Informed (Heuristic) Search Algorithms

Homework #1 assigneddue 10/4 before Exam 1

2

A

B

C

D

G

9

.1

.1

.1

25

Evaluating heuristic functions

No:A (0+7)

N1:B(.1+8.6) N2:G(9+0)

N3:C(.2+8.7)

N4:D(.3+25)

7

20

0

28

25

7

8.6

0

8.7

25

9

25.2

0

25.1

25

No:A (0)

N1:B(.1+25.2) N2:G(9+0)

Is pink-h admissible?

Is green-h admissible?

node A B C D Gh1(n) 7 8.6 8.7 25 0h2(n) 9 25.2 25.1 25 0

or h2 is more informed than h1

5

UCSA*

Uniform-cost and A*

h*

h1

h4

h5

Admissibility/Informedness

h2h3

Max(h2,h3)

Seach Nodes

Heu

ristic

Val

ue

7

8

9

10

11

12

13

Not required for HW and exam purposes

Consistency admissibilityAdmissibility consistency

Consistency

14

UCSg(n)

Greedyh(n)

A*f(n) = g(n) + h(n)

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16

17

18

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IDA*• Basicaly IDDFS, except instead of the iterations being

defined in terms of depth, we define it in terms of f-value– Start with the f cutoff equal to the f-value of the root

node– Loop

• Generate and search all nodes whose f-values are less than or equal to current cutoff.

– Use depth-first search to search the trees in the individual iterations

– Keep track of the node N’ which has the smallest f-value that is still larger than the current cutoff. Let this f-value be next-largest-f-value

-- If the search finds a goal node, terminate. If not, set cutoff = next-largest-f-value and go back to Loop

Properties: Linear memory. #Iterations in the worst case? =

bd (Happens when all nodes have distinct f-values.)

Who will give you admissible h(n)?

21

Relaxed problems

Obtain heuristic from relaxed problems The more relaxed, the easier to compute heuristic, but the less accurate it is

For 8-puzzle problem?

Assume ability to move the tile directly to the place distance= # misplaced tilesAssume ability to move only one position at a time distance = Sum of Manhattan distances.

A problem with fewer restrictions on the actions is called a relaxed problem.

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Different levels of abstraction for shortest path problems on the plane

I

G

I

G“circular abstraction”

I

G“Polygonal abstraction”

I

G

“disappearing-act abstraction”

hD

hC

hP

h*

The obstacles in the shortest path problem canbe abstracted in a variety of ways. --The more the abstraction, the cheaper it is to solve the problem in abstract space --The less the abstraction, the more “informed” the heuristic cost (i.e., the closer the abstract path length to actual path length)

ActualWhy are we inscribing the obstacles rather than circumscribing them?

hDhC hP

h*h0

Cost of computing the heuristic

Cost of searching with the heuristic

Total cost incurred in search

Not always clear where the total minimum occurs• Old wisdom was that the global min was

closer to cheaper heuristics• Current insights are that it may well be far

from the cheaper heuristics for many problems

How informed should the heuristic be?

I

G

I

G“circular abstraction”

I

G“Polygonal abstraction”

I

GhD

hC

hP

h*Actual

Reduced level of abstraction (i.e. more and more concrete)