dd* lite: efficient incremental search with state dominance

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

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 cost=19

S 1 2 3 4 5

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 cost=20… even when costs change.

S 1 2 3 4 5

Preview: DD* Lite

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

D* Lite path

DD* Lite path

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?

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?

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 < e2.

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.

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 we’ll 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.

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.

Time-cost map

Energy-cost map

1 2 3 4 5

• 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.

Time-cost map

Energy-cost map

1 2 3 4 5

• Goal is set to energy=0.• State (x,y,e)=(2,2,15) dominates (2,2,33).

– Less energy expenditure leads to a better solution.

Time-cost map

Energy-cost map

Path Time-cost at (2,2) Energy-cost at (2,2)

1 2 3 4 5

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.

• 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 < DOMINATED.

Solution cost

Dominancerelation

DD* Lite: Extra Bookkeeping

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.

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.

• Update rule for rhs from D* Lite:

• Update rule for rhs from DD* Lite:

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.

DD* Lite: New Update Rule for rhs

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.

DD* Lite Algorithm (1/3)

E = 1 E = 2 E = 1

E = 1 Goal E = 2 E = 2 Start

E = 1 E = 3 E = 1

Initialize()U = ØFor all s

rhs(s),g(s) [∞,NOT_DOMINATED]rhs(sgoal) [0,NOT_DOMINATED]U.insert(sgoal,CalculateKey(sgoal))

E = 1 E = 2 E = 1

E = 1 E = 3 E = 1

U = {}

E = 1 E = 2 E = 1

E = 1 E = 3 E = 1

U = {}

(∞,N,∞,N)

E = 1 Goal

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {}

(∞,N,∞,N)

E = 1 Goal

(0,N,∞,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { (1,2,0) – [0;0]}

(∞,N,∞,N)

E = 1 Goal

(0,N,∞,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { (1,2,0) – [0;0]}

(∞,N,∞,N)

E = 1 Goal

(0,N,∞,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

S = (1,2,0) – [0;0]

(∞,N,∞,N)

E = 1 Goal

(0,N,∞,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

S = (1,2,0) – [0;0]

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

S = (1,2,0) – [0;0]

DominanceNeighbors(s) U Pred(s) = {(1,1,1)(2,1,1)(2,2,1)(1,3,1)(2,3,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

F = (1,2,0) , (2,1,-1), (2,2,-1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

F = (1,2,0) , (2,1,-1), (2,2,-1)

Tempobjf = 1 (1,2,0)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

F = (1,2,0) , (2,1,-1), (2,2,-1)

Temp = [(1,2,0), NOT_DOMINATED]

(rhs(s),D,g(s),D,e)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

U = { }

s' = {}

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = { }

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1]}

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1]}

F = (1,1,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,1,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1]}

Tempobjf = 1 (1,2,0)F = (1,1,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,1,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1]}

Tempobjf = 1 (1,2,0)F = (1,1,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,1,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1]}

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;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)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;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)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(∞,N,∞,N)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;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)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1]}

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1]}

F = (1,2,0) , (2,1,-2), (2,2,-1)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1]}

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)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1]}

F = (1,2,0) , (2,1,-2), (2,2,-1) Tempobjf = 1 (1,2,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1](1,3,1) – [3;1]}

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1](1,3,1) – [3;1]}

F = (1,3,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,3,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1](1,3,1) – [3;1]}

F = (1,3,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,3,0) Tempobjf = 1 (1,2,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1](1,3,1) – [3;1]}

F = (1,3,0) , (1,2,0), (2,2,-1), (3,2,-1), (3,3,0) Tempobjf = 1 (1,2,0)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(1,1,1) – [3;1](2,1,1) – [2;1](2,2,1) – [2;1](1,3,1) – [3;1](2,3,1) – [2;1]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1]}

S = (2,1,1) – [2;1]

(1,N,∞,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1]}

S = (2,1,1) – [2;1]

(1,N,∞,N,1)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1]}

S = (2,1,1) – [2;1]

(1,N,∞,N,1)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1]}

F = (1,2,2) , (2,2,1), (2,1,1)

(1,N,∞,N,1)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1]}

F = (1,2,2) , (2,2,1), (2,1,1)

(1,N,∞,N,1)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1]}

Tempobjf = 1 (1,2,2)F = (1,2,2) , (2,2,1), (2,1,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

F = (1,2,2) , (2,2,1), (2,1,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;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)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;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) Tempobjf = 1 (1,2,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;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) Tempobjf = 1 (1,2,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

F = (3,1,2) , (2,1,1), (2,2,1), (2,3,0), (3,3,2)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

F = (3,1,2) , (2,1,1), (2,2,1), (2,3,0), (3,3,2) Tempobjf = 3 (2,1,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(∞,N,∞,N)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1]}

F = (3,1,2) , (2,1,1), (2,2,1), (2,3,0), (3,3,2) Tempobjf = 3 (2,1,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3]}

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3]}

F = (2,1,1) , (2,2,1), (3,2,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3]}

F = (2,1,1) , (2,2,1), (3,2,1) Tempobjf = 3 (2,1,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(∞,N,∞,N)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3]}

F = (2,1,1) , (2,2,1), (3,2,1) Tempobjf = 3 (2,1,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,2,1) – [2;1](2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1) , (2,2,1), (3,2,1)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

S = (2,2,1) – [2;1]

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,1) , (1,N,∞,N,3)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

S = (2,2,1) – [2;1]

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

(rhs(s),D,g(s),D,e)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

S = (2,2,1) – [2;1]

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

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)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

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)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

Tempobjf = 1 (1,2,2)S’ = {(2,2,3)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

Tempobjf = 1 (1,2,2)S’ = {(2,2,1)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,N,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

Tempobjf = 1 (1,2,2)S’ = {(2,2,1)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

Tempobjf = 1 (1,2,2)S’ = {(2,2,1)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,2,1), (2,1,1)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,2,1), (2,1,1) Tempobjf = 1 (1,2,2)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

Tempobjf = 1 (1,2,2)S’ = {(1,1,1), (1,1,3)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

Tempobjf = 1 (1,2,2)S’ = {(1,1,1), (1,1,3)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

S’ = {(1,1,1), (1,1,3)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,3,0), (2,2,1)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,3,0), (2,2,1) Tempobjf = 1 (1,2,2)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,3,0), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(1,3,1)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,3,0), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(1,3,1)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,3,0), (2,2,1) Tempobjf = 1 (1,2,2)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (2,3,0), (2,2,1)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(2,3,1)}

U = {(2,3,1) – [2;1](2,2,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(1,N,∞,N,1)

(∞,N,∞,N)

F = (1,2,2), (1,3,2), (3,3,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(2,3,1)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,3,0), (2,2,1), (3,2,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(∞,N,∞,N)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,3,0), (2,2,1), (3,2,1) Tempobjf = 4 (2,2,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,3,0), (2,2,1), (3,2,1) Tempobjf = 4 (2,2,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2) Tempobjf = 3 (2,1,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2) Tempobjf = 3 (2,1,1)S’ = {(3,2,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2) Tempobjf = 3 (2,1,1)S’ = {(3,2,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (2,3,0), (3,3,2), (3,1,2)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (3,2,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (3,2,1) Tempobjf = 3 (2,1,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (3,2,1) Tempobjf = 3 (2,1,1)S’ = {(3,1,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (3,2,1) Tempobjf = 3 (2,1,1)S’ = {(3,1,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (2,1,1), (2,2,1), (3,2,1) Tempobjf = 3 (2,1,1)S’ = {(3,1,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1), (2,2,1)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(3,1,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(3,1,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,D,∞,N,3) (1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

F = (1,2,2), (1,1,2), (2,1,1), (3,1,2), (3,2,1), (2,2,1) Tempobjf = 1 (1,2,2)S’ = {(3,1,3)}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,D,∞,N,3) (1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](2,1,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

(rhs(s),D,g(s),D,e)

(1,N,∞,N,1), (1,N,∞,N,3)

(1,D,∞,N,3) (1,N,1,N,1)

(3,N,∞,N,3)

E = 1 Goal

(0,N,0,N,0)

(1,D,∞,N,3) (1,N,1,N,1)

E = 2 Start

(3,N,∞,N,3)

(1,N,∞,N,1), (1,D,∞,N,3)

(4,N,∞,N,3)

U = {(2,3,1) – [2;1](2,2,3) – [2;1](2,3,3) – [2;1](2,1,3) – [2;1](1,1,1) – [3;1](1,3,1) – [3;1](1,1,3) – [3;1](1,3,3) – [3;1](3,2,3) – [3;3](3,1,3) – [4;3]}

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.

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.

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).

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. D* Lite path

DD* Lite path

Simulation Results (1/3)Comparison of planning efficiency with and without

dominance.

Simulation Results (2/3)Ratio of performance cost of planning from scratch

versus replanning, with and without dominance.

Simulation Results (3/3)Comparison of efficiency of planning from scratch

versus replanning, with and without dominance.

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.”

Questions?

dd* lite: efficient incremental search with state dominance

state u

state v

state s2

state s1

state dominancepaper

state dominancedefinition

dominating state

state dominance examplethere

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