Solving Problems by Searching - cs.uwaterloo.caakhtsang/cs486/02Searching.pdf · Otherwise, expand the node. Add resulting nodes to the tree 17. Generic Search Algorithm. Implementation

Solving Problems by

Searching

CS 486/686: Introduction to Artificial Intelligence

1

Announcements

• Enrollment: Wait list, Grads

• Projects

• Undergrads: Groups of 3

• Grads: Solo

• Stuck for ideas? Talk to us!

• My Office: DC 2306B (Accessed through AI Lab)

• Slide versions2

Previously on CS 486

• Fully Observable vs Partially

Observable

• Deterministic vs Stochastic

• Episodic vs Dynamic

• Discrete vs Continuous

• Single agent vs Multi agent

3

Introduction

• Search was one of the first topics

studied in AI

- Newell and Simon (1961) General Problem

Solver

• Central component to many AI systems

- Automated reasoning, theorem proving,

robot navigation, scheduling, game

playing,...

4

Search Problems

• Search problem consists of

• a state space

• a successor function (actions, cost)

• a start state

• Goal test

(N, 1.0)

(E, 1.0)

5

Search Problems

• Path: a sequence of states and operators

• May be associated with a cost c

• Solution: a sequence of actions from the start

state to a goal state

• Optimal solution: a path with minimum cost

…

6

Example: Traveling in Romania

Start

End

•States:

•Initial State:

•Successor Function:

•Goal test:

•Solution:

7

Examples of Search Problems

•States:

•Initial State:


•Goal test:

•Solution:

•States:

•Initial State:


•Goal test:

•Solution:8

Examples of Search Problems

9

Our Definition Excludes...

Chance

Continuous states

All of the above

Partial

Observability

Adversaries

10

What is is a state space?

The world state includes every last detail of the environment

A search state keeps only the details needed for planning (abstraction)

• Problem: Pathing

• States: (x,y) location

• Actions: NSEW

• Successor: update location

only

• Goal test: is (x,y)=END

• Problem: Eat-All-Dots

• States: {(x,y), dot booleans}

• Actions: NSEW

• Successor: update location and

possibly a dot boolean

• Goal test: dots all false

Adapted from UC Berkeley’s CS188 Course

11

Representing Search

• State space graph

- Vertices correspond to states

(one vertex for each state)

- Edges correspond to

successors

- Goal test is a set of goal nodes

• We search for a solution by

building a search tree and

traversing it to find a goal

state

S

G

d

b

pq

c

e

h

a

f

r

12

Search Tree

• A search tree:

• Start state is the root of

the tree

• Children are successors

• A plan is a path in the tree.

A solution is a path from

the root to a goal node.

• For most problems we do

not actually generate the

entire tree

S

B

A

G

S

A B

GB G

G13

Quiz

• Given this state graph, how large is the

search tree?

S

B

A

G

14

Expanding Nodes

• Expanding a node

- Applying all legal operators to the state

contained in the node

- Generating nodes for all corresponding

successor states

15

Example: Traveling in Romania

Start

End

16

Generic Search Algorithm

• Initialize with initial state of the problem

• Repeat

- If no candidate nodes can be expanded return failure

- Choose leaf node for expansion, according to search

strategy

- If node contains goal state, return solution

- Otherwise, expand the node. Add resulting nodes to the

tree

17

Generic Search Algorithm

Implementation Details

19

• Need to keep track of nodes to be expanded (frontier)

• Implement using a queue:

0. Insert node for initial state


20




• Repeat

1. If queue is empty, return failure

2. Dequeue a node


21




• Repeat


2. Dequeue a node

3. If node contains goal state, return solution

4. Expand node


22




• Repeat


2. Dequeue a node

3. If node contains goal state, return solution

4. Expand node

• Search algorithms differ in their queuing function!

Search Strategies

S

G

d

b

p

q

c

e

h

a

f

r

Adapted from UC Berkeley’s CS188 Course

23

Search Strategies

S

a

b

dp

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a 24

Depth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand deepest node first

Implementation: LIFO stack

25

S

a

b

d p

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

Depth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand deepest node first

Implementation: LIFO stack

26

S

a

b

d p

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

Key Properties

•Completeness: Is the alg. guaranteed to find a

solution if the solution exists?

•Optimality: Does the alg. find the optimal solution?

•Time complexity

•Space complexity (size of the fringe) …

b1 node

b nodes

b2 nodes

bm nodes

m tiers

b: branching factor

m: maximum depth

d: depth of shallowest goal node

Number of nodes in tree? 1+b+b2+…+bm=O(bm)27

DFS Properties

•Complete?

•Optimal?

•Time complexity

•Space complexity

…

b

1 node

b nodes

b2 nodes

bm nodes

m tiers

28

Breadth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand shallowest node first

Implementation: FIFO queue

29

S

a

b

dp

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

Breadth-First Search

S

G

d

b

p q

c

e

h

a

f

r

Strategy: Expand shallowest node first

Implementation: FIFO queue

30

S

a

b

dp

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

BFS Properties

•Complete?

•Optimal?

•Time complexity

•Space complexity

…

b

1 node

b nodes

b2 nodes

bm nodes

m tiers

31

depth d

Quiz: DFS vs BFS

…b

…b

32

Iterative Deepening Search

• Can we combine search methods to take advantage

of DFS space complexity and BFS

completeness/shallow solution advantage?

33

IDS Properties•Complete?

•Optimal?

•Time complexity

•Space complexity

…b

1 node

b nodes

b2 nodes

bm nodes

m tiers

Wasteful? Most nodes found in

lowest level of search so not too

bad

34

Cost-Sensitive Search

START

GOAL

d

b

pq

c

e

h

a

f

r

2

9 2

81

8

2

3

2

4

4

15

1

32

2

Recall that BFS was only optimal under some conditions

(i.e. we only cared about number of actions taken). What

can we do if actions have different costs? 35

Uniform Cost Search

Strategy: Expand node with lowest path cost first

Implementation: Priority queue

START

GOAL

d

b

pq

c

e

h

a

f

r

2

9 2

81

8

2

3

2

4

4

15

1

32

2

36

UCS Properties

•Complete?

•Optimal?

•Time complexity

•Space complexity

…

b

C*/𝜀 tiers

37

Summary

38

Criteria DFS BFS IDS UCS

Optimal No Yes Yes Yes

Complete No Yes Yes Yes

Time O(bm) O(bd) O(bd) O(b+1+floor(C*/ε))

Space O(bm) O(bd) O(bd) O(b+1+floor(C*/ε))

• Assumes no knowledge about the problem

- This makes them general but expensive

Figure 3.21 (Russell & Norvig)

Summary

• These algorithms are basically the

same except for the order in which they

expand nodes

• Basically all priority queues with different

ways to determining priorities

• How successful the search is depends

heavily on your model!

39

Questions?

• Next class: Informed search

“Can we make use of additional information?”

Why take a bus to London if we are going to Ottawa?

40

Solving Problems by Searching - cs.uwaterloo.caakhtsang/cs486/02Searching.pdf · Otherwise, expand the node. Add resulting nodes to the tree 17. Generic Search Algorithm. Implementation

Documents