Blind SearchBlind Search

Russell and Norvig: Chapter 3, Sections 3.4 – 3.6

Slides adapted from:robotics.stanford.edu/~latombe/cs121/2003/home.htmby Prof. Jean-Claude Latombe

Blind Search

Depth first searchBreadth first searchIterative deepeningNo matter where the goal is, these algorithms will do the same thing.

Depth-First1

fringe

Depth-First1

2

fringe

Depth-First1

2

3

fringe

Breadth First1

fringe

Breadth First1

2

fringe

Breadth First1

2

fringe

3

Breadth First1

2

fringe

3

4

Generic Search AlgorithmGeneric Search AlgorithmPath search(start, operators, is_goal) { fringe = makeList(start); while (state=fringe.popFirst()) { if (is_goal(state)) return pathTo(state); S = successors(state, operators);

fringe = insert(S, fringe); } return NULL;}

Depth-first: insert=prepend; Breadth-first: insert=append

Performance Measures of Performance Measures of Search AlgorithmsSearch Algorithms

CompletenessIs the algorithm guaranteed to find a solution when there is one?OptimalityIs this solution optimal?Time complexityHow long does it take?Space complexityHow much memory does it require?

Important ParametersImportant Parameters

Maximum number of successors of any state branching factor b of the search tree

Minimal length of a path in the state space between the initial and a goal state depth d of the shallowest goal node in the search tree

Evaluation of Breadth-Evaluation of Breadth-first Searchfirst Search

b: branching factor d: depth of shallowest goal node Complete Optimal if step cost is 1 Number of nodes generated: 1 + b + b2 + … + bd = (bd+1-1)/(b-1) = O(bd) Time and space complexity is O(bd)

Big O NotationBig O Notation

g(n) is in O(f(n)) if there exist two positive constants a and N such that:

for all n > N, g(n) af(n)

Time and Memory Time and Memory RequirementsRequirements

d #Nodes Time Memory

2 111 .01 msec 11 Kbytes

4 11,111 1 msec 1 Mbyte

6 ~106 1 sec 100 Mb

8 ~108 100 sec 10 Gbytes

10 ~1010 2.8 hours 1 Tbyte

12 ~1012 11.6 days 100 Tbytes

14 ~1014 3.2 years 10,000 Tb

Assumptions: b = 10; 1,000,000 nodes/sec; 100bytes/node

Evaluation of Depth-first Evaluation of Depth-first SearchSearch

b: branching factor d: depth of shallowest goal node m: maximal depth of a leaf node Complete only for finite search tree Not optimal Number of nodes generated: 1 + b + b2 + … + bm = O(bm) Time complexity is O(bm) Space complexity is O(bm) or O(m)

Depth-Limited StrategyDepth-Limited Strategy

Depth-first with depth cutoff k (maximal depth below which nodes are not expanded)

Three possible outcomes: Solution Failure (no solution) Cutoff (no solution within cutoff)

Iterative Deepening Iterative Deepening StrategyStrategy

Repeat for k = 0, 1, 2, …:Perform depth-first with depth cutoff k Complete Optimal if step cost =1Space complexity is: O(bd) or O(d)Time complexity is: (d+1)(1) + db + (d-1)b2 + … + (1) bd = O(bd) Same as BFS! WHY???

CalculationCalculation

db + (d-1)b2 + … + (1) bd

= bd + 2bd-1 + 3bd-2 +… + db= bd(1 + 2b-1 + 3b-2 + … + db-d)

bd(i=1,…,ib(1-i))

= bd (b/(b-1))2

Comparison of StrategiesComparison of Strategies

Breadth-first is complete and optimal, but has high space complexity Bad when branching factor is high

Depth-first is space efficient, but neither complete nor optimal Bad when search depth is infinite

Iterative deepening is asymptotically optimal

Uniform-Cost StrategyUniform-Cost Strategy• Each step has some cost > 0.• The cost of the path to each fringe node N is g(N) = costs of all steps.• The goal is to generate a solution path of minimal cost.• The queue FRINGE is sorted in increasing cost.

S0

1A

5B

15CS G

A

B

C

5

1

15

10

5

5

G11

G10

Repeated StatesRepeated States

8-queens

No

assembly planning

Few

1 2 34 5

67 8

8-puzzle and robot navigation

Many

search tree is finitesearch tree is infinite

Avoiding Repeated StatesAvoiding Repeated States

Requires comparing state descriptions Breadth-first strategy: Keep track of all generated states If the state of a new node already

exists, then discard the node

Avoiding Repeated StatesAvoiding Repeated States

Depth-first strategy: Solution 1:

Keep track of all states associated with nodes in current path

If the state of a new node already exists, then discard the node

Avoids loops Solution 2:

Keep track of all states generated so far If the state of a new node has already been

generated, then discard the node

Space complexity of breadth-first

Summary

Search strategies: breadth-first, depth-first, and variants Evaluation of strategies: completeness, optimality, time and space complexity Avoiding repeated states