artificialintelligence_4_informedsearch

Grundlagen der kunstlichen Intelligenz Informed Search

Matthias Althoff

TU Munchen

17. October 2014

Matthias Althoff Informed Search 17. October 2014 1 / 22

Organization

Informed search strategies

Heuristic functions

The content is covered in the AI book by the section Solving Problems bySearching, Sec. 5-6.


Informed Search Strategies

Informed Search

Requires problem-specific knowledge beyond the problem definition.

Can find solutions more efficiently than uninformed search.

Informed search uses indications whether a state is more promisingthan another to reach a goal.

The choice of the next node is based on an evaluation functionf (n), which itself is often based on a heuristic function h(n).

h(n) is problem specific with the only constraints that it isnonnegative and h(n) = 0, where n is a goal node.

All presented informed search strategies are implemented identically touniform-cost search, except that f instead of g is used in the priorityqueue.

Heuristic

Heuristic refers to the art to achieve good solutions with limitedknowledge and time based on experience.



Greedy Best-First Search: Idea

Expands the node that is closest to the goal by using just the heuristic functionso that f (n) = h(n).Example: Route-finding problem in Romania. We use the straight line distance tothe goal as h(n), which is listed below:

Bucharest

Giurgiu

Urziceni

Hirsova

Eforie

NeamtOradea

Zerind

Arad

Timisoara

LugojMehadia

DobretaCraiova

Sibiu

Fagaras

PitestiRimnicu Vilcea

Vaslui

Iasi

Straightline distanceto Bucharest

0160242161

77151

241

366

193

178

25332980

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

75120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86



Greedy Best-First Search: Example

Arad

366




Zerind

Arad

Sibiu Timisoara

253 329 374




Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

329 374

366 176 380 193




Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

Sibiu Bucharest

329 374

366 380 193

253 0

Note that the solution is not optimal since the path is 32 km longer thanthrough Rimnicu Vilcea and Pitesti.



Greedy Best-First Search: Another Example

Greedy best-first search is incomplete when not using an explored set,even on finite state spaces.E.g. when going from Iasi to Faragas, Neamt is first expanded due tocloser straight line distance, but this is a dead end.

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

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



Greedy Best-First Search: Performance

Reminder: Branching factor b, depth d, maximum length m of any path.

Completeness: Yes, if an explored set is used, otherwise no (seeprevious example).

Optimality: No (see previous example).

Time complexity: The worst-case is that the heuristics is misleadingthe search such that the solution is found last: O(bm). But a goodheuristic can provide dramatic improvement.

Space complexity: Equals time complexity since all nodes are stored:O(bm).



A* search: Idea

The most widely known informed search is A* search (pronouncedA-star search).

It evaluates nodes by combining the path cost g(n) and the estimatedcost to the goal h(n):

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

where h(n) has to be underestimated, i.e. it has to be less than theactual cost.

f (n) never overestimates the cost to the goal and thus thealgorithm keeps searching for paths that might have lower cost to thegoal than the previously found ones.



A* Search: Example

Straight line distance is an underestimation of the cost to the goal, whichare used for h(n).

Arad

366=0+366

Nodes are labeled with f = g + h



A* Search: Example


Zerind

Arad

Sibiu Timisoara

447=118+329 449=75+374393=140+253




A* Search: Example


Zerind

Arad

Sibiu

Arad

Timisoara

Rimnicu VilceaFagaras Oradea

447=118+329 449=75+374

646=280+366 413=220+193415=239+176 671=291+380




A* Search: Example


Zerind

Arad

Sibiu

Arad

Timisoara

Fagaras Oradea

447=118+329 449=75+374

646=280+366 415=239+176

Rimnicu Vilcea

Craiova Pitesti Sibiu

526=366+160 553=300+253417=317+100

671=291+380




A* Search: Example


Zerind

Arad

Sibiu

Arad

Timisoara

Sibiu Bucharest



447=118+329 449=75+374

646=280+366

591=338+253 450=450+0 526=366+160 553=300+253417=317+100

671=291+380




A* Search: Example


Zerind

Arad

Sibiu

Arad

Timisoara

Sibiu Bucharest



Bucharest Craiova Rimnicu Vilcea

418=418+0

447=118+329 449=75+374

646=280+366

591=338+253 450=450+0 526=366+160 553=300+253

615=455+160 607=414+193

671=291+380




A* Search: Effects of the Heuristics

The heuristics steers the search towards the goal.A* expands nodes in order of increasing f value, so that f -contoursof nodes are gradually added.Each contour i has all nodes with f = fi , where fi < fi+1.

O

Z

A

T

L

M

DC

R

F

P

G

BU

H

E

V

I

N

380

400

420

S



A* Search: Pruning

Given the cost C of the optimal path, it is obvious that only pathsare expanded with f (n) < C . A* never expands nodes, where f (n) C .

For instance, Timisoara is not expanded in the previous example. Wesay that the subtree below Timisoara is pruned.

The concept of pruning eliminating possibilities from considerationwithout having to examine them brings enormous time savings andis similarly done in other areas of AI.

Especially for large problems, pruning becomes important.



A* Search: Performance

Reminder: Branching factor b, depth d, maximum length m of any path.

Time and space complexity of A* is quite involved and we will not derivethe results. They are presented in terms of the relative error = (h h)/h, where h is the estimated and h is the actual cost fromthe root to the goal.

Completeness: Yes, if costs are greater than 0 (otherwise infiniteoptimal paths of zero cost exist).

Optimality: Yes (if cost are positive).

Time complexity: We only consider the easiest case: The statespace has a single goal and all actions are reversible: O(bd) (withoutproof)

Space complexity: Equals time complexity since all nodes are stored.



Alternatives of A* Search

One of the big disadvantages of A* search is the possibly huge spaceconsumption. This can be alleviated by extensions, which we only mentionhere:

Iterative-deepening A*: adapts the idea of iterative deepening toA*. The main difference is that the f -cost (g + h) is used for cutoff,rather than the depth.

Recursive best-first search: a simple recursive algorithm with linearspace complexity. Its structure is similar to the one ofrecursive-depth-first search, but keeps track of the f -value of the bestalternative path.

Memory-bounded A* and simplified memory-bounded A*: Thosealgorithms work just as A* until the memory is full. The algorithmsdrop less promising paths to free memory.


Heuristic Functions

Heuristic Functions

For the shortest route in Romania, the straight line distance is an obviousunderapproximating heuristics.

2

Start State Goal State

1

3 4

6 7

5

1

2

3

4

6

7

8

5

8

This is not always so easy, as we will show for the 8-puzzle. Problem size:

On average 22 steps to a solution.Average branching factor is about 3 (4 moves when empty tile is inthe middle, 2 in a corner, and 3 on an edge).An exhaustive search visits about 322 3.1 1010 states.Remembering previously visited states reduces to 9!/2 = 181, 440distinct states, but there are already 15!/2 1013 for the 15-puzzle.Matthias Althoff Informed Search 17. October 2014 14 / 22

Heuristic Functions

Heuristic Functions for the 8-Puzzle

Two commonly used candidates that underestimate the costs to the goal:

h1: the number of misplaced tiles (e.g. in the figure h1 = 8).Admissible since any misplaced tile has to be moved at least once.

h2: the sum of the distances to the goal positions using horizontaland vertical movement. Admissible since all a move can do is bringthe tile one step closer. In the figure:

tile 1 2 3 4 5 6 7 8 sumsteps 3 1 2 2 2 3 3 2 18

2


1

3 4

6 7

5

1

2

3

4

6

7

8

5

8

(the true cost is 26)


Heuristic Functions

Effective Branching Factor

One way of characterizing the quality of a heuristics is the effectivebranching factor b.Given:

Number of nodes N generated by the A* search.

A uniform tree with depth d (each node has the same fractionalnumber b of children)

Thus,N + 1 = 1 + b + (b)2 + . . .+ (b)d .

E.g. if A* generates a solution at depth 5 using 52 nodes, b = 1.92 since

53 1 + 1.92 + (1.92)2 + . . .+ (1.92)5.

The branching factor makes it possible to compare heuristics applied toproblems of different size. Why?


Heuristic Functions

Comparison of the Heuristics for the 8-Puzzle

1200 random problems with solution length from 2 24 (100 for each evennumber):

Search Cost (nodes generated) Effective Branching Factord IDS A*(h1) A*(h2) IDS A*(h1) A*(h2)

2 10 6 6 2.45 1.79 1.79

4 112 13 12 2.87 1.48 1.45

6 680 20 18 2.73 1.34 1.30

8 6384 39 25 2.80 1.33 1.24

10 47127 93 39 2.79 1.38 1.22

12 3644035 227 73 2.78 1.42 1.24

14 539 113 1.44 1.23

16 1301 211 1.45 1.25

18 3056 363 1.46 1.26

20 7276 676 1.47 1.27

22 18094 1219 1.48 1.28

24 39135 1641 1.48 1.26


Heuristic Functions

Domination of a Heuristics

Question: Is h2 always better than h1?

Answer: Yes.

Reason:

For every node we have that h2(n) > h1(n). We say that h2dominates h1.

A* using h2 will never expand more nodes than with h1 (exceptspossibly for some nodes with f (n) = C ):

A* expands all nodes with

f (n) < C h(n) < C g(n),

where g(n) is fixed. Since h2(n) > h1(n), less nodes are expanded.


Heuristic Functions

Heuristics from Relaxed ProblemsIdea: The previous heuristics h1 and h2 are perfectly accurate for simplifiedversions of the 8-puzzle.

Method: Formalize a problem definition and remove restrictions.

Result:

One obtains a relaxed problem, i.e. a problem with more freedom, whosestate-space graph is a supergraph of the original problem (see figure).

An optimal solution in the original problem is automatically a solution in therelaxed problem, but the relaxed problem might have better solutions due toadded edges in the state-space graph.

Hence, the cost of an optimal solution in the relaxed problem is anunderapproximative heuristic for the original problem.

supergraph

graph


Heuristic Functions

Heuristics from Relaxed Problems: 8-Puzzle Example

A tile can move from square A to B if 1 2, where

1: B is blank

2: A is adjacent to B

We generate three relaxed problems by removing one or two conditions:

1 remove 1: A tile can move from square A to B if A is adjacent to B.2 remove 2: A tile can move from square A to B if B is blank.3 remove 1 and 2: A tile can move from square A to B.

From the first relaxed problem, we can generate h2 and from the thirdrelaxed problem, we can derive h1.

If one is not sure which heuristics is better, one can apply in each step

h(n) = max{h1(n), h2(n), . . . , hM(n)}.

Why?


Heuristic Functions

Heuristics from Pattern Databases

Underapproximative heuristics can also be obtained from subproblems.

Those solution costs are underapproximations and can be stored in adatabase.

The number of subproblems has to be much less than the original problemsto not exceed storage capacities.

The figure shows a subproblem of an 8-puzzle, where only the first 4 tiles have tobe brought into a goal position (the cost is obtained by optimal backwards searchfrom the goal):


1

2

3

4

6

8

5

21

3 6

7 8

54


Summary

Summary

Informed search methods require a heuristic function that estimatesthe cost h(n) from a node n to the goal:

Greedy best-first search expands nodes with minimal h(n), which isnot always optimal.

A* search expands nodes with minimal f (n) = g(n) + h(n). A* iscomplete and optimal for underapproximative h(n).

The performance of informed search depends on the quality of theheuristics. Possibilities to obtain good heuristics are relaxedproblems and pattern databases.


Informed Search StrategiesHeuristic FunctionsSummary

artificialintelligence_4_informedsearch

Documents

uninformed search

touniformcost search

examplegreedy best

goal node

routefinding problem

problem definition

evaluation functionf

heuristic functionso