3_Informed_search_algorithms.pdf

7/24/2019 3_Informed_search_algorithms.pdf

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Informed search algorithms

Chapter 3 Section 3.5 & 3.6


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Outline

Best-first search

Greedy best-first search

A*search

Heuristics


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Best-first search

Idea: use an evaluation functionf(n) for each node

estimate of "desirability"

Expand most desirable unexpanded node

Implementation:Order the nodes in fringe in decreasing order of desirability

Special cases: greedy best-first search

A*search


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Romania with step costs in km


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Greedy best-first search

Evaluation functionf(n) = h(n) (heuristic) = estimate of cost from

nto goal

e.g., hSLD(n)= straight-line distance from nto Bucharest

Greedy best-first search expands the node that appearsto be

closest to goal


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Greedy best-first search example


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Properties of greedy best-first search

Complete?Nocan get stuck in loops, e.g., IasiNeamt

IasiNeamt

Time?O(b

m

), but a good heuristic can give dramaticimprovement

Space?O(bm) -- keeps all nodes in memory

Optimal?No


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A*search

Idea: avoid expanding paths that are already

expensive

Evaluation functionf(n) = g(n) + h(n)

g(n) = cost so far to reach n

h(n)= estimated cost from nto goal

f(n) = estimated total cost of path through nto goal


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A*search example


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A*search example


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A*search example


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A*search example


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A*search example


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A*search example


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Admissible heuristics

A heuristic h(n)is admissibleif for every node n,

h(n) h*(n), where h*(n)is the true cost to reach the goal

state from n.

An admissible heuristic never overestimatesthe cost to reach

the goal, i.e., it is optimistic

Example: hSLD(n) (never overestimates the actual road

distance)

Theorem: If h(n) is admissible, A*using TREE-SEARCHis

optimal


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Optimality of A*(proof by contradiction)

Suppose some suboptimal goal G2has been generated and is in the fringe.Let nbe an unexpanded node in the fringe such that n is on a shortestpath to an optimal goal G.

f(G2) = g(G2) since h(G2) = 0

> g(G) since G2is suboptimal

f(n) since h is admissible

Since f(G2) f(n), A* will NEVER select G2for expansion.


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Consistent heuristics

A heuristic is consistentif for every node n, every successor n'of ngeneratedby any action a,

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

If his consistent, we havef(n') = g(n') + h(n')

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

i.e.,f(n)is non-decreasing along any path.

Theorem: If h(n)is consistent, A*using GRAPH-SEARCHis optimal


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Optimality of A*

A*expands nodes in order of increasingfvalue

Gradually adds "f-contours" of nodes

Contour ihas all nodes withf=fi, wherefi< fi+1


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Properties of A*

Complete?Yes (unless there are infinitely many nodes with f

f(G) )

Time?Exponential

Space?Keeps all nodes in memory

Optimal?Yes


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E.g., for the 8-puzzle:

h1(n) = number of misplaced tiles

h2(n) = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

h1(S) = ?

h2(S) = ?


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E.g., for the 8-puzzle:

h1(n) = number of misplaced tiles

h2(n) = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

h1(S) = ?8

h2(S) = ?3+1+2+2+2+3+3+2 = 18

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3_Informed_search_algorithms.pdf