Lecture 3: State-spaces and Uninformed Search

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Lecture 3 State-spaces and Uninformed Search

• ICS 271 Fall 2006

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Overview

• Intelligent agents problem solving as search
• State-space problem formulation consists of
• state space
• operators, successor function
• start state
• goal states
• Cost function
• The search graph
• A Search Tree is an efficient way to represent
  the search process
• There are a variety of search algorithms,
  including
• Depth-First Search
• Breadth-First Search
• Others which use heuristic knowledge (in future
  lectures)

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Robot block world

• Given a set of blocks in a certain configuration,
• Move the blocks into a goal configuration.
• Example
• (c,b,a) ? (b,c,a)

A
A
Move (x,y)
B
C
C
B
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Operator Description
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Problem-Solving Agents

• Intelligent agents can solve problems by
  searching a state-space
• State-space Model
• the agents model of the world
• usually a set of discrete states
• e.g., in driving, the states in the model could
  be towns/cities
• Goal State(s)
• a goal is defined as a desirable state for an
  agent
• there may be many states which satisfy the goal
• e.g., drive to a town with a ski-resort
• or just one state which satisfies the goal
• e.g., drive to Mammoth
• Operators
• operators are legal actions which the agent can
  take to move from one state to another

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The Traveling Salesperson Problem

• Find the shortest tour that visits all cities
  without visiting any city twice and return to
  starting point.
• State sequence of cities visited
• S0 A

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The Traveling Salesperson Problem

• Find the shortest tour that visits all cities
  without visiting any city twice and return to
  starting point.
• State sequence of cities visited
• S0 A

• SG a complete tour

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Example 8-queen problem
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The Sliding Tile Problem
Up Down Left Right
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State space of the 8 puzzle problem
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Searching the search space

• Uninformed Blind search
• Breadth-first
• uniform first
• depth-first
• Iterative deepening depth-first
• Bidirectional
• Branch and Bound
• Informed Heuristic search
• Greedy search, hill climbing, Heuristics
• Important concepts
• Completeness
• Time complexity
• Space complexity
• Quality of solution

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Breadth-first search

• Expand shallowest unexpanded node
• Fringe nodes waiting in a queue to be explored,
  also called OPEN
• Implementation
• fringe is a first-in-first-out (FIFO) queue,
  i.e., new successors go at end of the queue.

Is A a goal state?
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Breadth-first search

• Expand shallowest unexpanded node
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end

Expand fringe B,C Is B a goal state?
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Breadth-first search

• Expand shallowest unexpanded node
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end

Expand fringeC,D,E Is C a goal state?
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Breadth-first search

• Expand shallowest unexpanded node
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end

Expand fringeD,E,F,G Is D a goal state?
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Example Map Navigation
S start, G goal, other nodes
intermediate states, links legal transitions
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Initial BFS Search Tree
S

D
A

B
D
E
E
C
Note this is the search tree at some particular
point in in the search.
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Search Method 1 Breadth First Search
S
D
A
B
A
E
D
E
S
E
C
F
B
B
S
(Use the simple heuristic of not generating a
child node if that node is a parent to avoid
obvious loops this clearly does not avoid all
loops and there are other ways to do this)
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Breadth-First Search
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Breadth-First-Search ()

• 1. Put the start node s on OPEN
• 2. If OPEN is empty exit with failure.
• 3. Remove the first node n from OPEN and place
  it on CLOSED.
• 4. If n is a goal node, exit successfully with
  the solution obtained by tracing back pointers
  from n to s.
• 5. Otherwise, expand n, generating all its
  successors attach to them pointers back to n,
  and put them at the end of OPEN in some order.
• Go to step 2.
• For Shortest path or uniform cost
• 5 Otherwise, expand n, generating all its
  successors attach to them pointers back to n,
  and put them in OPEN in order of shortest cost
  path.
• This simplified version does not check for
  loops

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What is the Complexity of Breadth-First Search?

• Time Complexity
• assume (worst case) that there is 1 goal leaf at
  the RHS
• so BFS will expand all nodes 1 b b2
  ......... bd O (bd)
• Space Complexity
• how many nodes can be in the queue (worst-case)?
• at depth d-1 there are bd unexpanded nodes in
  the Q O (bd)

d0
d1
d2
G
d0
d1
d2
G
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Examples of Time and Memory Requirements for
Breadth-First Search
Depth of Nodes Solution Expanded Time Memory
0 1 1 millisecond 100 bytes 2 111 0.1
seconds 11 kbytes 4 11,111 11 seconds 1
megabyte 8 108 31 hours 11 giabytes 12 1012
35 years 111 terabytes
Assuming b10, 1000 nodes/sec, 100 bytes/node
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Breadth-First Search (BFS) Properties

• Solution Length optimal
• Expand each node once (can check for duplicates)
• Search Time O(Bd)
• Memory Required O(Bd)
• Drawback requires exponential space

1
2
3
7
4
5
6
15
14
13
12
11
10
9
8
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe Last In First Out (LIPO) queue, i.e.,
  put successors at front

Is A a goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueB,C Is B a goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueD,E,C Is D goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueH,I,E,C Is H goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueI,E,C Is I goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueE,C Is E goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueJ,K,C Is J goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueK,C Is K goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueC Is C goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueF,G Is F goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueL,M,G Is L goal state?
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Depth-first search

• Expand deepest unexpanded node
• Implementation
• fringe LIFO queue, i.e., put successors at front

queueM,G Is M goal state?
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Search Method 2 Depth First Search (DFS)
S
D
A
B
D
E
C
Here, to avoid repeated states assume we dont
expand any child node which appears already in
the path from the root S to the parent. (Again,
one could use other strategies)
F
D
G
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Depth-First Search
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Depth-First-Search ()

• 1. Put the start node s on OPEN
• 2. If OPEN is empty exit with failure.
• 3. Remove the first node n from OPEN and place
  it on CLOSED.
• 4. If n is a goal node, exit successfully with
  the solution obtained by tracing back pointers
  from n to s.
• 5. Otherwise, expand n, generating all its
  successors attach to them pointers back to n,
  and put them at the top of OPEN in some order.
• 6. Go to step 2.

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What is the Complexity of Depth-First Search?

• Time Complexity (d is deepest path)
• assume (worst case) that there is 1 goal leaf at
  the RHS
• so DFS will expand all nodes 1 b b2
  ......... bd O (bd)
• Space Complexity
• how many nodes can be in the queue (worst-case)?
• at depth l lt d we have b-1 nodes
• at depth d we have b nodes
• total (d-1)(b-1) b O(bd)

d0
d1
d2
G
d0
d1
d2 d3 d4

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Techniques for Avoiding Repeated States
S
B
S
B
C
C
S
C
B
S
State Space
Example of a Search Tree

• Method 1
• do not create paths containing cycles (loops)
• Method 2
• never generate a state generated before
• must keep track of all possible states (uses a
  lot of memory)
• e.g., 8-puzzle problem, we have 9! 362,880
  states
• Method 1 is most practical, work well on most
  problems

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Example, diamond networks
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Depth-First Search (DFS) Properties

• Non-optimal solution path
• Incomplete unless there is a depth bound
• Reexpansion of nodes (when the search space is a
  graph),
• Exponential time
• Linear space

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Comparing DFS and BFS

• Same worst-case time Complexity, but
• In the worst-case BFS is always better than DFS
• Sometime, on the average DFS is better if
• many goals, no loops and no infinite paths
• BFS is much worse memory-wise
• DFS is linear space
• BFS may store the whole search space.
• In general
• BFS is better if goal is not deep, if infinite
  paths, if many loops, if small search space
• DFS is better if many goals, not many loops,
• DFS is much better in terms of memory

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Iterative Deepening Search (DFS)

• Every iteration is a DFS with a depth cutoff.
• Iterative deepening (ID)
• i 1
• While no solution, do
• DFS from initial state S0 with cutoff i
• If found goal, stop and return solution, else,
  increment cutoff
• Comments
• IDS implements BFS with DFS
• Only one path in memory
• BFS at step i may need to keep 2i nodes in OPEN

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Iterative deepening search L0
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Iterative deepening search L1
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Iterative deepening search L2
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Iterative Deepening Search L3
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Iterative deepening search
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Comments on Iterative Deepening Search

• Complexity
• Space complexity O(bd)
• (since its like depth first search run different
  times)
• Time Complexity
• 1 (1b) (1 bb2) .......(1 b....bd)
• O(bd) (i.e., asymptotically the same as BFS
  or DFS to limited depth d in the worst case)
• The overhead in repeated searching of the same
  subtrees is small relative to the overall time
• e.g., for b10, only takes about 11 more time
  than BFS
• A useful practical method
• combines
• guarantee of finding an optimal solution if one
  exists (as in BFS)
• space efficiency, O(bd) of DFS
• But still has problems with loops like DFS

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Iterative Deepening (DFS)

• Time

• BFS time is O(bn)
• b is the branching degree
• IDS is asymptotically like BFS,
• For b10 d5 dcut-off
• DFS 110100,,111,111
• IDS 123,456
• Ratio is

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Bidirectional Search

• Idea
• simultaneously search forward from S and
  backwards from G
• stop when both meet in the middle
• need to keep track of the intersection of 2 open
  sets of nodes
• What does searching backwards from G mean
• need a way to specify the predecessors of G
• this can be difficult,
• e.g., predecessors of checkmate in chess?
• what if there are multiple goal states?
• what if there is only a goal test, no explicit
  list?
• Complexity
• time complexity is best O(2 b(d/2)) O(b (d/2))
• memory complexity is the same

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Bi-Directional Search
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Uniform Cost Search

• Expand lowest-cost OPEN node (g(n))
• In BFS g(n) depth(n)

• Requirement
• g(successor)(n)) ? g(n)

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Uniform cost search

• 1. Put the start node s on OPEN
• 2. If OPEN is empty exit with failure.
• 3. Remove the first node n from OPEN and place
  it on CLOSED.
• 4. If n is a goal node, exit successfully with
  the solution obtained by tracing back pointers
  from n to s.
• 5. Otherwise, expand n, generating all its
  successors attach to them pointers back to n,
  and put them in OPEN in order of shortest cost
• Go to step 2.

DFS Branch and Bound
At step 4 compute the cost of the solution
found and update the upper bound U. at step 5
expand n, generating all its successors attach to
them pointers back to n, and put on top of
OPEN. Compute cost of partial path to node and
prune if larger than U. .
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Comparison of Algorithms
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Summary

• A review of search
• a search space consists of states and operators
  it is a graph
• a search tree represents a particular exploration
  of search space
• There are various strategies for uninformed
  search
• breadth-first
• depth-first
• iterative deepening
• bidirectional search
• Uniform cost search
• Depth-first branch and bound
• Repeated states can lead to infinitely large
  search trees
• we looked at methods for for detecting repeated
  states
• All of the search techniques so far are blind
  in that they do not look at how far away the goal
  may be next we will look at informed or
  heuristic search, which directly tries to
  minimize the distance to the goal. Example we
  saw greedy search