED lt DOMINATED.

15
DD Lite Tracking Dominance

• Definition of dominance requires us to define
  comparisons between objective functions, which
  are now ordered pairs.
• Define less-than operator
• Note that gdom values only matter when gobjf
  values are equal.

16
DD Lite Tracking Consistency

• Similarly, the definition of consistency requires
  us to define comparisons between g and rhs, which
  are now ordered pairs.
• Define less-than operator
• Just as before, dom values only matter when
  objf values are equal.

17
DD Lite New Update Rule for rhs

• Update rule for rhs from D Lite
• Update rule for rhs from DD Lite

18
DD Lite New Update Rule for rhs

19
DD Lite New Update Rule for rhs

• F(s) is the family of all states which can
  potentially lower the objective function at s.

• D(s) is the set of all states that can cause s to
  be dominated.

20
Domain-Dependent Functions

• Dominate(s, s) returns TRUE iff the state s
  dominates the state s.
• DominanceNeighbors(s) returns the set of all
  states s in S for which
• Dominate(s, s) ? Dominate(s, s)
• is TRUE.
• Note that the set of dominance neighbors need not
  intersect with the sets of predecessors and
  successors.

21
DD Lite Algorithm (1/3)
22
DD Lite Algorithm (2/3)
23
DD Lite Algorithm (3/3)
24
1 2 3
1 2 3
25
1 2 3
1 2 3
Initialize() U Ø For all s rhs(s),g(s) ?
8,NOT_DOMINATED rhs(sgoal) ?
0,NOT_DOMINATED U.insert(sgoal,CalculateKey(sgoa
l))
26
1 2 3
1 2 3
Initialize() U Ø For all s rhs(s),g(s) ?
8,NOT_DOMINATED rhs(sgoal) ?
0,NOT_DOMINATED U.insert(sgoal,CalculateKey(sgoa
l))
U
27
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
Initialize() U Ø For all s rhs(s),g(s) ?
8,NOT_DOMINATED rhs(sgoal) ?
0,NOT_DOMINATED U.insert(sgoal,CalculateKey(sgoa
l))
U
28
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
Initialize() U Ø For all s rhs(s),g(s) ?
8,NOT_DOMINATED rhs(sgoal) ?
0,NOT_DOMINATED U.insert(sgoal,CalculateKey(sgoa
l))
U
29
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U (1,2,0) 00
Initialize() U Ø For all s rhs(s),g(s) ?
8,NOT_DOMINATED rhs(sgoal) ?
0,NOT_DOMINATED U.insert(sgoal,CalculateKey(sgoa
l))
30
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U (1,2,0) 00
31
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
S (1,2,0) 00
32
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
S (1,2,0) 00
33
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
S (1,2,0) 00
34
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,2,0) , (2,1,-1), (2,2,-1)
35
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,2,0) , (2,1,-1), (2,2,-1)
Tempobjf 1 (1,2,0)
36
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,2,0) , (2,1,-1), (2,2,-1)
Temp (1,2,0), NOT_DOMINATED
37
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
s'
Temp (1,2,0), NOT_DOMINATED
38
1 2 3
(rhs(s),D,g(s),D,e)
1 2 3
U
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
Temp (1,2,0), NOT_DOMINATED
39
U (1,1,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
40
U (1,1,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,1,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,1,0)
41
U (1,1,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
Tempobjf 1 (1,2,0)
F (1,1,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,1,0)
42
U (1,1,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
Tempobjf 1 (1,2,0)
F (1,1,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,1,0)
43
U (1,1,1) 31 (2,1,1) 21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
44
U (1,1,1) 31 (2,1,1) 21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,1,0) , (1,2,0), (1,3,0), (2,3,-2),
(3,3,0), (3,2,-1), (3,1,0), (2,1,-1)
45
U (1,1,1) 31 (2,1,1) 21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,1,0) , (1,2,0), (1,3,0), (2,3,-2),
(3,3,0), (3,2,-1), (3,1,0), (2,1,-1)
Tempobjf 1 (1,2,0)
46
U (1,1,1) 31 (2,1,1) 21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,1,0) , (1,2,0), (1,3,0), (2,3,-2),
(3,3,0), (3,2,-1), (3,1,0), (2,1,-1)
Tempobjf 1 (1,2,0)
47
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
48
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,2,0) , (2,1,-2), (2,2,-1)
49
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,2,1) (1,2,2) (1,2,3) (1,2,4) (1,1,1) (2,1,1)
(2,2,1) (1,3,1) (2,3,1)
F (1,2,0) , (2,1,-2), (2,2,-1)
Tempobjf 1 (1,2,0)
50
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,2,0) , (2,1,-2), (2,2,-1)
Tempobjf 1 (1,2,0)
51
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
52
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,3,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,3,0)
53
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,3,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,3,0)
Tempobjf 1 (1,2,0)
54
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
F (1,3,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,3,0)
Tempobjf 1 (1,2,0)
55
U (1,1,1) 31 (2,1,1) 21 (2,2,1)
21 (1,3,1) 31 (2,3,1) 21
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,1) (2,1,1) (2,2,1) (1,3,1) (2,3,1)
56
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
S (2,1,1) 21
57
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
S (2,1,1) 21
58
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
S (2,1,1) 21
59
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
F (1,2,2) , (2,2,1), (2,1,1)
60
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2) , (2,2,1), (2,1,1)
61
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2) , (2,2,1), (2,1,1)
62
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
F (1,2,2) , (2,2,1), (2,1,1)
63
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
64
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
65
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
F (1,1,2) , (1,2,1), (1,3,1), (2,3,0),
(3,3,2), (3,2,1), (3,1,2), (2,1,1)
66
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 1 (1,2,1)
F (1,1,2) , (1,2,1), (1,3,1), (2,3,0),
(3,3,2), (3,2,1), (3,1,2), (2,1,1)
67
U (2,2,1) 21 (2,3,1) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 1 (1,2,1)
F (1,1,2) , (1,2,1), (1,3,1), (2,3,0),
(3,3,2), (3,2,1), (3,1,2), (2,1,1)
68
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
69
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
F (3,1,2) , (2,1,1), (2,2,1), (2,3,0), (3,3,2)
70
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 3 (2,1,1)
F (3,1,2) , (2,1,1), (2,2,1), (2,3,0), (3,3,2)
71
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 3 (2,1,1)
F (3,1,2) , (2,1,1), (2,2,1), (2,3,0), (3,3,2)
72
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (3,2,3) 33
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
73
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (3,2,3) 33
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
F (2,1,1) , (2,2,1), (3,2,1)
74
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (3,2,3) 33
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 3 (2,1,1)
F (2,1,1) , (2,2,1), (3,2,1)
75
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (3,2,3) 33
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
Tempobjf 3 (2,1,1)
F (2,1,1) , (2,2,1), (3,2,1)
76
U (2,2,1) 21 (2,3,1) 21 (2,2,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (3,2,3) 33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(1,1,3) (1,2,3) (2,2,3) (3,2,3) (3,1,3)
F (2,1,1) , (2,2,1), (3,2,1)
77
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
S (2,2,1) 21
78
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
S (2,2,1) 21
79
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (2,2,1) 21
80
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
F (1,1,2), (1,2,2), (1,2,2), (2,3,0), (3,3,2),
(3,2,1), (3,1,2), (2,1,1)
81
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,1,2), (1,2,2), (1,2,2), (2,3,0), (3,3,2),
(3,2,1), (3,1,2), (2,1,1)
82
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (2,2,3)
Tempobjf 1 (1,2,2)
83
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (2,2,1)
Tempobjf 1 (1,2,2)
84
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (2,2,1)
Tempobjf 1 (1,2,2)
85
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (2,2,1)
Tempobjf 1 (1,2,2)
86
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
87
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
F (1,2,2), (2,2,1), (2,1,1)
88
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (2,2,1), (2,1,1)
89
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (1,1,1), (1,1,3)
Tempobjf 1 (1,2,2)
90
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (1,1,1), (1,1,3)
Tempobjf 1 (1,2,2)
91
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
S (1,1,1), (1,1,3)
92
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
93
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
94
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
95
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
F (1,2,2), (2,3,0), (2,2,1)
96
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (2,3,0), (2,2,1)
97
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (2,3,0), (2,2,1)
S (1,3,1)
98
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (2,3,0), (2,2,1)
S (1,3,1)
99
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (2,3,0), (2,2,1)
100
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (1,3,3)
31 (3,2,3) 33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
F (1,2,2), (2,3,0), (2,2,1)
101
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (1,3,3)
31 (3,2,3) 33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
F (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1)
102
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (1,3,3)
31 (3,2,3) 33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1)
103
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (1,3,3)
31 (3,2,3) 33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1)
S (2,3,1)
104
U (2,3,1) 21 (2,2,3) 21 (1,1,1)
31 (1,3,1) 31 (1,1,3) 31 (1,3,3)
31 (3,2,3) 33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1)
S (2,3,1)
105
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
106
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
F (2,3,0), (2,2,1), (3,2,1)
107
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 4 (2,2,1)
F (2,3,0), (2,2,1), (3,2,1)
108
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 4 (2,2,1)
F (2,3,0), (2,2,1), (3,2,1)
109
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
F (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)
110
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)
111
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)
S (3,2,3)
112
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)
S (3,2,3)
113
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
F (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)
114
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
F (2,1,1), (2,2,1), (3,2,1)
115
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (3,2,1)
116
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (3,2,1)
S (3,1,3)
117
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (3,2,1)
S (3,1,3)
118
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,1)
Tempobjf 3 (2,1,1)
F (2,1,1), (2,2,1), (3,2,1)
S (3,1,3)
119
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
F (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1),
(2,2,1)
120
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1),
(2,2,1)
121
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1),
(2,2,1)
S (3,1,3)
122
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1),
(2,2,1)
S (3,1,3)
123
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (1,1,1) 31 (1,3,1) 31 (1,1,3)
31 (1,3,3) 31 (3,2,3) 33 (3,1,3)
43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
Tempobjf 1 (1,2,2)
F (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1),
(2,2,1)
S (3,1,3)
124
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (2,1,3) 21 (1,1,1) 31 (1,3,1)
31 (1,1,3) 31 (1,3,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
125
U (2,3,1) 21 (2,2,3) 21 (2,3,3)
21 (2,1,3) 21 (1,1,1) 31 (1,3,1)
31 (1,1,3) 31 (1,3,3) 31 (3,2,3)
33 (3,1,3) 43
1 2 3
1 2 3
(rhs(s),D,g(s),D,e)
DominanceNeighbors(s) U Pred(s)
(2,2,3) (1,1,3) (1,2,3) (1,3,3) (2,3,3) (3,3,3)
(3,2,3) (3,1,3) (2,1,3)
126
Properties of DD Lite (1/3)

• Theorem. ComputeShortestPath() expands a
  non-dominated state in the space at most twice
  namely once when it is locally underconsistent
  and once when it is locally overconsistent.

127
Properties of DD Lite (2/3)

• Theorem. ComputeShortestPath() expands a
  dominated state in the space at most four times
  namely at most once when it is underconsistent
  and not dominated, once when it is overconsistent
  and not dominated, once when it is
  underconsistent and dominated, and once when it
  is overconsistent and dominated.

128
Properties of DD Lite (3/3)

• Theorem. After termination of ComputeShortestPath
  (), one can follow an optimal path from sstart to
  sgoal by always moving from the current state s,
  starting at sstart, to any non-dominated
  successor s that minimizes c(s s) gobjf (s)
  until sgoal is reached (breaking ties
  arbitrarily).

129
Simulation

• A set of square worlds ranging from 8x8 to 64x64
  were solved using D Lite and DD Lite.
• Costs were set randomly.
• Start and goal states at opposite corners.
• 10 trials for each size.
• Discretized energy levels.

DD Lite path
D Lite path
130
Simulation Results (1/3)

• Comparison of planning efficiency with and
  without dominance.

131
Simulation Results (2/3)

• Ratio of performance cost of planning from
  scratch versus replanning, with and without
  dominance.

132
Simulation Results (3/3)

• Comparison of efficiency of planning from scratch
  versus replanning, with and without dominance.

133
Summary

• DD Lite is an incremental search algorithm that
  reasons about state dominance.
• DD Lite allows search to be extended into higher
  dimensional state spaces without the full cost
  that it would normally entail.
• Requires you to know which states are dominance
  neighbors.
• DD Lite is sound, complete, optimal, and
  efficient.

134
Questions?