State space representations and search strategies Spring 2007, Juris Vīksna.

State space representations and search strategies

Spring 2007, Juris Vīksna

Sample problems - 8-puzzle

[Adapted from C.Arany, R.Barak, M.Dang]

Sample problems - 8 queens


Sample problems - Cryptarithmetic


Sample problems - River crossing puzzles


Real world problems


Real world problems - Route finding


Real world problems - Travelling salesman


Real world problems - VLSI layout


State spaces

<S,P,I,G,W> - state space

S - set of statesP= {(x,y)|x,yS} - set of production rulesIS - initial stateGS - set of goal states W: PR+ - weight function

State spaces


The problem

• find a goal state• find a path from the initial state to a goal state• find a shortest path from the initial state to a goal state• find a path with minimal weight from the initial state to a goal state

River crossing puzzle

[Adapted from R.Shinghal]

River crossing puzzle[Adapted from R.Shinghal]

River crossing puzzle


8-puzzle


8-puzzle

[Adapted from G.Luger, W.Stubblefield]

Search strategies - Data Driven


Search strategies - Goal Driven


Search strategies - Bi-directional


Search strategies -

BFS


Search strategies - BFS


Search strategies - BFS

BreadthFirstSearch(state space =<S,P,I,G,W>)Open {I}Closed while Open do

x DeQueue(Open)if Goal(x, ) then return xInsert(x,Closed)for y Child(x ,) do

if yClosed and yOpen thenEnQueue(y,Open)

return fail

Open is implemented as queue (FIFO); Closed can be an arbitrary data structure for sets

Search strategies - Uniform Cost Search

UniformCostSearch(state space =<S,P,I,G,W>)Open {<0,I>}Closed while Open do

<wx,x> ExtractMin(Open)if Goal(x, ) then return xInsert(x,Closed)for y Child(x ,) do

if yClosed thenwy wx + W((x,y))Insert(<wy,y>,Open)

return fail

Open is implemented as a priority queue

Search strategies - DFS


Search strategies -

DFS



[Adapted from J.Pearl]


DepthFirstSearch(state space =<S,P,I,G,W>)Open {I}Closed while Open do

x Pop(Open)if Goal(x, ) then return xInsert(x,Closed)for y Child(x ,) do

if yClosed and yOpen thenPush(y,Open)

return fail

Open is implemented as stack (LIFO); Closed can be an arbitrary data structure for sets


DepthFirstSearch(state space =<S,P,I,G,W>)Closed return DFSVisit(I,Closed,)

DFSVisit(x,Closed,) if Goal(x, ) then return xInsert(x,Closed)

for y Child(x ,) doif y Closed then

value DFSVisit(y,Closed,) if value fail then return value

return fail

Search strategies - Backtracking


Search strategies - Backtracking

BacktrackingSearch(state space =<S,P,I,G,W>)Open {I}Closed while Open do

x Pop(Open)if Goal(x, ) then return xInsert(x,Closed)if there exists y Child(x ,) and yClosed

and yOpen then Push(y,Open)

return fail

Open is implemented as stack (LIFO); Closed can be an arbitrary data structure for sets

Heuristic function


S - set of statesP= {(x,y)|x,yS} - set of production rulesIS - initial stateGS - set of goal states W: PR+ - weight function

d: SR+

d(x) - a shortest distance fromx to the closest goal state

f: SR+ - heuristic estimate of d

(if xG we usually assume f(x)=0)

Search strategies - Hill climbing

HCSearch(state space =<S,P,I,G,W>,f)HCVisit(I,,f)

HCVisit(x,,f) if Goal(x, ) then return xfind y Child(x ,) with the smallest f(y) valueHCVisit(y,,f)

Search strategies - Best first


Search strategies - Best first

BestFirstSearch(state space =<S,P,I,G,W>,f)Open {<f(I),I>}Closed while Open do

<wx,x> ExtractMin(Open)if Goal(x, ) then return xInsert(x,Closed)for y Child(x ,) do

if yClosed thenInsert(<f(y),y>,Open)

return fail

Open is implemented as a priority queue

Search strategies


State space representations and search strategies Spring 2007, Juris Vīksna.

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