Informed Search CSE 473 University of Washington.

Informed Search

CSE 473University of Washington

© Daniel S. Weld 2

Last Time

• Agents• Problem Spaces• Blind Search

DFS BFS Iterative Deepening Iterative Broadening

© Daniel S. Weld 3

Best-first SearchGeneralization of breadth first searchPriority queue of nodes to be exploredCost function f(n) applied to each node

Add initial state to priority queueWhile queue not empty Node = head(queue) If goal?(node) then return node

Add children of node to queue

© Daniel S. Weld 4

Old Friends

• Breadth first = best first With f(n) = depth(n)

• Dijkstra’s Algorithm = best first With f(n) = g(n) Where g(n) = sum of edge costs from start to

n Space bound (stores all generated nodes)

© Daniel S. Weld 5

A* Search

• Hart, Nilsson & Rafael 1968 Best first search with f(n) = g(n) + h(n) Where g(n) = sum of edge costs from start to n And h(n) = estimate of lowest cost path n-->goal If h(n) is admissible then search will find optimal

Underestimates cost of any solution which can reached from node{

Space bound since must maintain queue

© Daniel S. Weld 6

European Example

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end

© Daniel S. Weld 7

A* Example

© Daniel S. Weld 8

A* Example

© Daniel S. Weld 9

A* Example

© Daniel S. Weld 10

A* Example


A* Example


Optimality of A*


Optimality Continued


A* Summary

• Pros

• Cons


Iterative-Deepening A*• Like iterative-deepening depth-first, but...• Depth bound modified to be an f-limit

Start with limit = h(start) Prune any node if f(node) > f-limit Next f-limit=min-cost of any node pruned

a

b

c

d

e

fFL=15

FL=21


IDA* Analysis

• Complete & Optimal (ala A*)• Space usage depth of solution• Each iteration is DFS - no priority queue!• # nodes expanded relative to A*

Depends on # unique values of heuristic function In 8 puzzle: few values close to # A* expands In traveling salesman: each f value is unique

1+2+…+n = O(n2) where n=nodes A* expandsif n is too big for main memory, n2 is too long to wait!

• Generates duplicate nodes in cyclic graphs


SMA*• Problem with IDA*: f-limit increases slowly• Must do iterative search again and again• Storing little state between each iteration

Just one number: next highest contour level• SMA*

Uses all available memory to store state• Keeps bounded-size priority queue• When needs memory, drops node with highest f-cost• In ancestor of forgotten nodes, retains cost of best path which

has been dropped. This focuses regeneration. Duplicates minimal work Optimal in a number of nice ways


No

O(b^d)

O(b + N)

Beam Search

• Idea Best first but only keep N best items on

priority queue• Evaluation

Complete?

Time Complexity?

Space Complexity?


Hill Climbing• Idea

Always choose best child; no backtracking

Beam search with |queue| = 1

• Problems? Local maxima

Plateaus

Diagonal ridges

“Gradient ascent”


Randomizing Hill Climbing

• Randomly disobeying heuristic• Random restarts

• [Add something on heavy tailed distribution]


Simulated Annealing• Objective: avoid local minima• Technique:

For the most part use hill climbing When no improvement possible

• Choose random neighbor• Let be the decrease in quality• Move to neighbor with probability e --/T

Reduce “temperature” (T) over time• Pros & cons

Optimal?

temp

If T decreased slowly enough, will reach optimal state

• Widely used• See also

WalkSAT


Limited Discrepancy Search

• Discrepancy bound indicates how often to violate heuristic

• Iteratively increase...

a

b

c

d

e

f

g

h

Assume that heuristic says go left


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Genetic Algorithms • Start with random population

Representation serialized States are ranked with “fitness function”

• Produce new generation Select random pair(s):

• probability ~ fitness Randomly choose “crossover point”

• Offspring mix halves Randomly mutate bits

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Crossover Mutation


Properties

• Importance of careful reprsentation

Informed Search CSE 473 University of Washington.

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