Search Strategies Reading: Russell’s Chapter 3

Search StrategiesSearch Strategies

Reading: Russell’s Chapter 3Reading: Russell’s Chapter 3

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Search strategiesSearch strategies

A search strategy is defined by picking the A search strategy is defined by picking the order of node order of node expansionexpansion

Strategies are evaluated along the following dimensions:Strategies are evaluated along the following dimensions:– completenesscompleteness: does it always find a solution if one exists?: does it always find a solution if one exists?– time complexitytime complexity: number of nodes generated/expanded: number of nodes generated/expanded– space complexityspace complexity: maximum number of nodes in memory: maximum number of nodes in memory– optimalityoptimality: does it always find a least-cost solution?: does it always find a least-cost solution?

Time and space complexity are measured in terms of Time and space complexity are measured in terms of – b:b: maximum branching factor of the search tree maximum branching factor of the search tree– d: d: depth of the least-cost solutiondepth of the least-cost solution– mm: maximum depth of the state space (may be ∞): maximum depth of the state space (may be ∞)

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Uninformed search strategiesUninformed search strategies

UninformedUninformed search strategies use only the search strategies use only the information available in the problem definitioninformation available in the problem definition

Breadth-first searchBreadth-first search

Uniform-cost searchUniform-cost search

Depth-first searchDepth-first search

Depth-limited searchDepth-limited search

Iterative deepening searchIterative deepening search

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Expand shallowest unexpanded nodeExpand shallowest unexpanded node

ImplementationImplementation::– fringefringe is a FIFO queue, i.e., new successors go is a FIFO queue, i.e., new successors go

at endat end

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at endat end

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Properties of breadth-first searchProperties of breadth-first search

Complete?Complete? Yes (if Yes (if bb is finite) is finite)

Time?Time? 1+b+b1+b+b22+b+b33+… ++… +bbdd + ( + (bbd+1d+1-b-b) = O(b) = O(bd+1d+1))

Space?Space? O(bO(bd+1d+1)) (keeps every node in memory) (keeps every node in memory)

Optimal?Optimal? Yes (if cost = 1 per step) Yes (if cost = 1 per step)

SpaceSpace is the bigger problem (more than time) is the bigger problem (more than time)

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BFS; evaluationBFS; evaluation Two lessons:Two lessons:

– Memory requirements are a bigger problem than its execution time.Memory requirements are a bigger problem than its execution time.– Exponential complexity search problems cannot be solved by uninformed search Exponential complexity search problems cannot be solved by uninformed search

methods for any but the smallest instances.methods for any but the smallest instances.

Table is based on: b = 10; 10000/second; 1000 bytes/node Table is based on: b = 10; 10000/second; 1000 bytes/node

DEPTH2DEPTH2 NODESNODES TIMETIME MEMORYMEMORY

22 11001100 0.11 seconds0.11 seconds 1 megabyte1 megabyte

44 111100111100 11 seconds11 seconds 106 megabytes106 megabytes

66 101077 19 minutes19 minutes 10 gigabytes10 gigabytes

88 101099 31 hours31 hours 1 terabyte1 terabyte

1010 10101111 129 days129 days 101 terabytes101 terabytes

1212 10101313 35 years35 years 10 petabytes10 petabytes

1414 10101515 3523 years3523 years 1 exabyte1 exabyte

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Extension of BFS:Extension of BFS:– Expand node with Expand node with lowest path costlowest path cost

Implementation: Implementation: fringefringe = queue ordered by path cost. = queue ordered by path cost.

UCS is the same as BFS when all step-costs are equal.UCS is the same as BFS when all step-costs are equal.

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Completeness: Completeness: – YES, if step-cost > YES, if step-cost > (small positive constant) (small positive constant)

Time complexity:Time complexity:– Assume C* the cost of the optimal solution.Assume C* the cost of the optimal solution.

– Assume that every action costs at least Assume that every action costs at least – Worst-case: Worst-case:

Space complexity:Space complexity:– The same as time complexityThe same as time complexity

Optimality: Optimality: – nodes expanded in order of increasing path cost.nodes expanded in order of increasing path cost.– YES, if complete.YES, if complete.

)( /* CbO

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Expand deepest unexpanded nodeExpand deepest unexpanded node ImplementationImplementation::

– fringe fringe = LIFO queue, i.e., put successors at front= LIFO queue, i.e., put successors at front

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Properties of depth-first searchProperties of depth-first search

Complete?Complete? No: fails in infinite-depth spaces, spaces No: fails in infinite-depth spaces, spaces with loopswith loops– Modify to avoid repeated states along pathModify to avoid repeated states along path

complete in finite spacescomplete in finite spaces

Time?Time? O(bO(bmm)): terrible if : terrible if mm is much larger than is much larger than dd– but if solutions are dense, may be much faster than but if solutions are dense, may be much faster than

breadth-firstbreadth-first

Space?Space? O(bm), O(bm), i.e., linear space!i.e., linear space! Optimal?Optimal? No No

–

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Depth-limited search

= depth-first search with depth limit l,

i.e., nodes at depth l have no successors

Recursive implementation:

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Is DFS with depth limit Is DFS with depth limit ll..– i.e. nodes at depth i.e. nodes at depth l l have no successors.have no successors.– Problem knowledge can be usedProblem knowledge can be used

Solves the infinite-path problem.Solves the infinite-path problem.

If If l < dl < d then incompleteness results. then incompleteness results.

If If l > dl > d then not optimal. then not optimal.

Time complexity:Time complexity:

Space complexity:Space complexity:

Depth-limited searchDepth-limited search

O(bl )

O(bl)26

Iterative deepening search

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Iterative deepening search

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Properties of iterative deepening searchProperties of iterative deepening search

Complete?Complete? Yes Yes

Time?Time? (d+1)b(d+1)b00 + d b + d b11 + (d-1)b + (d-1)b22 + … + b + … + bdd = O(b = O(bdd))

Space?Space? O(bd)O(bd)

Optimal?Optimal? Yes, if step cost = 1 Yes, if step cost = 1

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ID search, evaluation

Completeness:– YES (no infinite paths)

Time complexity:– Algorithm seems costly due to repeated generation of certain states.– Node generation:

– level d: once– level d-1: 2– level d-2: 3– …– level 2: d-1– level 1: d

N(IDS) (d)b (d 1)b2 ... (1)bd

N(BFS) b b2 ... bd (bd 1 b)

O(bd )

N(IDS) 50 400 3000 20000100000 123450

N(BFS) 10 100 100010000100000 999990 1111100

Num. Comparison for b=10 and d=5 solution at far right

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Summary of algorithmsSummary of algorithms

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Repeated statesRepeated states

Failure to detect repeated states can turn a Failure to detect repeated states can turn a linear problem into an exponential one!linear problem into an exponential one!

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Graph searchGraph search

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Bidirectional search

Two simultaneous searches from start an goal.– Motivation:

Check whether the node belongs to the other fringe before expansion. Space complexity is the most significant weakness. Complete and optimal if both searches are BFS.

bd / 2 bd / 2 bd

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How to search backwards?

The predecessor of each node should be efficiently computable.– When actions are easily reversible.

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Search Strategies Reading: Russell’s Chapter 3