ADijkstra-LikeScenario - - PDF document

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HeuristicGraphSearch: AnotherReasonAIisCool

NickDeibel QuizSection– 5/31/02

ADijkstra-LikeScenario

Yourcompanyownsadeliverytruckthatwillhavetomakemany tripsinadaytovariouswarehouses,alwaysstartingatwarehouseA andendingatwarehouseB.However,itwillneverrepeatvisitthe samewarehousetwiceinoneday.Becauseofdemand,youcannot wastetimerefuelingthetruckuntilitreachesB.Thetruckcan travelatmostKmilesonasingletankofgas.Youaregivena graphwherethenodesrepresentthewarehousesandthedirected edgesrepresentthehighwaysconnectingthetwowarehouses.Each edgeisweightedaccordingtothelengthofthathighway. AsyouwantyourdriverstoavoidtakingpathsoflongerthanK miles,designanalgorithmthattellsyouifthereexistsanysimple pathfromAtoB(nonodes/warehousesrepeated)whoselengthis greaterthanKmiles.

Whoops,That’sREALLYHard

• LongestPathisanNP-Completeproblem.
• Noknownpolynomial-timealgorithmsolvesit.

– Apoly-timealgorithmdoesexistforDAGS.

• Thisisadifficultproblemforevensmallgraphs.

HugeGraphs

• Considersomereallyhuge graphs…

– AllcitiesandtownsintheWorldAtlas – AllstarsintheGalaxy – Allways10blockscanbestacked Huh???

ImplicitlyGeneratedGraphs

• Ahugegraphmaybeimplicitlyspecified byrulesfor

generatingiton-the-fly

• Blocksworld:

– vertex=relativepositionsofallblocks – edge=robotarmstacksoneblock

stack(blue,red) stack(green,red) stack(green,blue) stack(blue,table) stack(green,blue)

BlocksWorld

• Source=initialstateoftheblocks
• Goal=desiredstateoftheblocks
• Pathsourcetogoal=sequenceofactions

(program)forrobotarm!

• nblocks nn vertices
• 10blocks 10billionvertices!

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Problem:BranchingFactor

• Cannotsearchsuchhugegraphsexhaustively.

Supposeweknowthegoalisonlyd stepsaway.

• Dijkstra’s algorithmisbasicallybreadth-first

search(modifiedtohandlearcweights)

• Breadth-firstsearch(orforweightedgraphs,

Dijkstra’s algorithm)– Ifout-degreeofeachnode is10,potentiallyvisits10d vertices

– 10stepplan=10billionverticesvisited!

AnEasierCase

• SupposeyouliveinManhattan;whatdoyoudo?

52nd St 51st St 50th St

10th Ave 9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave 3rd Ave 2nd Ave

S G

Best-FirstSearch

• TheManhattandistance ( x+ y)isanestimate
• fthedistancetothegoal

– aheuristicvalue

• Best-FirstSearch

– Ordernodesinprioritytominimizeestimateddistance tothegoalh(n)

• Compare:BFS/Dijkstra

– Ordernodesinprioritytominimizedistancefromthe start

BestFirstinAction

• SupposeyouliveinManhattan;whatdoyoudo?

52nd St 51st St 50th St

10th Ave 9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave 3rd Ave 2nd Ave

S G

Problem1:LedAstray

• Eventuallywillexpandvertextogetbackonthe

righttrack

52nd St 51st St 50th St

10th Ave 9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave 3rd Ave 2nd Ave

S G

Problem2:Optimality

• WithBest-FirstSearch,areyouguaranteed a

shortestpathisfoundwhen

– goalisfirstseen? – whengoalisremovedfrompriorityqueue(aswith Dijkstra?)

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Sub-OptimalSolution

• No!Goalisbydefinitionatdistance0:willbe

removedfrompriorityqueueimmediately,evenif ashorterpathexists!

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

h=2 h=1 h=4 h=5

Synergy?

• Dijkstra/BreadthFirstguaranteedtofindoptimal

solution

• BestFirstoftenvisitsfarfewer vertices,butmay

notprovideoptimalsolution – Canwegetthebestofboth?

Heuristics

• Aruleofthumb,simplification,oreducatedguess.
• Reducesthesearchforsolutionsinlargesolution

spaces

• Unlikealgorithms,heuristicsdonotguarantee
• ptimal,orevenfeasible,solutions.

A*(“Astar”)

• Orderverticesinpriorityqueuetominimize

(distancefromstart)+(estimateddistancetogoal) f(n)=g(n) +h(n) f(n)=priorityofanode g(n) =truedistancefromstart h(n) =heuristicdistancetogoal

Optimality

• Supposetheestimateddistance(h)is

always lessthanorequalto thetrue distancetothe goal – heuristicisalowerboundontruedistance

• Then:whenthegoalisremoved fromthepriority

queue,weareguaranteed tohavefoundashortest path!

Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 5 5 52nd &9th f(n) h(n) g(n) vertex

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Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 5 4 1 51st &9th 7 2 5 52nd &4th f(n) h(n) g(n) vertex

Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 7 5 2 50th &9th 5 3 2 51st &8th 7 2 5 52nd &4th f(n) h(n) g(n) vertex

Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 7 4 3 50th &8th 7 5 2 50th &9th 5 2 3 51st &7th 7 2 5 52nd &4th f(n) h(n) g(n) vertex

Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 7 3 4 50th &7th 7 4 3 50th &8th 7 5 2 50th &9th 5 1 4 51st &6th 7 2 5 52nd &4th f(n) h(n) g(n) vertex

Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 7 3 4 50th &7th 7 4 3 50th &8th 7 5 2 50th &9th 5 5 51st &5th 7 2 5 52nd &4th f(n) h(n) g(n) vertex

Problem2Revisited

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 7 3 4 50th &7th 7 4 3 50th &8th 7 5 2 50th &9th 7 2 5 52nd &4th f(n) h(n) g(n) vertex

DONE!

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WhatWouldDijkstraHave Done?

52nd St 51st St

9th Ave 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave

S G

(5blocks)

50th St 49th St 48th St 47th St

ProofofA*Optimality

• A*terminateswhenGispoppedfromtheheap.
• SupposeGispoppedbutthepathfoundisn’toptimal:

priority(G)>optimalpathlengthc

• LetPbeanoptimalpathfromStoG,andletNbethelast

vertexonthatpaththathasbeenvisitedbutnotyetpopped.

TheremustbesuchanN,otherwisetheoptimalpathwouldhavebeen found. priority(N)=g(N)+h(N) c

• SoNshouldhavepoppedbeforeGcanpop.Contradiction.

S N G

non-optimalpathtoG portionofoptimal pathfoundsofar undiscoveredportion

• fshortestpath

WhatAboutThoseBlocks?

• “Distancetogoal”isnotalwaysphysicaldistance
• Blocksworld:

– distance=numberofstackstoperform – heuristiclowerbound=numberofblocksoutofplace #outofplace=1,truedistancetogoal=3

OtherReal-WorldApplications

• Routingfinding– computernetworks,airline

routeplanning

• VLSIlayout– celllayoutandchannelrouting
• Productionplanning– “justintime”optimization
• Proteinsequencealignment
• Manyother“NP-Hard”problems

– Aclassofproblemsforwhichnoexactpolynomial timealgorithmsexist– soheuristicsearchisthebest wecanhopefor