Graduate AI Lecture 2: Search I Instructors: Nihar B. Shah (this - - PowerPoint PPT Presentation

Graduate AI

Lecture 2: Search I

Instructors: Nihar B. Shah (this time)

• J. Zico Kolter

EXAMPLE: PATHFINDING

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Best route?

15780 Spring 2019: Lecture 2

EXAMPLE: 8-PUZZLE

3 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2

Fewest “moves”?

15780 Spring 2019: Lecture 2

SEARCH PROBLEMS

• A search problem has:
• States (configurations)
• Start state and goal states
• Successors: mapping of states to

(action,state,cost) triples

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High-level objective: Find minimum-cost path from s to t in a computationally efficient manner

EXAMPLE: PATHFINDING

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! "

EXAMPLE: PATHFINDING

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! "

EXAMPLE: PATHFINDING

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! "

EXAMPLE: PATHFINDING

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! " 3 2 4 1

15780 Spring 2019: Lecture 2

GRAPH REPRESENTATION

Cost ≥ 0 between each pair of vertices x & y

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# $ % & '

1 3 2 5 1 1 1

! "

High-level objective: Find minimum-cost path from s to t in a computationally efficient manner

starting state goal

15780 Spring 2019: Lecture 2

EXAMPLE: 8-PUZZLE

10 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2

s t

1 1

15780 Spring 2019: Lecture 2

TREE SEARCH

Inputs:

• Problem instance
• “Expansion strategy”

Output:

• Path from s to t

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15780 Spring 2019: Lecture 2

RECALL: BREADTH FIRST SEARCH

• Graph with each cost 1

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# $ % & '

1 1 1 1 1 1

, = 0 , = 1

#

, = 2

$

, = 2

&

, = 2

'

1

#1 #2 #3

! " !

…

• Define , 0 = cost till 0
• Expansion strategy: Expand node with minimum g

, = 3

%

15780 Spring 2019: Lecture 2

TREE SEARCH

function TREE-SEARCH(problem, strategy) set of frontier nodes contains the start state of problem loop

• if there are no frontier nodes then return failure
• choose a frontier node for expansion using strategy
• if the chosen node is t then return the corresponding solution
• else expand the node and add the resulting nodes to the set
• f frontier nodes

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15780 Spring 2019: Lecture 2

UNIFORM COST SEARCH ALGORITHM

Define , 0 = cost till 0 Strategy: Expand node with smallest ,

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# $ % & '

1 3 2 5 1 1

, = 0 , = 1

#

, = 2

$

, = 4

&

, = 6

'

, = 3

%

, = 6

1

, = 7

&

#1 #2 #3 #4 #5 #6 #7

! " ! "

s a d s a e d Frontier:

15780 Spring 2019: Lecture 2

UNINFORMED VS. INFORMED

• Uniform cost search uses no information about

the problem other than the edge costs

• “Uninformed” search
• Often we may have more information…
• “Informed” search

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EXAMPLE: PATHFINDING

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! "

“Going this direction is generally a good idea”

15780 Spring 2019: Lecture 2

EXAMPLE: 8-PUZZLE

17 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2

“Having more blocks in their correct position is generally a good idea”

15780 Spring 2019: Lecture 2

INFORMED SEARCH

• Additional information: For each vertex 0, given

ℎ 0 = heuristic evaluation of cost from 0 to t

• ℎ " = 0

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# $ % & '

1 3 2 5 1 1 1

ℎ = 6 ℎ = 5 ℎ = 2 ℎ = 0 ℎ = 1 ℎ = 6 ℎ = 7

! "

15780 Spring 2019: Lecture 2

GREEDY SEARCH USING HEURISTIC

• Strategy: Expand node with min. value of ℎ

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# $ % & '

1 3 2 5 1 1

ℎ = 6 ℎ = 5

#

ℎ = 6

$

ℎ = 2

&

ℎ = 1

'

ℎ = 2

&

ℎ = 0

1

#1 #2 #3 #4 #5

ℎ = 6 ℎ = 5 ℎ = 2 ℎ = 0 ℎ = 1 ℎ = 6 ℎ = 7

! " ! "

15780 Spring 2019: Lecture 2

A* TREE SEARCH

• Strategy: Expand node with min. value of

6 0 = , 0 + ℎ 0

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# $ % & '

1 3 2 5 1 1 1

ℎ = 6 ℎ = 5 ℎ = 2 ℎ = 0 ℎ = 1 ℎ = 6 ℎ = 7

! "

6 = 6 6 = 6

#

6 = 8

$

6 = 6

&

6 = 7

'

6 = 6

"

#1 #2 #3 #4

!

• Question: Which node is expanded fourth?

15780 Spring 2019: Lecture 2

A* TREE SEARCH

• Should we stop when we discover a goal?

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# $

2 2

ℎ = 3 ℎ = 2 ℎ = 0 ℎ = 1

2 3

Slide adapted from Dan Klein

! "

• No: Only stop when we expand a goal

6 = 3 6 = 4

#

6 = 3

$

6 = 5

"

6 = 4

"

#1 #2 #4

!

#3

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A* PERFORMANCE

Find minimum-cost path from s to t in a computationally efficient manner

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today

15780 Spring 2019: Lecture 2

A* DOESN’T ALWAYS WORK

• Issue: Good path has pessimistic estimate
• Circumvent this issue by being optimistic!

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#

5 1

ℎ = 7 ℎ = 0 ℎ = 6

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Slide adapted from Dan Klein

! "

6 = 7 6 = 7

#

6 = 5

"

#1 #2

!

15780 Spring 2019: Lecture 2

ADMISSIBLE HEURISTICS

ℎ is admissible if for all 0, ℎ 0 ≤ ℎ∗ 0 , where ℎ∗ is the cost of the optimal path from 0 to "

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# $

2 2

ℎ = 3 ℎ = 2 ℎ = 0 ℎ = 1

2 3

! "

§ Aerial distance in pathfinding § ℎ ≡ 0

#

5 1

ℎ = 7 ℎ = 0 ℎ = 6

3

! "

§ §

✅ ❌ ✅ ✅

15780 Spring 2019: Lecture 2

OPTIMALITY OF A* TREE SEARCH

Theorem: If the heuristic is admissible, then the path returned by A* tree search has minimum cost.

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PROOF

• Recall: A* stops when goal " is expanded
• For contradiction, assume " with suboptimal path

is expanded before " with optimal path

• There is a node 0 on the optimal path to " that has

been discovered but not expanded

• 0 should have been expanded before "! ∎

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Adapted from Dan Klein

• 6 0 = , 0 + ℎ 0

≤ , 0 + ℎ∗ 0 = , ">?@ < , "BC?DEFBF = 6("BC?DEFBF)

15780 Spring 2019: Lecture 2

8-PUZZLE HEURISTICS

• ℎ1: #tiles in wrong position
• ℎ2: sum of Manhattan distances of

tiles from goal

• Question: Which of these is

admissible?

• Answer: Both
• Heuristic for designing admissible

heuristics: relax the problem!

27 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2

Example state Goal state

15780 Spring 2019: Lecture 2

DOMINANCE OF HEURISTICS

• ℎ dominates ℎ′ iff ∀0, ℎ 0 ≥ ℎ′(0)
• ℎ1: #tiles in wrong position
• ℎ2: sum of Manhattan distances of

tiles from goal

• Question: What is the dominance

relation between ℎL and ℎM?

• Answer: ℎM dominates ℎL

28 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2

Example state Goal state

15780 Spring 2019: Lecture 2

8-PUZZLE HEURISTICS

• The following table gives the number of nodes

expanded by A* with the two heuristics, averaged

• ver random 8-puzzles, for various solution lengths
• Moral: Good heuristics are crucial!

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Length N∗(OP) N∗(OQ) 16 1301 211 18 3056 363 20 7276 676 22 18094 1219 24 39135 1641

15780 Spring 2019: Lecture 2

TREE SEARCH

• Tree search can expand many nodes

corresponding to the same state

• In a rectangular grid:
• Search tree of depth & has 4R leaves
• There are only 4& states at Manhattan

distance exactly & from any given state

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15780 Spring 2019: Lecture 2

GRAPH SEARCH

function GRAPH-SEARCH(problem, strategy) set of frontier nodes contains the start state of problem loop

• if there are no unexpanded frontier nodes then return failure
• choose an unexpanded frontier node for expansion using strategy,

and add it to the expanded set

• if the node contains a goal then return the corresponding solution
• else expand the node and add the resulting nodes to the set of

frontier nodes, only if not in the expanded set

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Graph search is the same as tree search, except that it never expands a node twice

15780 Spring 2019: Lecture 2

A* GRAPH SEARCH

• Does A* graph search always find the
• ptimal path under an admissible heuristic?

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# $

1 1

ℎ = 2 ℎ = 4 ℎ = 1 ℎ = 1

1 2

%

3

ℎ = 0

Adapted from Dan Klein

! "

6 = 2 6 = 5

#

6 = 2

$

#1 #2

!

6 = 4

%

6 = 6

"

#3 #4

6 = 3

%

#5

• No!

15780 Spring 2019: Lecture 2

CONSISTENT HEURISTICS

• % 0, S = cost of cheapest path from 0 to S
• ℎ is consistent if for every two nodes 0, S,

ℎ 0 ≤ %(0, S) + ℎ(S)

• Question: What is the relation between

admissibility and consistency?

1.

Admissible ⇒ consistent

2.

Consistent ⇒ admissible

3.

They are equivalent

4.

They are incomparable

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S "

• Set y = " above
• Graph in previous slide

15780 Spring 2019: Lecture 2

8-PUZZLE HEURISTICS, CONSISTENT?

• ℎ1: #tiles in wrong position
• ℎ2: sum of Manhattan distances of

tiles from goal

• Poll: Which of these is consistent?
• Answer: Both
• Heuristic for designing admissible

heuristics: relax the problem!

34 5 4 6 1 8 7 3 2 5 4 6 1 8 7 3 2

Example state Goal state

Consistent heuristics yield guarantees for A*graph search (next class)

15780 Spring 2019: Lecture 2

SUMMARY

• Terminology and algorithms:
• Search problems
• Uninformed vs. informed search
• Tree search, graph search,

uniform cost search, greedy, A*

• Admissible and consistent heuristics
• Theorems:
• A* tree search is optimal with admissible ℎ
• A* graph search is optimal with consistent ℎ
• Big ideas:
• Don’t be too pessimistic!

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