problem solving as state space search · pdf fileearly ai: what are the ... problem solving as...
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Problem Solving as State Space Search
Brian C.Williams16.410-13Sep 14th, 2004
Slides adapted from:6.034 Tomas Lozano Perez,Russell and Norvig AIMA Brian Williams, Spring 04 1
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Assignments
• Remember: Problem Set #1: Simple Scheme and Search due Monday, September 20th, 2003.
• Reading: – Solving problems by searching: AIMA Ch. 3
Brian Williams, Spring 04 2
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Brian Williams, Spring 04 3
Complex missions must carefully:
• Plan complex sequences of actions
• Schedule actions
• Allocate tight resources
• Monitor and diagnose behavior
• Repair or reconfigure hardware.
Most AI problems, like these, may be formulated as state space search.
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Outline
• Problem Formulation– Problem solving as state space search
• Mathematical Model– Graphs and search trees
• Reasoning Algorithms– Depth and breadth-first search
Brian Williams, Spring 04 4
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AstronautGooseGrainFox
Rover
Can the astronaut get its supplies safely across the Martian canal?
• Astronaut + 1 item allowed in the rover.
• Goose alone eats Grain• Fox alone eats Goose
Early AI: What are the universal problem solving methods?
TrivialSimpleBrian Williams, Spring 04 5
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Problem Solving as State Space Search
• Formulate Goal– State
• Astronaut, Fox, Goose & Grain across river
• Formulate Problem– States
• Location of Astronaut, Fox, Goose & Grain at top or bottom river bank
– Operators• Astronaut drives rover and 1 or 0 items
to other bank.
• Generate Solution– Sequence of Operators (or States)
• Move(goose,astronaut), Move(astronaut), . . .Brian Williams, Spring 04 6
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Brian Williams, Spring 04 7
AstronautGooseGrainFox
GrainFox
AstronautGoose
GooseGrain
AstronautFox
GooseFox
AstronautGrain
GooseGrainFox
Astronaut
AstronautGooseGrainFox
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Brian Williams, Spring 04 8
AstronautGooseGrainFox
GrainFox
AstronautGoose
AstronautGooseGrainFox
GooseFox
AstronautGrain
AstronautGrainFox
Goose
AstronautGoose
GrainFox
GooseGrain
AstronautFox
AstronautGooseFox
Grain
Grain
AstronautGooseFox
Fox
AstronautGooseGrain
Goose
AstronautFox
Grain
GooseGrainFox
Astronaut
AstronautGrain
GooseFox
AstronautFox
GooseGrain
Astronaut
GooseGrainFox
AstronautGooseGrain
Fox
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Brian Williams, Spring 04 9
AstronautGooseGrainFox
GrainFox
AstronautGoose
AstronautGrainFox
Goose
Goose
AstronautFox
Grain
AstronautGoose
GrainFox
AstronautGooseGrainFox
Grain
AstronautGooseFox
Fox
AstronautGooseGrain
GooseFox
AstronautGrain
GooseGrain
AstronautFox
AstronautGooseFox
Grain
AstronautGrain
GooseFox
AstronautFox
GooseGrain
GooseGrainFox
Astronaut
Astronaut
GooseGrainFox
AstronautGooseGrain
Fox
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Example: 8-Puzzle
5 4
6 1
7 3
8
2
1 2
8
3
7 6
4
5
Start Goal
• States: • Operators: • Goal Test:
integer location for each tile AND …move empty square up, down, left, rightgoal state as given
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Example: Planning Discrete Actions
courtesy of NASA
• Swaggert & Lovell work on Apollo 13 emergency rig lithium hydroxide unit. – Assembly
• Mattingly works in ground simulator to identify new sequence handling severe power limitations.– Planning & Resource Allocation
• Mattingly identifies novel reconfiguration, exploiting LEM batteries for power.– Reconfiguration and Repair
Brian Williams, Spring 04 11
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Planning as State Space Search:STRIPS Operator Representation
Effects specify how to change the set of assertions.
Initial state:(and (hose a)
(clamp b)(hydroxide-unit c) (on-table a) (on-table b) (clear a) (clear b) (clear c) (arm-empty))
goal (partial state):(and (connected a b)
(attached b a)))
precondition: (and (clear hose)(on-table hose)
(empty arm))
effect: (and (not (clear hose))(not (on-table hose))(not (empty arm))
(holding arm hose)))
pickup hose
Note: strips doesn’t
allow derived effects;
you must be complete!}
Available actions Strips operators
Brian Williams, Spring 04 12
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STRIPS Action Assumptions
• Atomic time.
• Agent is omniscient (no sensing necessary).
• Agent is sole cause of change.
• Actions have deterministic effects.
• No indirect effects.
precondition: (and (clear hose)(on-table hose)
(empty arm))
effect: (and (not (clear hose))(not (on-table hose))(not (empty arm))
(holding arm hose)))
pickup hose
Brian Williams, Spring 04 13
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STRIPS Action Schemata• Instead of defining:
pickup-hose and pickup-clamp and …
• Define a schema (with variables ?v):
Brian Williams, Spring 04 14
(:operator pick-up:parameters ((hose ?ob1)):precondition (and (clear ?ob1)
(on-table ?ob1) (empty arm))
:effect (and (not (clear ?ob1))(not (on-table ?ob1))(not (empty arm))(holding arm ?ob1)))
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Outline
• Problem Formulation– Problem solving as state space search
• Mathematical Model– Graphs and search trees
• Reasoning Algorithms– Depth and breadth-first search
Brian Williams, Spring 04 15
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Problem Formulation: A GraphState
Node(vertex)
Operator
Link(edge)
de
Brian Williams, Spring 04 16
DirectedGraph(one-way streets)
neighbors(adjacent)
Incident to edge
Start Vertex of Edge
End Vertex of Edge
UndirectedGraph(two-way streets)
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Problem Formulation: A Graph
In Degree (2) Degree (1)
bOut Degree (1)
b
UndirectedGraph(two-way streets)
DirectedGraph(one-way streets)
Brian Williams, Spring 04 17
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Problem Formulation: A GraphStrongly connected graph
Directed path between all vertices.Connected graph
Path between all vertices.
Complete graphAll vertices are adjacent.
ac
ed
b
ac
de
b
UndirectedGraph(two-way streets)
Sub graphSubset of verticesedges between vertices in Subset
CliqueA complete subgraphAll vertices are adjacent.
DirectedGraph(one-way streets)
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Examples of Graphs
San Fran
Boston
LA Dallas
Wash DC
Roadmap
Brian Williams, Spring 04 19
A B C
A B
C
A B
C
A
B
C
Put C on B
Put C on A
Put B on C
Put C on A
A
B
CPut A on C
Planning Actions
(graph of possible states of the world)
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A Graph
SFO
Bos
LA Dallas
Wash DC
Airline Routes
Brian Williams, Spring 04 20
A Graph G is represented as a pair <V,E>, where:
• V is a set of vertices {v1 …}
• E is a set of (directed) edges {e1, …}
An edge is a pair <v1, v2> of vertices, where
• v2 is the head of the edge,
•and v1 is the tail of the edge
< {Bos, SFO, LA, Dallas, Wash DC}
{<SFO, Bos>,
<SFO, LA>
<LA, Dallas>
<Dallas, Wash DC>
. . .} >
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A Solution is a State Sequence:Problem Solving Searches Paths
S
D
A
C
C
S
B
GA
D start
end
A path is a sequence of edges (or vertices)<S, A, D, C>
Brian Williams, Spring 04 21
Simple path has no repeated vertices.
For a cycle, start = end.
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A Solution is a State Sequence:Problem Solving Searches Paths
S
D
A
C G
C
S
B
GA
D
Represent searched paths using a tree.
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A Solution is a State Sequence:Problem Solving Searches Paths
S
D
BA
C G
C G
D
C G
C
S
B
GA
D
Represent searched paths using a tree.
Brian Williams, Spring 04 23
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Search Trees
Root
Branch(Edge)
Node(vertex)
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Search Trees
Parent(Ancestor)
Child(Descendant)
Siblings
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Search Trees
Ancestors
Descendants
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Outline
• Problem Formulation– Problem solving as state space search
• Mathematical Model– Graphs and search trees
• Reasoning Algorithms– Depth and breadth-first search
Brian Williams, Spring 04 27
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Classes of SearchBlind Depth-First Systematic exploration of whole tree
(uninformed) Breadth-First until the goal is found.
Iterative-Deepening
Heuristic Hill-Climbing Uses heuristic measure of goodness
(informed) Best-First of a node,e.g. estimated distance to.
Beam goal.
Optimal Branch&Bound Uses path “length” measure. Finds
(informed) A* “shortest” path. A* also uses heuristic
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Classes of SearchBlind Depth-First Systematic exploration of whole tree
(uninformed) Breadth-First until the goal is found.
Iterative-Deepening
Brian Williams, Spring 04 29
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Depth First Search (DFS)Idea: After visiting node•Visit children, then siblings•Visit siblings left to right
S
D
BA
C G
C G
D
C G
1
2
3
5
6
7
8 11
Brian Williams, Spring 04 30
4 9 10
S
A
D
C G
C
B
D
C G
G
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Breadth First Search (BFS)Idea: After visiting node• Visit siblings, then children• Visit relatives left to right (top to bottom)
S
D
BA
C G
C G
D
C G
1
2
4
9
5
3
6
10 11
7
S
A
D
C G
C
B
D
C G
G
8
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Elements of Algorithm DesignDescription: (today)
– stylized pseudo code, sufficient to analyze and implement the algorithm (next Monday).
Analysis: (next Wednesday)• Soundness:
– when a solution is returned, is it guaranteed to be correct?• Completeness:
– is the algorithm guaranteed to find a solution when there is one?
• Time complexity: – how long does it take to find a solution?
• Space complexity: – how much memory does it need to perform search?
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Outline
• Problem Formulation: State space search• Model: Graphs and search trees• Reasoning Algorithms: DFS and BFS
– A generic search algorithm– Depth-first search example– Handling cycles– Breadth-first search example
Brian Williams, Spring 04 33
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Simple Search AlgorithmGoing Meta: Search as State Space SearchHow do we maintain the search state?
• A set of partial paths explored thus far.
• An ordering on which partial path to expand next • called a queue Q.
How do we perform search? • Repeatedly:
• Select next partial path from Q. • Expand it.• Add expansions to Q.
• Terminate when goal found.
State Space
S
D
BA
C G
C G
D
C G
Operator
Goal-TestBrian Williams, Spring 04 34
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Simple Search Algorithm
Brian Williams, Spring 04 35
• S denotes the start node• G denotes the goal node.• A partial path is a path from S to some node D,
• e.g., (D A S)
• The head of a partial path is the most recent node of the path, • e.g., D.
• The Q is a list of partial paths, • e.g. ((D A S) (C A S) …).
S
D
BA
C G
C G
D
C G
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Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
Brian Williams, Spring 04 36
1. Initialize Q with partial path (S)2. If Q is empty, fail. Else, pick a partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else:
a) Remove N from Qb) Find all children of head(N) and
create all one-step extensions of N to each child.c) Add all extended paths to Qd) Go to step 2.
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Outline
• Problem Formulation: State space search• Model: Graphs and search trees• Reasoning Algorithms: DFS and BFS
– A generic search algorithm– Depth-first search example– Handling cycles– Breadth-first search example
Brian Williams, Spring 04 37
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Depth First Search (DFS)Idea: • Visit children, then siblings• Visit siblings left to right, (top to bottom).
S
D
BA
C G
C G
D
C G
1
2
3
5
6
7
8 11
Brian Williams, Spring 04 38
4 9 10
S
A
D
C G
C
B
D
C G
G
Assuming that we pick the first element of Q,Then where do we add path extensions to the Q?
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Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
Brian Williams, Spring 04 39
1. Initialize Q with partial path (S)2. If Q is empty, fail. Else, pick a partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else:
a) Remove N from Qb) Find all children of head(N) and
create all the one-step extensions of N to each child.c) Add all extended paths to Qd) Go to step 2.
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Depth-FirstPick first element of Q; Add path extensions to front of Q
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C
S
B
GA
D
Q
1 (S)2345
1
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Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
Brian Williams, Spring 04 41
1. Initialize Q with partial path (S)2. If Q is empty, fail. Else, pick a partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else:
a) Remove N from Qb) Find all children of head(N) and
create all the one-step extensions of N to each child.c) Add all extended paths to Qd) Go to step 2.
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Q
1 (S)2345
C
S
B
GA
D1
Brian Williams, Spring 04 42
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Q
1 (S)2345
(A S)
C
S
B
GA
D1
Added paths in blue
Brian Williams, Spring 04 43
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Q
1 (S)2345
(A S) (B S)
C
S
B
GA
D1
Added paths in blue
Brian Williams, Spring 04 44
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Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
Brian Williams, Spring 04 45
1. Initialize Q with partial path (S)2. If Q is empty, fail. Else, pick a partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else:
a) Remove N from Qb) Find all children of head(N) and
create all the one-step extensions of N to each child.c) Add all extended paths to Qd) Go to step 2.
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Q
1 (S)2345
(A S) (B S)
C
S
B
GA
D1
2
Added paths in blue
Brian Williams, Spring 04 46
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Q
1 (S)2345
(A S) (B S)(C A S) (D A S) (B S)
C
S
B
GA
D
Brian Williams, Spring 04 47
1
2
Added paths in blue
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Q
1 (S)2345
(A S) (B S)(C A S) (D A S) (B S)
C
S
B
GA
D1
2
Added paths in blue
Brian Williams, Spring 04 48
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Brian Williams, Spring 04 49
C
S
B
GA
D1
2
3Q
1 (S)2345
(A S) (B S)(C A S) (D A S) (B S)
Added paths in blue
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Depth-FirstPick first element of Q; Add path extensions to front of Q
Brian Williams, Spring 04 50
C
S
B
GA
D1
2
3Q
1 (S)2345
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)
Added paths in blue
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C
S
B
GA
D
Q
1 (S)2345
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)
1
2
3
4
Depth-FirstPick first element of Q; Add path extensions to front of Q
Added paths in blue
Brian Williams, Spring 04 51
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C
S
B
GA
D
Q
1 (S)234
5
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)(C D A S)(G D A S)(B S)
1
2
3
4
Depth-FirstPick first element of Q; Add path extensions to front of Q
Brian Williams, Spring 04 52
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Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
Brian Williams, Spring 04 53
1. Initialize Q with partial path (S)2. If Q is empty, fail. Else, pick a partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else:
a) Remove N from Qb) Find all children of head(N) and
create all the one-step extensions of N to each child.c) Add all extended paths to Qd) Go to step 2.
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C
S
B
GA
D
Q
1 (S)234
5
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)(C D A S)(G D A S)(B S)
1
2
3
4
Depth-FirstPick first element of Q; Add path extensions to front of Q
Brian Williams, Spring 04 54
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C
S
B
GA
D
Q
6 (G D A S)(B S)
1 (S)234
5
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)(C D A S)(G D A S)(B S)
1
2
3
4
Depth-FirstPick first element of Q; Add path extensions to front of Q
Brian Williams, Spring 04 55
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C
S
B
GA
D
Q
6 (G D A S)(B S)
1 (S)234
5
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)(C D A S)(G D A S)(B S)
1
2
3
4
Depth-FirstPick first element of Q; Add path extensions to front of Q
Brian Williams, Spring 04 56
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Outline
• Problem Formulation: State space search• Model: Graphs and search trees• Reasoning Algorithms: DFS and BFS
– A generic search algorithm– Depth-first search example– Handling cycles– Breadth-first search example
Brian Williams, Spring 04 57
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Issue: Starting at S and moving top to bottom, will depth-first search ever reach G?
C
S
B
GA
D
Brian Williams, Spring 04 58
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C
S
B
GA
D
Q
6 (G D A S)(B S)
1 (S)234
5
(A S) (B S)(C A S) (D A S) (B S)(D A S) (B S)(C D A S)(G D A S)(B S)
1
2
3
4
Depth-FirstEffort can be wasted in more mild cases
• C visited multiple times• Multiple paths to C, D & G
How much wasted effort can be incurred in the worst case?Brian Williams, Spring 04 59
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How Do We Avoid Repeat Visits?
Brian Williams, Spring 04 60
Idea:
• Keep track of nodes already visited.
• Do not place visited nodes on Q.
Does this maintain correctness?
• Any goal reachable from a node that was visited a second time would be reachable from that node the first time.
Does it always improve efficiency?
• Visits only a subset of the original paths, suc thateach node appears at most once at the head of a path in Q.
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How Do We ModifySimple Search Algorithm
Let Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
1. Initialize Q with partial path (S) as only entry; 2. If Q is empty, fail. Else, pick some partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else
a) Remove N from Qb) Find all children of head(N) and
create all the one-step extensions of N to each child.c) Add to Q all the extended paths; d) Go to step 2.
Brian Williams, Spring 04 61
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Simple Search AlgorithmLet Q be a list of partial paths, Let S be the start node and Let G be the Goal node.
Brian Williams, Spring 04 62
1. Initialize Q with partial path (S) as only entry; set Visited = ( )2. If Q is empty, fail. Else, pick some partial path N from Q3. If head(N) = G, return N (goal reached!)4. Else
a) Remove N from Qb) Find all children of head(N) not in Visited and
create all the one-step extensions of N to each child.c) Add to Q all the extended paths; d) Add children of head(N) to Visitede) Go to step 2.
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Testing for the Goal• This algorithm stops (in step 3) when head(N) = G.
• We could have performed this test in step 6 as each extended path is added to Q. This would catch termination earlier and be perfectly correct for all the searches we have covered so far.
• However, performing the test in step 6 will be incorrect for the optimal searches we look at later. We have chosen to leave the test in step 3 to maintain uniformity with these future searches.
Brian Williams, Spring 04 63
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Outline
• Problem Formulation: State space search• Model: Graphs and search trees• Reasoning Algorithms: DFS and BFS
– A generic search algorithm– Depth-first search example– Handling cycles– Breadth-first search example
Brian Williams, Spring 04 64
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Breadth First Search (BFS)Idea: • Visit siblings before their children• Visit relatives left to right
S
D
BA
C G
C G
D
C G
1
2
4
9
5
3
6
10 11
7
S
A
D
C G
C
B
D
C G
G
8
Assuming that we pick the first element of Q,Then where do we add path extensions to the Q?
Brian Williams, Spring 04 65
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
C
S
B
GA
D1
Brian Williams, Spring 04 66
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
C
S
B
GA
D1
Brian Williams, Spring 04 67
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S
C
S
B
GA
D1
Brian Williams, Spring 04 68
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S
C
S
B
GA
D1
2
Brian Williams, Spring 04 69
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S
C
S
B
GA
D1
2
Brian Williams, Spring 04 70
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S
C
S
B
GA
D1
2
3
Brian Williams, Spring 04 71
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S(C A S) (D A S) (G B S)* G,C,D,B,A,S
C
S
B
GA
D1
2
3
* We could stop here, when the first path to the goal is generated.Brian Williams, Spring 04 72
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S(C A S) (D A S) (G B S)* G,C,D,B,A,S
C
S
B
GA
D1
2
3
4
* We could stop here, when the first path to the goal is generated.Brian Williams, Spring 04 73
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
C
S
B
GA
D
Q Visited
6
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S(C A S) (D A S) (G B S)* G,C,D,B,A,S(D A S) (G B S) G,C,D,B,A,S
1
2
3
4
5
Brian Williams, Spring 04 74
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
C
S
B
GA
D
Q Visited
6 (G B S) G,C,D,B,A,S
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S(C A S) (D A S) (G B S)* G,C,D,B,A,S(D A S) (G B S) G,C,D,B,A,S
1
2
3
4
5
6
Brian Williams, Spring 04 75
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Breadth-FirstPick first element of Q; Add path extensions to end of Q
C
S
B
GA
D
Q Visited
6 (G B S) G,C,D,B,A,S
1 (S) S2345
(A S) (B S) A,B,S(B S) (C A S) (D A S) C,D,B,A,S(C A S) (D A S) (G B S)* G,C,D,B,A,S(D A S) (G B S) G,C,D,B,A,S
1
2
3
4
5
6
Brian Williams, Spring 04 76
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Depth-first with visited listPick first element of Q; Add path extensions to front of Q
C
S
B
GA
D
Q Visited
1 (S) S2345
(A S) (B S) A, B, S(C A S) (D A S) (B S) C,D,B,A,S(D A S) (B S) C,D,B,A,S(G D A S) (B S) G,C,D,B,A,S
1
2
3
4
5
Brian Williams, Spring 04 77
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Depth First Search (DFS)S
D
BA
C G
C G
D
C G
Depth-first:
Add path extensions to front of Q
Pick first element of Q
S
D
BA
C G
C G
D
C G
Breadth First Search (BFS)Breadth-first:
Add path extensions to back of Q
Pick first element of Q
For each search type, where do we place the children on the queue?Brian Williams, Spring 04 78
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What You Should Know• Most problem solving tasks may be
formulated as state space search.• Mathematical representations for search are
graphs and search trees.• Depth-first and breadth-first search may be
framed, among others, as instances of a generic search strategy.
• Cycle detection is required to achieve efficiency and completeness.
Brian Williams, Spring 04 79
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Appendix
Brian Williams, Spring 04 80
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
67
1 (S)2345
C
S
B
GA
D1
Brian Williams, Spring 04 81
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
67
1 (S)2345
(A S) (B S)
C
S
B
GA
D1
2
Added paths in blue
Brian Williams, Spring 04 82
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
67
1 (S)2345
(A S) (B S)(B S) (C A S) (D A S)
C
S
B
GA
D1
2
3
Added paths in blue
Brian Williams, Spring 04 83
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
67
1 (S)2345
(A S) (B S)(B S) (C A S) (D A S)(C A S) (D A S) (D B S) (G B S)*
C
S
B
GA
D1
2
3
4
Added paths in blueRevisited nodes in pink* We could have stopped here, when the first path to the goal was generated.
Brian Williams, Spring 04 84
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
67
1 (S)2345
(A S) (B S)(B S) (C A S) (D A S)(C A S) (D A S) (D B S) (G B S)*(D A S) (D B S) (G B S)
C
S
B
GA
D1
2
3
4
5
Brian Williams, Spring 04 85
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
6 (D B S) (G B S) (C D A S) (G D A S)7
1 (S)2345
(A S) (B S)(B S) (C A S) (D A S)(C A S) (D A S) (D B S) (G B S)*(D A S) (D B S) (G B S)
C
S
B
GA
D1
2
3
4
56
Brian Williams, Spring 04 86
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Breadth-First (without Visited list)Pick first element of Q; Add path extensions to end of Q
Q
6 (D B S) (G B S) (C D A S) (G D A S)7 (G B S) (C D A S) (G D A S)
1 (S)2345
(A S) (B S)(B S) (C A S) (D A S)(C A S) (D A S) (D B S) (G B S)*(D A S) (D B S) (G B S)
C
S
B
GA
D1
2
3
4
56
7
Brian Williams, Spring 04 87