Heuristic-search PDF Practice

7/24/2019 Heuristic-search PDF Practice

1/4

1

Heuristic Search

2

Heuristic Search

Heuristicorinformed searchexploits additionalknowledge about the problem that helps direct search tomore promising paths.

Aheuristic function, h(n), provides an estimate of the cosof the path from a given node to the closest goal state.

Must be zero if node represents a goal state.-Example: Straight-line distance from current location to

the goal location in a road navigation problem.

Many search problems are NP-complete so in the worstcase still have exponential time complexity; however a gooheuristic can:

-Find a solution for an average problem efficiently.

-Find a reasonably good but not optimal solutionefficiently.

3

Best-First Search

At each step, best-first search sorts the queue according to aheuristic function.

function BEST-FIRST-SEARCH(problem,EVAL-FN)returnsa solution sequence

inputs: problem, a problem

Eval-Fn, an evaluation function

Queueing-Fn

a function that orders nodes by EVAL-FN

returnGENERAL-SEARCH(problem,Queueing-Fn)

Bucharest

Giurgiu

Urziceni

Hirsova

Eforie

NeamtOradea

Zerind

Arad

Timisoara

LugojMehadia

DobretaCraiova

Sibiu

Fagaras

PitestiRimnicu Vilcea

Vaslui

Iasi

Straightline distanceto Bucharest

0160

242

161

77

151

241

366

193

178

253

329

80

199

244

380

226

234

374

98

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

Dobreta

Craiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

4

Best-First Example

Does not find shortest path to goal (through Rimnicu) since itonly focused on the cost remaining rather than the total cost.

Arad

h=366

Arad

Timisoara ZerindSibiu

h=253 h=329 h=374

Arad


h=329 h=374

Arad OradeaFagaras Rimnicu

h=380 h=193h=366 h=178

Arad

Timisoara ZSibiu

h=329 h=37


h=380 h=193h=366

Sibiu

h=253

Bucharest

h=0


2/4

5

Best-First Properties

Not complete, may follow infinite path if heuristic rates eachstate on such a path as the best option. Most reasonableheuristics will not cause this problem however.

Worst case time complexity is still O(b

m

) where m is themaximum depth.

Since must maintain a queue of all unexpanded states,space-complexity is also O(bm).

However, a good heuristic will avoid this worst-casebehavior for most problems.

6

Beam Search

Space and time complexity of storing and sorting the complequeue can be too inefficient.

Beam search trims queue to the best noptions (nis called tbeam width) at each point.

Focuses search more but may eliminate solution even for fin

seach graphs

Example for n=2.

Arad

h=366

Arad


h=253 h=329 h=374

Arad


h=329 h=374


h=380 h=193h=366 h=178

Arad

Timisoara ZSibiu

h=329 h=37

Arad OradeaFagaras Rimnicuh=380 h=193h=366

Sibiu

h=253

Bucharest

h=0

7

Hill-Climbing

Beam search with a beam-width of 1 is calledhill-climbing.

Pursues locally best option at each point, i.e. the bestsuccessor to the current best node.

Subject to local maxima, plateaux, and ridges.

evaluation

current

state

8

Minimizing Total Path Cost: A* Search

A* combines features of uniform cost search (complete,optimal, inefficient) with best-first (incomplete, non-optimaefficient).

Sort queue by estimated total cost of the completion of apath.

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

If the heuristic function always underestimates the distancto the goal, it is said to beadmissible.

If his admissible, then f(n) never overestimates the actualcost of the best solution through n.


3/4

9

A* Example

Finds the optimal path to Bucharest through Rimnicu and Pitesti

AradArad


Sibiu


Arad

Timisoara Zerind

Sibiu


Arad

Timisoara Zerind

Craiova Pitesti Sibiu

f=0+366

=366

f=140+253

=393

f=118+329

=447

f=75+374

=449

f=280+366

=646

f=239+178

=417

f=146+380

=526

f=118+329

=447

f=220+193

=413

f=75+374

=449

f=280+366

=646

f=239+178

=417

f=146+380

=526

f=118+329

=447

f=75+374

=449

f=300+253

=553

f=317+98

=415

f=366+160

=526

10

Optimality of A*

If his admissible, A* will always find a least cost path to thgoal.

Proof by contradiction:

Let Gbe an optimal goal state with a path cost f*

Let G2be a suboptimal goal state supposedly found by A*Let nbe a current leaf node on an optimal path to G

Since his admissible:f*>=f(n)

If G2is chosen for expansion over nthen:f(n) >= f(G2)

Therefore:f*>=f(G2)

Since G2is a goal state, h(G2)=0, thereforef(G2) = g(G2)f*>= g(G2)

Therefore G2is optimal.Contradiction.Q.E.D.

11

Other Properties of A*

A* is complete as long as

-Branching factor is always finite

-Every operator adds cost at least > 0

Time and space complexity still O(bm) in the worst casesince must maintain and sort complete queue of unexploredoptions.

However, with a good heuristic can find optimal solutions formany problems in reasonable time.

Again, space complexity is a worse problem than time.

12

Heuristic Functions

8-puzzle search space

-Typical solution length: 20 steps

-Average branching factor: 3

-Exhaustive search: 320=3.5 x 109

-Bound on unique states: 9! = 362,880

Admissible Heuristics:

-Number of tiles out of place (h1): 7

-City-block (Manhattan) distance (h2):2+3+3+2+4+2+0+2=18

Start State Goal State

2

45

6

7

8

1 2 3

4

67

81

23

45

6

7

81

23

45

6

7

8

5


4/4

Heuristic-search PDF Practice

Documents

Transcript of Heuristic-search PDF Practice