03 - Uninformed Search - povinelli.eece.mu.edupovinelli.eece.mu.edu/teaching/eece6820/slides/03 - Uninformed... · function Tree-Search( problem, strategy) ... Uninformed search strategies

Copyright Richard J. Povinelli Page 1rev 1.1, 9/3/2003

Uninformed Search

Dr. Richard J. Povinelli

Copyright Richard J. Povinelli rev 1.1, 9/3/2003 Page 2

Objectives

You shouldbe able to formulate a search problem. be able to evaluate a search algorithm on the basis of completeness, optimality, time complexity, and space complexity. be able to compare, contrast, classify, and implement various search algorithms including breadth-first, uniform-cost, depth-first, depth-limited, iterative deepening, bidirectional.


Problem-solving agentsRestricted form of general agentNote: this is offline problem solving. Online problem solving involves acting without complete knowledge of the problem and solution.function Simple-Problem-Solving-Agent( percept) returns an action static: seq, an action sequence, initially empty

state, some description of the current world state goal, a goal, initially null problem, a problem formulation

state ← Update-State(state,percept) if seq is empty thengoal ← Formulate-Goal(state) problem ← Formulate-Problem(state, goal) seq ← Search( problem) action ← Recommendation(seq, state) seq ← Remainder(seq, state) return action

AIMA Slides © Stuart Russell and Peter Norvig, 1998


Example: Romania

On holiday in Romaniacurrently in AradFlight leaves tomorrow from Bucharest

Formulate goal: be in Bucharest

Formulate problem:states: various citiesoperators: drive between cities

Find solution: sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest



Example: Romania



Problem typesDeterministic, fully observable => single-state problem

Agent knows exactly which state it will be in; solution is a sequence

Non-observable => conformant (sensorless) problemAgent may have no idea where it is; solution (if any) is a sequence

Nondeterministic and/or partially observable=> contingency problem

Percepts provide new information about current statesolution is a tree or policyoften interleave search, execution

Unknown state space => exploration problem (“online”)



Example: vacuum worldSingle-state, start in #5

Solution??Conformant, start in {1, 2, 3, 4, 5, 6, 7, 8}

e.g., Right goes to {2, 4, 6, 8}.Solution??

Contingency, start in #5Murphy's Law: Suck can dirty a clean carpetLocal sensing: dirt, location only.Solution??



Single-state problem formulationA problem is defined by four items:

initial state “at Arad”

successor function S(x) = set of action-state pairsS(Arad) = {

<Arad → Zerind, Zerind>, <Arad → Sibiu, Zerind>,<Arad → Timisoara, Timisoara >}

goal testexplicit, e.g., x = “at Bucharest”implicit, e.g., NoDirt(x)

path cost (additive)sum of distances, number of operators executed, or others

A solution is a sequence of actions leading from the initial state to a goal state



Selecting a state space Real world is absurdly complex

State space must be abstracted for problem solving(Abstract) state = set of real states(Abstract) action = complex combination of real actions

“Arad → Zerind” represents a complex set of possible routes, detours, rest stops, and other states.

For guaranteed realizability, any real state “in Arad” must get to some real state “in Zerind”(Abstract) solution = set of real paths that are solutions in the real worldEach abstract action should be “easier” than the original problem!



CAT – 8 PuzzleDescribe the following for the 8 puzzle

statesoperatorsgoal testpath cost

Work with a partner for 5 minutes

Start State Goal State


Example: The 8-puzzle

states??: integer locations of tiles (ignore intermediate positions)operators??: move blank left, right, up, down (ignore unjamming)goal test??: = goal state (given)path cost??: 1 per move[Note: optimal solution of n-Puzzle family is NP-hard]


Start State Goal State


Example: vacuum world


states??operators??goal test??path cost??


Example: vacuum world


states??: integer dirt and robot locations (ignore dirt amounts)operators??: Left, Right, Suck, NoOp goal test??: no dirtpath cost??: 1 per operator (0 for NoOp)


Example: robotic assembly


states??: real-valued coordinates of robot joint angles, parts of the object to be assembledoperators??: continuous motions of robot jointsgoal test??: complete assembly with no robot included!path cost??: time to execute


Tree Search algorithms

Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states)


function Tree-Search( problem, strategy) returns a solution, or failure initialize the search tree using the initial state of problemloop do

if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategyif the node contains a goal state then return the corresponding solution else expand the node and add the resulting nodes to the search tree

end


Implementation: states vs. nodesA state is a (representation of) a physical configurationA node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(x)States do not have parents, children, depth, or path cost!


The Expand function creates new nodes, filling in the various fields and using the Operators (or SuccessorFn) of the problem to create the corresponding states.


Implementation of general tree search

function Tree-Search( problem, fringe) returns a solution or failurefringe ← Insert(Make-Node(Initial-State[problem]), fringe) loop do

if fringe is empty then return failure node ← Remove-Front(fringe ) if Goal-Test[problem] applied to State(node) succeeds

then return node fringe ← InsertAll (Expand(node, problem), fringe)

function Expand(node, problem) returns a set of nodessuccessors ← the empty setfor each action, result in Successor-Fn [problem](State[node]) do

s ← a new NodeParent-Node [s] ← node; Action[s] ← action; State[s] ← resultPath-Cost[s] ← Path-Cost[node] + Step-Cost(node, action, s)Depth[s] ← Depth[node] + 1add s to successors

return successorsAIMA Slides © Stuart Russell and Peter Norvig, 2003


Tree search example



Search strategiesA strategy is defined by picking the order of node expansionStrategies are evaluated along the following dimensions

completeness – does it always find a solution if one exists?time complexity – number of nodes generated/expandedspace complexity – maximum number of nodes in memoryoptimality – does it always find a least-cost solution?

Time and space complexity are measured in terms ofb – maximum branching factor of the search treed – depth of the least-cost solutionm – maximum depth of the state space (may be ∞)



Uninformed search strategiesUninformed strategies use only the information available in the problem definitionBreadth-first searchUniform-cost searchDepth-first searchDepth-limited searchIterative deepening search



CAT – Manual search I

Find the shortest cost route from Bucharest to Zerind in 15 s.


CAT – Manual search II

Find the shortest cost route from Pitesti to Timisoaro in 1 minute


CAT – Manual search III

Bucharest to Zerind: 101 + 97 + 80 + 140 + 75 = 493Pitesti to Timisoaro: 97 + 80 + 140 + 118 = 435


Breadth-first searchExpand shallowest unexpanded node Implementation:

fringe is a FIFO queue, I.e. new successors go at end



Properties of breadth-first searchComplete??Time??Space??Optimal??



Properties of breadth-first searchComplete?? Yes (if b is finite)Time?? 1 + b + b2 + b3 + … + bd = O(bd ), i.e., exponential in dSpace?? O(bd ) (keeps every node in memory)Optimal?? Yes (if cost = 1 per step), not optimal in general Space is the big problem, can easily generate nodes at 10MB/sec so 24hrs = 860GB.



Romania with step costs in km



Uniform-cost searchExpand least-cost unexpanded nodeImplementation: fringe = queue ordered by path cost



Properties of uniform-cost searchExpand least-cost unexpanded nodeImplementation:

fringe = queue ordered by path costEquivalent to breadth-first if step costs all equalComplete?? Yes, if step cost ≥ εTime?? # of nodes with g ≤ cost of optimal solution, O(b^(ceil(C*/ε)), where C* is the cost of the optimal solutionSpace?? # of nodes with g ≤ cost of optimal solution, O(b^(ceil(C*/ε)) Optimal?? Yes, nodes expanded in increasing order of g(n)



DFS on a depth-3 binary tree


Expand deepest unexpanded nodeImplementation

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


Properties of depth-first searchComplete??Time??Space??Optimal??



Properties of depth-first searchComplete??

No: fails in infinite-depth spaces, spaces with loopsModify to avoid repeated states along path ⇒ complete in finite spaces

Time?? O(bm ): terrible if m (maximum depth of the state space) is much larger than d, (depth of the least-cost solution)but if solutions are dense, may be much faster than breadth-first

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



Depth-limited searchDepth-first search with depth limitImplementation: Nodes at depth l have no successors

function Depth-Limited-Search (problem, limit) returns soln/fail/cutoffRecursive-DLS(Make-Node(Initial-State[problem]), problem, limit)

function Recursive-DLS (node, problem, limit) returns soln/fail/cutoffcutoff-occurred? ← falseif Goal-Test[problem](State[node]) then return nodeelse if Depth[node] = limit then return cutoffelse for each successor in Expand(node, problem) do

result ← Recursive-DLS(successor, problem, limit)if result = cutoff then cutoff-occurred? ← trueelse if result ≠ failure then return result

if cutoff-occurred? then return cutoff else return failure



Iterative deepening search

function Iterative-Deepening-Search( problem) returns a solutioninputs: problem, a problem for depth ← 0 to ∞ do

result ← Depth-Limited-Search( problem, depth) if result ≠ cutoff then return result

end



Iterative deepening search l = 0


Arad




Arad

Sibiu TimisoaraZerind




Arad

Sibiu TimisoaraZerind

OradeaArad OradeaArad Fagaras Rimnicu Vilcea Arad Lugoj


Properties of iterative deepening search

Complete??Time??Space??Optimal??



Properties of iterative deepening search

Complete?? YesTime?? (d + 1)b0 + db1 + (d - 1)b2 + … + bd = O(bd )Space?? O(bd)Optimal?? Yes, if step cost = 1

Can be modified to explore uniform-cost treeNumerical comparison for b=10 and d=5, solution at far right:

N(IDS) = 50 + 400 + 3,000 + 20,000 + 100,000 = 123,450N(BFS) = 10 + 100 + 1,000 + 10,000 + 100,000 + 999,990 = 1,111,100



CAT – Search StrategiesWork out the tree for the four queens search problems. Each team will use a different search strategy

Breadth-firstUniform-costDepth-firstIterative deepening

Work with a partner for 5 minutes, write on board


Summary of algorithms

a – complete if b is finiteb – complete if step costs ≥ ε e for positive ec – optimal if step costs are all identicald – if both directions use breadth-first search

Criterion Breadth- Uniform- Depth- Depth- IterativeFirst Cost First Limited Deepening

Complete? Yesa Yesa,b No No Yesa

Time O (b d +1) O (b 1+floor(C* /ε )) O (b m ) O (b l ) O (b d )Space O (b d +1) O (b 1+floor(C* /ε )) O (bm ) O (bl ) O (bd ) Optimal? Yesc Yes No No Yesc


Repeated states

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


Graph searchfunction Graph-Search(problem, fringe) returns a solution, or

failureclosed ← an empty setfringe ← Insert(Make-Node(Initial-State[problem]), fringe)loop do

if fringe is empty then return failurenode ← Remove-Front(fringe)if Goal-Test[problem](State[node]) then return nodeif State[node] is not in closed then

add State[node] to closedfringe ← InsertAll(Expand(node, problem), fringe)

end


SummaryProblem formulation usually requires abstracting away real-world details to define a state space that can feasibly be exploredVariety of uninformed search strategiesIterative deepening search uses only linear space and not much more time than other uninformed algorithms


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03 - Uninformed Search - povinelli.eece.mu.edupovinelli.eece.mu.edu/teaching/eece6820/slides/03 - Uninformed... · function Tree-Search( problem, strategy) ... Uninformed search strategies