Informed Search & Explorationweb.ntnu.edu.tw/~tcchiang/ai/3_Informed_search-1.pdf · Informed...

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Artificial Intelligence, Spring, 2009

Informed Search &Exploration

Instructor: Tsung-Che [email protected]

Department of Computer Science and Information EngineeringNational Taiwan Normal University

2Artificial Intelligence, Spring, 2009

Outline

Informed Search StrategiesHeuristic FunctionsLocal Search & Optimization ProblemsOnline Search Agents and Unknown

EnvironmentsSummary

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Informed Search Strategies

Can we select the node from the fringe in amore efficient way?

Artificial Intelligence: A Modern Approach, 2nd ed., Figure 3.19



Best-first search selects the node forexpansion according to an evaluationfunction f(n).

There is a whole family of best-firstsearch algorithms with different evaluationfunctions.

A key component of them is a heuristicfunction h(n).

h(n) = estimated cost of the cheapest pathfrom node n to a goal node

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Example of a heuristic functionThe straight line distance



Heuristic functions are the most commonform to impart additional knowledge.

There is one constraint:for a goal node n, h(n) = 0.

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Greedy Best-first Search

Greedy best-first search expands the nodethat is closest to the goal, namely usingf(n) = h(n).

Assume we use the straight-line distanceas the heuristic function.

Exercise: Apply the greedy best-first search.



253

176


366

h(n), straight-line distance

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The solution is not optimal.Notice that the name reveals that it just tries

to get as close to the goal as it can.



ProblemsFrom Iasi to Fagaras

Dead end?Infinite loop?

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Analysis It is neither optimal nor complete.The worst-case time and space complexity is

O(bm), where m is the maximum depth of thesearch space.

The complexity can be reduced substantially,and the amount of reduction depends on theproblem and the quality of the heuristic.


A* Search

It is the most well-known form of best-first search.

It evaluates the nodes by

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

g(n) = the cost to reach the node h(n) = the estimated cost from the node to the

goal

f(n) = estimated cost of thecheapest solution through n

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A* Search

Exercise


A* Search

366=366+0

393=140+253

415=239+176 413=220+193

417=317+100


1

23

4

5

0

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A* Search

A* using TREE-SEARCH is optimal if h(n) isadmissible. It requires that h(n) never overestimates the

cost to reach the goal. Consequently, f(n) never overestimates the true

cost of a solution through n.

In our route-finding example, the straight-line distance is an admissible heuristic.


A* Search

Proof of optimalityG2 is a suboptimal goal node appearing on the

fringe. n is a fringe node on an optimal path.C* is the cost of the optimal solution.

f(G2) = g(G2) + h(G2) = g(G2) > C*f(n) = g(n) + h(n) C* < f(G2)

G2 will not be expanded and A*must return an optimal solution.

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A* Search

What if we use GRAPH-SEARCH?There are two ways to keep the optimality.

(1) Extend GRAPH-SEARCH so that it discardsthe more expensive of any two paths found tothe same node. The extra bookkeeping is messy.

(2) Ensure that the optimal path to anyrepeated state is always the FIRST onefollowed. An extra requirement of h(n) –consistency

(monotonicity)


A* Search

A* using GRAPH-SEARCH is optimal if h(n)is consistent. n: node n: successor generated from n by action a c(n, a, n): cost from n to n

h(n) c(n, a, n) + h(n)

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A* Search

Every consistent heuristic function is alsoadmissible.

It is quite hard to find heuristics that areadmissible but not consistent.

The straight-line distance heuristic isconsistent.

h(n) c(n, a, n) + h(n)

n

goal

n

h(n)

h(n)

c(n, a, n)


A* Search

If h(n) is consistent, then the values of f(n) alongany path are nondecreasing.

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

It follows that A* using GRAPH-SEARCH expandsthe nodes in nondecreasing order of f(n). Hence,the first goal being expanded must be an optimalsolution.

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A* Search

The fact that f-cost are nondecreasing along anypath means that we can draw contours in the statespace.


All nodes inside thecontour labeled 420 havef(n) be at most 420.

380

400

420


A* Search

With uniform-cost search (A* using h(n) =0), the bands will be “circular”around thestart state.

With more accurate heuristics, the bandbecome more narrowly focused around theoptimal path.

A* expands all nodes with f(n) < C*. It might expand some of the nodes right on the

contour with f(n) = C* before selecting a goalnode.

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A* Search

Intuitively, the first solution found mustbe an optimal one.Nodes in all subsequent contours will have

higher f-cost and thus higher g-cost (h=0).

It is also intuitive that A* search iscomplete. It requires that there is only finitely many

nodes with cost no greater than C*. step cost exceeding some finite finite b


A* Search

A* is optimally efficient for any heuristicfunction.No optimal algorithm can guarantee to expand

fewer nodes than A*.This is because any algorithm that does not

expand all nodes with f(n) < C* runs the risk ofmissing the optimal solution.

A* is complete, optimal, and optimallyefficient. What else can you ask for?

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A* Search

Complexity For most problems, the number of nodes within

the goal contour is still exponential in thelength of the solution,

unless …

))((log)()( ** nhOnhnh h*(n) is the true costfrom n to the goal.


A* Search

Space requirement is more serious thanthe time requirement.

It’s impractical to insist on the optimality.One can use heuristic functions that are more

accurate but not admissible.

Next, we will discuss algorithms thatovercome the space problem withoutsacrificing the optimality.

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Demonstration

BFS: http://www.youtube.com/watch?v=NpeEWq09mMw

DFS: http://www.youtube.com/watch?v=Xjppe7LfvnQ

A* Algorithm with Python-Pygame:http://www.youtube.com/watch?v=FNRfSQDF7TA

A-Star Pathfind Test:http://www.youtube.com/watch?v=vxX5A3-I88I

A* Personality-guided Pathfinding :http://www.youtube.com/watch?v=GA5vhBkH29I

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