Problem Solving
as State Space Search
Brian C. Williams
16.410-13
Sept 13th, 2010
Slides adapted from:
6.034 Tomas Lozano Perez,
Russell and Norvig AIMA
Brian Williams, Fall 10
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Assignments
•  Remember:
Problem Set #1: Java warm up
Out last Wednesday,
Due this Wednesday, September 15th
•  Reading:
–  Today: Solving problems through search [AIMA] Ch. 3.1-4
–  Wednesday: Asymptotic Analysis Lecture 2 Notes of 6.046J;
Recurrences, Lecture 12 Notes of 6.042J.
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1
Recap - Course Objectives
1.  Understand the major types of agents and
architectures:
–  goal-directed vs utility-based
–  deliberative vs reactive
–  model-based vs model-free
2.  Learn the modeling and algorithmic building
blocks for creating agents:
–  Model problem in an appropriate
formal representation.
–  Specify, analyze, apply and implement reasoning
algorithms to solve the problem formulations.
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Recap – Agent Architectures
Mission Goals
Map
Plan
Locate in
World
Monitor &
Diagnosis
Execute
Plan Routes
Maneuver and Track
Functions: Robust, coordinated operation + mobility
It Begins with State Space Search!
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2
Problem Solving as
State Space Search
•  Problem Formulation (Modeling)
–  Problem solving as state space search
•  Formal Representation
–  Graphs and search trees
•  Reasoning Algorithms
–  Depth and breadth-first search
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Most Agent Building Block
Implementations Use Search
Robust Operations:
•  Activity Planning
•  Diagnosis
•  Repair
•  Scheduling
•  Resource Allocation
Mobility:
•  Path Planning
•  Localization
•  Map Building
•  Control Trajectory
Design
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3
Example: Outpost
Logistics Planning
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Image produced for NASA by John Frassanito and Associates.
Can the astronaut get its supplies
safely across a Lunar crevasse?
Astronaut
Goose
Grain
Fox
• Astronaut + 1 item
allowed in the rover.
• Goose alone eats Grain
• Fox alone eats Goose
Rover
Early AI: What are the universal problem solving methods?
Simple
Trivial
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4
Problem Solving as
State Space Search
•  Formulate Goal
–  State
•  Astronaut, Fox, Goose & Grain below crevasse.
•  Formulate Problem
–  States
•  Astronaut, Fox, Goose & Grain
above or below the crevasse.
–  Operators
•  Move: Astronaut drives rover and 1 or 0 items
to other side of crevasse.
–  Initial State
•  Astronaut, Fox, Goose & Grain above crevasse.
•  Generate Solution
–  Sequence of Operators (or States)
•  Move(goose,astronaut), Move(astronaut), . . .
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Goose
Grain
Astronaut
Fox
Astronaut
Goose
Grain
Fox
Grain
Fox
Astronaut
Goose
Grain
Fox
Astronaut
Goose
Goose
Grain
Fox
Astronaut
Goose
Fox
Astronaut
Grain
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5
Goose
Grain
Grain
Astronaut
Fox
Astronaut
Goose
Grain
Fox
Astronaut
Goose
Fox
Grain
Fox
Astronaut
Goose
Goose
Grain
Fox
Goose
Fox
Astronaut
Astronaut
Grain
Goose
Grain
Astronaut
Goose
Grain
Fox
Astronaut
Fox
Grain
Fox
Goose
Fox
Astronaut
Grain
Goose
Grain
Goose
Astronaut
Fox
Grain
Astronaut
Goose
Grain
Fox
Grain
Fox
Astronaut
Goose
Fox
Astronaut
Grain
Fox
Goose
Fox
Astronaut
Goose
Grain
Goose
Grain
Fox
Astronaut
Goose
Astronaut
Grain
Astronaut
Grain
Goose
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10
Grain
Astronaut
Goose
Fox
Astronaut
Goose
Goose
Grain
Fox
Goose
Astronaut Astronaut
Fox
Fox
Astronaut
Grain
Fox
Grain
Astronaut
Fox
Astronaut
Goose
Grain
Goose
Fox
11
Astronaut Astronaut
Fox
Goose
Grain
Fox
Goose
Astronaut
Fox
Grain
Astronaut
Goose
Fox
Astronaut
Grain
Goose
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10
Grain
Goose
Grain
Fox
Astronaut
Goose
Grain
Fox
Astronaut
Goose
Grain
Fox
Astronaut
Grain
Goose
Fox
12
6
Formulation Example: 8-Puzzle
5
4
1
6
1
8
8
7
3
2
7
Start
2
3
4
6
5
Goal
•  States:
integer location for each tile AND …??
•  Operators: move empty square up, down, left, right
•  Initial and
as shown above
Goal States:
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Languages for Expressing
States and Operators for Complex Tasks
•  Swaggert & Lovell assemble
emergency rig for Apollo 13
lithium hydroxide unit.
Image source: NASA.
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7
Example: STRIPS Planning Language
Initial state:
(and (hose a) (clamp b)
(hydroxide-unit c) (on-table a) (on-table b)
(on-table c) (clear a) (clear b) (clear c) (empty arm))
Operators
precondition: (and (clear hose)
(on-table hose)
(empty arm))
pickup hose
effect: (and (not (clear hose))
(not (on-table hose))
(not (empty arm))
(holding arm hose)))
goal (partial state):
(and (connected a b) (connected b c)))
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Problem:
Find a Route from home to MIT
States? Operators?, Initial and Goal State?
16
© MapQuest, Navigation Technologies. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
8
Problem: Compensating for Error Online
1
2
8
7
3
4
6
5
•  Policy, π(v) → e, dictates how to act in all states.
•  Policy π corresponds to a shortest path tree from all vertices to the destination.
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How do we Map Path Planning to
State Space Search?
Start
position
Vehicle translates,
but no rotation
Goal
position
18
9
1. Create Configuration Space
Start
position
Vehicle translates,
but no rotation
Idea: Transform to equivalent
problem of navigating a point.
Goal
position
19
2. Map From Continuous Problem to
Graph Search: Create Visibility Graph
Start
position
Goal
position
20
10
2. Map From Continuous Problem to
Graph Search: Create Visibility Graph
Start
position
Goal
position
21
3. Find Shortest Path (e.g., A*)
Start
position
Goal
position
22
11
Resulting Solution
Start
position
Goal
position
23
A Visibility Graph is a Kind of Roadmap
Start
position
What are the strengths / weaknesses of roadmaps?
What are some other types of roadmaps?
Goal
position
24
12
Voronoi Diagrams
Lines equidistant from CSpace obstacles
25
Roadmaps: Approximate Fixed Cell
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13
Roadmaps: Approximate Fixed Cell
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Roadmaps: Approximate Variable Cell
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14
Roadmaps: Exact Cell Decomposition
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How do we handle large state spaces?
Start
position
RRT,
Ta
RRT,
Tb
Goal
position
Rapid exploring Random Trees
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15
Problem Solving as
State Space Search
•  Problem Formulation (Modeling)
–  Problem solving as state space search
•  Formal Representation
–  Graphs and search trees
•  Reasoning Algorithms
–  Depth and breadth-first search
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Problem Formulation: A Graph
Operator
State
Edge
Vertex
t
h
Head Vertex
of Edge
Tail Vertex
of Edge
neighbors
(adjacent)
Undirected
Graph
Directed
Graph
(two-way streets)
(one-way streets)
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16
Problem Formulation: A Graph
In Degree (2)
Degree (3)
b
b
Out Degree (1)
Undirected
Graph
Directed
Graph
(two-way streets)
(one-way streets)
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Problem Formulation: A Graph
Strongly connected graph
Directed path between all vertices.
Connected graph
Path between all vertices.
Complete graph
All vertices are adjacent.
b
b
a
a
c
d
Directed
Graph
(one-way streets)
c
e
d
e
Sub graph
Subset of vertices
edges between vertices in Subset
Clique
A complete subgraph
(All vertices are adjacent).
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Undirected
Graph
(two-way streets)
34
17
Specifying a Graph: G = <V, E>
Notation:
<a, b, … n> an ordered list of elements a, b …
{a, b, … n} an unordered set of distinguished elements.
Vertices V = {a, b, c, d, e}
b
a
c
Edges E = {<a, b>, <a, c>,
<b, e>,
<c, b>, <c,d>,
<e, d>}
e
d
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Examples of Graphs
Boston
San Fran
Roadmap
Wash DC
LA
Dallas
B
Planning Actions
Put B on C
(graph of possible
states of the world)
C
A
C
A
B
Put C on A
Put C on A
A
B
C
A
C
Put C on B
A
B
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Put A on C
C
B
36
18
Formalizing State Space Search
Input: A search problem S = <g, S, G> where
•  graph g = <V, E>,
•  start vertex S in V, and
•  goal vertex G in V.
Output: A simple path P = <S, v2, … G> in g from S to G.
C
G
A
D
S
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B
37
Simple Paths of Graph g = <V, E>
C
G
A
<S, A, D, C>
D
S
start
end
B
A (directed) path P of graph g is
a sequence of vertices <v1, … vn> in V
such that each successive pair <vi,vi+1> is a (directed) edge in E
A simple path is a path that has no cycles.
A cycle is a subpath where start = end (i.e., repeated vertices).
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19
A Problem Solver Searches
through all Simple Paths
S
C
A
G
A
D
D
C
S
G
B
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A Search Tree Denotes All Simple Paths
S
C
A
B
G
A
D
D
S
C
C
G
D
C
G
G
B
Enumeration is:
•  Complete
•  Systematic
•  Sound
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20
Search Trees
think of a tree as a “family” tree
A tree T is a directed graph, such that
•  there exists exactly one undirected path between any
pair of vertices.
• In degree of each vertex is 1
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Search Trees
Node
(Vertex)
Root
Branch
(Edge)
Leaf
think of a tree as a “family” tree
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21
Search Trees
Parent
Siblings
Child
think of a tree as a “family” tree
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Search Trees
Ancestors
think of a tree as a “family” tree
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22
Search Trees
Descendants
think of a tree as a “family” tree
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Problem Solving as
State Space Search
•  Problem Formulation (Modeling)
–  Problem solving as state space search
•  Formal Representation
–  Graphs and search trees
•  Reasoning Algorithms
–  Depth and breadth-first search
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23
Classes of Search
Blind
Depth-First
Systematic exploration of whole tree
(uninformed)
Breadth-First
until the goal is found.
Iterative-Deepening
Optimal
A*
Use path “length” measure. Find
(informed)
Branch&Bound
“shortest” path.
Heuristic
Hill-Climbing
Use heuristic measure of goodness
(informed)
Best-First
of a node.
Beam
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Depth First Search (DFS)
Local Rule: After visiting node…
•  Visit its children before its siblings
•  Visit its children left to right
1
S
2
A
3
4
C
6
D
5
G
7
B
11
8
C
D
9
C
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G
G
10
48
24
Breadth First Search (BFS)
Local Rule: After visiting node…
•  Visit its siblings, before its children
•  Visit its children left to right
1
S
2
A
4
8
C
5
D
9
G
3
B
7
6
C
D
10
C
G
G
11
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Elements of Algorithm Design
Algorithm Description: (Today)
–  stylized pseudo code, sufficient to analyze and implement the algorithm
(implementation next Wednesday).
Algorithm Analysis: (Wednesday & Monday)
•  Time complexity:
–  how long does it take to find a solution?
•  Space complexity:
–  how much memory does it need to perform search?
•  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?
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25
Problem Solving as
State Space Search
•  Problem Formulation (Modeling)
•  Formal Representation
•  Reasoning Algorithms
–  A generic search algorithm description
–  Depth-first search example
–  Handling cycles
–  Breadth-first search example
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Solve <g = <V, E>, S, G>
using State Space Search
Search States:
•  All simple paths <S, …v> in g starting at S
Initial State:
•  <S>
Operator:
•  Extend a path <S, … v> to <S, … v, u>
for each <v, u> in E
–  call u a child of v
Goal:
•  A simple path <S, …, G> in g
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26
Solve <g = <V, E>, S, G>
using State Space Search
How do we maintain the search state?
•
An ordering on partial paths yet to be expanded
(called a queue Q).
How do we perform search?
•  Repeatedly:
1.  Select next partial path from Q.
2.  Expand it.
3.  Add expansions to Q.
C
•  Terminate when goal G is found.
S
A
D
B
C
D
G
C
G
G
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Simple Search Algorithm: Preliminaries
•  A partial path from S to D is listed in reverse order,
•  e.g., <D, A, S>
•  The head of a partial path is its most recent visited node,
•  e.g., D.
S
A
•  The Q is a list of partial paths,
•  e.g. (<D, A, S>, <C, A, S> …>.
D
C
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B
C
G
D
C
G
G
54
27
Simple Search Algorithm
Let Q be a list of partial paths,
S be the Start node and
G be the Goal node.
1.  Initialize Q with partial path <S>
2.  If Q is empty, fail. Else, pick a partial path N from Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else:
a)  Remove N from Q
b)  Find all children of head(N) and
create a one-step extension of N to each child
c)  Add all extended paths to Q
d)  Go to step 2
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Problem Solving as
State Space Search
•  Problem Formulation (Modeling)
•  Formal Representation
•  Reasoning Algorithms
–  A generic search algorithm description
–  Depth-first search example
–  Handling cycles
–  Breadth-first search example
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28
Depth First Search (DFS)
Idea: After visiting node
•  Visit its children left to right (or top to bottom)
•  Visit its children before its siblings
1
S
2
A
3
4
C
6
D
5
7
B
11
8
C
G
D
9
C
G
G
10
Assume we remove the first element of Q,
Where to Q do we add the path extensions?
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Simple Search Algorithm
Let Q be a list of partial paths,
S be the start node and
G be the Goal node.
1.  Initialize Q with partial path <S>
2.  If Q is empty, fail. Else, pick a partial path N from Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else:
a)  Remove N from Q
b)  Find all children of head(N) and
create a one-step extension of N to each child
c)  Add all extended paths to Q
d)  Go to step 2
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29
Depth-First
Pick first element of Q; Add path extensions to front of Q
C
Q
1
(S)
G
A
2
D
1
3
S
4
B
5
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Simple Search Algorithm
Let Q be a list of partial paths,
S be the Start node and
G be the Goal node.
1.  Initialize Q with partial path <S>
2.  If Q is empty, fail. Else, pick a partial path N from Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else:
a)  Remove N from Q
b)  Find all children of head(N) and
create a one-step extension of N to each child
c)  Add all extended paths to Q
d)  Go to step 2
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30
Depth-First
Pick first element of Q; Add path extensions to front of Q
C
Q
1
(S)
G
A
2
D
1
3
S
4
B
5
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Depth-First
Pick first element of Q; Add path extensions to front of Q
C
Q
1
(S)
2
(A S)
G
A
D
1
3
S
4
B
5
Added paths in blue
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31
Depth-First
Pick first element of Q; Add path extensions to front of Q
C
Q
1
(S)
2
(A S) (B S)
G
A
D
1
3
S
4
B
5
Added paths in blue
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Simple 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>
2.  If Q is empty, fail. Else, pick a partial path N from Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else:
a)  Remove N from Q
b)  Find all children of head(N) and
create a one-step extension of N to each child
c)  Add all extended paths to Q
d)  Go to step 2
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32
Depth-First
Pick first element of Q; Add path extensions to front of Q
C
Q
1
(S)
2
(A S) (B S)
2
G
A
D
1
3
S
4
B
5
Added paths in blue
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Depth-First
Pick first element of Q; Add path extensions to front of Q
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
2
G
A
D
1
S
4
B
5
Added paths in blue
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33
Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
2
G
A
D
1
S
4
B
5
Added paths in blue
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Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
2
G
A
D
1
S
B
5
Added paths in blue
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34
Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
2
G
A
4
D
1
S
B
5
Added paths in blue
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Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
5
(C D A S)(G D A S)
(B S)
2
G
A
4
D
1
S
B
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35
Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
5
(C D A S)(G D A S)
(B S)
2
G
A
4
D
1
S
B
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Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
5
(C D A S)(G D A S)
(B S)
6
(G D A S)(B S)
2
G
A
4
D
1
S
B
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36
Depth-First
Pick first element of Q; Add path extensions to front of Q
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
5
(C D A S)(G D A S)
(B S)
6
(G D A S)(B S)
2
G
A
4
D
1
S
B
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Simple 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>
2.  If Q is empty, fail. Else, pick a partial path N from Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else:
a)  Remove N from Q
b)  Find all children of head(N) and
create a one-step extension of N to each child
c)  Add all extended paths to Q
d)  Go to step 2
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37
Problem Solving as
State Space Search
•  Problem Formulation (Modeling)
•  Formal Representation
•  Reasoning Algorithms
–  A generic search algorithm description
–  Depth-first search example
–  Handling cycles
–  Breadth-first search example
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Issue: Starting at S and moving top to bottom,
will depth-first search ever reach G?
C
G
A
D
S
B
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38
Depth-First
Effort can be wasted in more mild cases
3
C
Q
1
(S)
2
(A S) (B S)
3
(C A S) (D A S) (B S)
4
(D A S) (B S)
5
(C D A S)(G D A S)
(B S)
6
(G D A S)(B S)
2
G
A
4
D
1
S
B
•  C visited multiple times
•  Multiple paths to C, D & G
How much wasted effort can be incurred in the worst case?
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How Do We Avoid Repeat Visits?
Idea:
•  Keep track of nodes already visited.
•  Do not place expanded path on Q if head is a visited node.
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 this always improve efficiency?
•  Visits only a subset of the original paths, such that
each node appears at most once at the head of a visited path.
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39
How Do We Modify The
Simple 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 Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else
a)  Remove N from Q
b)  Find all children of head(N) and
create a one-step extension of N to each child
c)  Add to Q all the extended paths
d)  Go to step 2
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Simple 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; set Visited = {}
2.  If Q is empty, fail. Else, pick some partial path N from Q
3.  If head(N) = G, return N
(goal reached!)
4.  Else
a)  Remove N from Q
b)  Find all children of head(N) not in Visited and
create a one-step extension of N to each child
c)  Add to Q all the extended paths
d)  Add children of head(N) to Visited
e)  Go to step 2
Brian Williams, Fall 10
80
40
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 search algorithms that we
look at later. We have chosen to leave the test in step 3
to maintain uniformity with these future searches.
Brian Williams, Fall 10
81
Problem Solving as
State Space Search
•  Problem Formulation (modeling)
•  Formal Representation
•  Reasoning Algorithms
–  A generic search algorithm description
–  Depth-first search example
–  Handling cycles
–  Breadth-first search example
Brian Williams, Fall 10
82
41
Breadth First Search (BFS)
Idea: After visiting node
•  Visit its children left to right
•  Visit its siblings, before its children
1
S
2
A
4
8
C
5
D
9
3
B
7
6
C
D
10
G
C
G
11
G
Assume we remove the first element of Q,
Where to Q do we add the path extensions?
Brian Williams, Fall 10
83
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
1
Q
Visited
(S)
S
C
2
G
A
3
D
1
S
4
B
5
6
Brian Williams, Fall 10
84
42
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
1
Q
Visited
(S)
S
C
2
G
A
3
D
1
S
4
B
5
6
Brian Williams, Fall 10
85
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
C
G
A
D
1
S
4
B
5
6
Brian Williams, Fall 10
86
43
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
C
2
G
A
D
1
S
4
B
5
6
Brian Williams, Fall 10
87
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
C
2
G
A
D
1
S
4
B
5
6
Brian Williams, Fall 10
88
44
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
C
2
G
A
D
1
S
4
B
5
3
6
Brian Williams, Fall 10
89
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
4
(C A S) (D A S) (G B S)*
G,C,D,B,A,S
C
2
G
A
D
1
S
B
5
3
6
* We could stop here, when the first path to the goal is generated.
Brian Williams, Fall 10
90
45
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
4
(C A S) (D A S) (G B S)*
4
C
2
G
A
D
1
S
G,C,D,B,A,S
B
5
3
6
* We could stop here, when the first path to the goal is generated.
Brian Williams, Fall 10
91
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
4
(C A S) (D A S) (G B S)*
G,C,D,B,A,S
5
(D A S) (G B S)
G,C,D,B,A,S
4
C
2
G
A
5
D
1
S
B
3
6
Brian Williams, Fall 10
92
46
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
4
(C A S) (D A S) (G B S)*
G,C,D,B,A,S
5
(D A S) (G B S)
G,C,D,B,A,S
6
(G B S)
G,C,D,B,A,S
4
C
6
G
2
A
5
D
1
S
B
3
Brian Williams, Fall 10
93
Breadth-First with Visited List
Pick first element of Q; Add path extensions to end of Q
Q
Visited
1
(S)
S
2
(A S) (B S)
A,B,S
3
(B S) (C A S) (D A S)
C,D,B,A,S
4
(C A S) (D A S) (G B S)*
G,C,D,B,A,S
5
(D A S) (G B S)
G,C,D,B,A,S
6
(G B S)
G,C,D,B,A,S
Brian Williams, Fall 10
4
C
6
G
2
A
5
D
1
S
B
3
94
47
Depth-first with Visited List
Pick first element of Q; Add path extensions to front of Q
3
C
Q
Visited
1
(S)
S
2
(A S) (B S)
A, B, S
3
(C A S) (D A S) (B S)
C,D,B,A,S
4
(D A S) (B S)
C,D,B,A,S
5
(G D A S) (B S)
G,C,D,B,A,S
5
2
G
A
4
D
1
S
B
Brian Williams, Fall 10
95
For each search type, where do we place the children on the queue?
Depth First Search (DFS)
S
A
D
Depth-first:
B
C
D
G
Add path extensions to front of Q
Pick first element of Q
C
G
C
G
Breadth First Search (BFS)
S
A
D
Breadth-first:
B
C
D
G
Add path extensions to back of Q
Pick first element of Q
C
G
C
G
Brian Williams, Fall 10
96
48
What You Should Know
•  Most problem solving tasks may be
formulated as state space search.
•  State space search is formalized using
graphs, simple paths, search trees, and
pseudo code.
•  Depth-first and breadth-first search are
framed, among others, as instances of a
generic search strategy.
•  Cycle detection is required to achieve
efficiency and completeness.
Brian Williams, Fall 10
97
Appendix
Brian Williams, Fall 10
98
49
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
1
(S)
C
2
G
A
3
D
1
4
S
5
B
6
7
Brian Williams, Fall 10
99
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
1
(S)
2
(A S) (B S)
C
2
G
A
3
D
1
4
S
5
B
6
7
Added paths in blue
Brian Williams, Fall 10
100
50
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
1
(S)
2
(A S) (B S)
3
(B S) (C A S) (D A S)
C
2
G
A
D
1
4
S
5
B
3
6
7
Added paths in blue
Brian Williams, Fall 10
101
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
1
(S)
2
(A S) (B S)
3
(B S) (C A S) (D A S)
4
(C A S) (D A S) (D B S) (G B S)*
4
C
2
G
A
D
1
S
5
B
3
6
7
Added paths in blue
Revisited nodes in pink
* We could have stopped here, when the first path to the goal was generated.
Brian Williams, Fall 10
102
51
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
4
1
(S)
2
(A S) (B S)
3
(B S) (C A S) (D A S)
4
(C A S) (D A S) (D B S) (G B S)*
5
(D A S) (D B S) (G B S)
C
2
G
A
5
D
1
S
B
3
6
7
Brian Williams, Fall 10
103
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
4
1
(S)
2
(A S) (B S)
3
(B S) (C A S) (D A S)
4
(C A S) (D A S) (D B S) (G B S)*
5
(D A S) (D B S) (G B S)
6
(D B S) (G B S) (C D A S) (G D A S)
C
2
A
G
6
5
D
1
S
B
3
7
Brian Williams, Fall 10
104
52
Breadth-First (without Visited list)
Pick first element of Q; Add path extensions to end of Q
Q
4
1
(S)
2
(A S) (B S)
3
(B S) (C A S) (D A S)
4
(C A S) (D A S) (D B S) (G B S)*
5
(D A S) (D B S) (G B S)
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)(C D B S)(G D B S)
C
7
G
2
A
6
5
D
1
S
B
Brian Williams, Fall 10
3
105
53
MIT OpenCourseWare
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16.410 / 16.413 Principles of Autonomy and Decision Making
Fall 2010
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