Search Gone Wrong? CS 188: Artificial Intelligence Fall Today. Announcements. General Tree Search. Recap: Search. Lecture 3: A* Search 9/3/2009

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1 C 88: Artiicil Intelligence Fll 009 erch one Wrong? Lecture : A* erch 9//009 Pieter Aeel UC Berkeley Mny slides rom Dn Klein Announcements Assignments: Project 0 (Python tutoril): due Fridy /8 t 4:59m Project (erch): due Fridy /4 t 4:59m Wtch or oice hour seciics --- I roject Czr! till looking or roject rtners? --- Come to ront ter lecture. Try ir rogrmming, not divide-nd-conuer Account orms ville u ront during rek nd ter lecture Lecture Videos: will e linked rom lecture schedule ections strt tomorrow Hve un solving exercises! olutions will e osted online on Fridy ter lst section. Ater weeks o section we will evlute otentil overcrowdedness issues nd ind solution Tody Time nd sce comlexity o nd BF Itertive deeening --- est o oth worlds Uniorm cost serch reedy serch A* serch Heuristic design Admissiility, Consistency Tree serch rh serch Rec: erch enerl Tree erch erch rolem: ttes (conigurtions o the world) uccessor unction: unction rom sttes to lists o (stte, ction, cost) triles; drwn s grh trt stte nd gol test erch tree: Nodes: reresent lns or reching sttes Plns hve costs (sum o ction costs) erch Algorithm: ystemticlly uilds serch tree Chooses n ordering o the ringe (unexlored nodes) Imortnt ides: Fringe Exnsion Exlortion strtegy Detiled seudocode is in the ook! Min uestion: which ringe nodes to exlore?

2 Exmle erch Tree erch Algorithm Proerties Comlete? urnteed to ind solution i one exists? Otiml? urnteed to ind the lest cost th? Time comlexity? ce comlexity? Vriles: erch: Exnd out ossile lns Mintin ringe o unexnded lns Try to exnd s ew tree nodes s ossile n Numer o sttes in the rolem The verge rnching ctor B (the verge numer o successors) C* Cost o lest cost solution s m Deth o the shllowest solution Mx deth o the serch tree Algorithm Comlete Otiml Time ce Deth First erch N N N N O(B Ininite LMAX ) O(LMAX) Ininite With cycle checking, is comlete.* m tiers node nodes nodes TART OAL m nodes Ininite ths mke incomlete How cn we ix this? Algorithm Comlete Otiml Time ce w/ Pth Checking Y N O( m ) O(m) When is otiml? * Or grh serch next lecture. BF Algorithm Comlete Otiml Time ce BF w/ Pth Checking Y N O( m ) O(m) Y N* O( s+ ) O( s+ ) Comrisons When will BF outerorm? s tiers node nodes nodes When will outerorm BF? s nodes m nodes When is BF otiml?

3 Itertive Deeening Itertive deeening uses s suroutine:. Do which only serches or ths o length or less.. I iled, do which only serches ths o length or less.. I iled, do which only serches ths o length or less..nd so on. Algorithm Comlete Otiml Time ce BF ID w/ Pth Checking Y N O( m ) O(m) Y N* O( s+ ) O( s+ ) Y N* O( s+ ) O(s) TART Costs on Actions d 5 8 OAL Notice tht BF inds the shortest th in terms o numer o trnsitions. It does not ind the lest-cost th. We will uickly cover n lgorithm which does ind the lest-cost th. c 9 8 h 4 4 e r Uniorm Cost erch Priority Queue Reresher Exnd cheest node irst: Fringe is riority ueue Cost contours d 4 c 6 e 9 e 5 h r 7 c 0 8 c 8 e d 9 h 8 r 5 h 7 r 0 c 6 A riority ueue is dt structure in which you cn insert nd retrieve (key, vlue) irs with the ollowing oertions:.ush(key, vlue).o() inserts (key, vlue) into the ueue. returns the key with the lowest vlue, nd removes it rom the ueue. You cn decrese key s riority y ushing it gin Unlike regulr ueue, insertions ren t constnt time, usully O(log n) We ll need riority ueues or cost-sensitive serch methods Uniorm Cost erch Uniorm Cost Issues Algorithm Comlete Otiml Time (in nodes) ce BF UC w/ Pth Checking Y N O( m ) O(m) Y N O( s+ ) O( s+ ) Y* Y O( C*/ε ) O( C*/ε ) Rememer: exlores incresing cost contours The good: UC is comlete nd otiml! c c c C*/ε tiers * UC cn il i ctions cn get ritrrily che The d: Exlores otions in every direction No inormtion out gol loction trt ol

4 Uniorm Cost erch Exmle erch Heuristics Any estimte o how close stte is to gol Designed or rticulr serch rolem Exmles: Mnhttn distnce, Eucliden distnce 0 5. Heuristics Best First / reedy erch Exnd the node tht seems closest Wht cn go wrong? Best First / reedy erch reedy A common cse: Best-irst tkes you stright to the (wrong) gol Worst-cse: like dlyguided in the worst cse Cn exlore everything Cn get stuck in loos i no cycle checking Like in comleteness (inite sttes w/ cycle checking) Uniorm Cost 4

5 Comining UC nd reedy Uniorm-cost orders y th cost, or ckwrd cost g(n) Best-irst orders y gol roximity, or orwrd cost h(n) d h=6 h=5 h= c h=7 h=6 A* erch orders y the sum: (n) = g(n) + h(n) 5 e h= h=0 Exmle: Teg renger When should A* terminte? hould we sto when we enueue gol? A h = h = B h = No: only sto when we deueue gol h = 0 h = 7 Is A* Otiml? A h = 6 5 h = 0 Wht went wrong? Actul d gol cost < estimted good gol cost We need estimtes to e less thn ctul costs! Admissile Heuristics Otimlity o A*: Blocking A heuristic h is dmissile (otimistic) i: where Exmles: 66 is the true cost to nerest gol Coming u with dmissile heuristics is most o wht s involved in using A* in rctice. 5 Proo: Wht could go wrong? We d hve to hve to o suotiml gol o the ringe eore * This cn t hen: Imgine suotiml gol is on the ueue ome node n which is suth o * must lso e on the ringe (why?) n will e oed eore 5

6 Proerties o A* UC vs A* Contours Uniorm-Cost A* Uniorm-cost exnded in ll directions trt ol A* exnds minly towrd the gol, ut does hedge its ets to ensure otimlity trt ol Exmle: Exlored ttes with A* Comrison reedy Uniorm Cost Heuristic: mnhttn distnce ignoring wlls A str Creting Admissile Heuristics Exmle: 8 Puzzle Most o the work in solving hrd serch rolems otimlly is in coming u with dmissile heuristics Oten, dmissile heuristics re solutions to relxed rolems, with new ctions ( some cheting ) ville 66 Indmissile heuristics re oten useul too (why?) 5 Wht re the sttes? How mny sttes? Wht re the ctions? Wht sttes cn I rech rom the strt stte? Wht should the costs e? 6

7 Heuristic: Numer o tiles mislced Why is it dmissile? h(strt) = 8 This is relxedrolem heuristic 8 Puzzle I Averge nodes exnded when otiml th hs length 4 stes 8 stes stes UC 6,00.6 x 0 6 TILE 9 7 Wht i we hd n esier 8-uzzle where ny tile could slide ny direction t ny time, ignoring other tiles? Totl Mnhttn distnce Why dmissile? h(strt) = = 8 8 Puzzle II Averge nodes exnded when otiml th hs length 4 stes 8 stes stes TILE 9 7 MANHATTAN Puzzle III How out using the ctul cost s heuristic? Would it e dmissile? Would we sve on nodes exnded? Wht s wrong with it? Trivil Heuristics, Dominnce Dominnce: h h c i Heuristics orm semi-lttice: Mx o dmissile heuristics is dmissile With A*: trde-o etween ulity o estimte nd work er node! Trivil heuristics Bottom o lttice is the zero heuristic (wht does this give us?) To o lttice is the exct heuristic Other A* Alictions Pthing / routing rolems Resource lnning rolems Root motion lnning Lnguge nlysis Mchine trnsltion eech recognition Tree erch: Extr Work! Filure to detect reeted sttes cn cuse exonentilly more work. Why? 7

8 rh erch In BF, or exmle, we shouldn t other exnding the circled nodes (why?) d c h e r c h e r c rh erch Ide: never exnd stte twice How to imlement: Tree serch + list o exnded sttes (closed list) Exnd the serch tree node-y-node, ut Beore exnding node, check to mke sure its stte is new Python trick: store the closed list s set, not list Cn grh serch wreck comleteness? Why/why not? How out otimlity? Otimlity o A* rh erch Proo: New ossile rolem: nodes on th to * tht would hve een in ueue ren t, ecuse some worse n or the sme stte s some n ws deueued nd exnded irst (disster!) Tke the highest such n in tree Let e the ncestor which ws on the ueue when n ws exnded Assume () < (n) (n) < (n ) ecuse n is suotiml would hve een exnded eore n o n would hve een exnded eore n, too Contrdiction! Consistency Wit, how do we know rents hve etter -vles thn their successors? Couldn t we o some node n, nd ind its child n to hve lower vlue? YE: h = 0 h = 8 B g = 0 Wht cn we reuire to revent these inversions? Consistency: A h = 0 Rel cost must lwys exceed reduction in heuristic Otimlity Tree serch: A* otiml i heuristic is dmissile (nd nonnegtive) UC is secil cse (h = 0) rh serch: A* otiml i heuristic is consistent UC otiml (h = 0 is consistent) Consistency imlies dmissiility ummry: A* A* uses oth ckwrd costs nd (estimtes o) orwrd costs A* is otiml with dmissile heuristics Heuristic design is key: oten use relxed rolems In generl, nturl dmissile heuristics tend to e consistent 8