Announcements. CS 188: Artificial Intelligence Fall Recap: Search. Today. General Tree Search. Uniform Cost. Lecture 3: A* Search 9/4/2007

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1 CS 88: Artificil Intelligence Fll 2007 Lecture : A* Serch 9/4/2007 Dn Klein UC Berkeley Mny slides over the course dpted from either Sturt Russell or Andrew Moore Announcements Sections: New section 06: Tu 5-6pm You cn go to ny section, if there s spce Sections strt this week Homework Project on the we, due 9/2 New written homework formt: One or two questions hnded out end of section (nd online) Due the next week in section, grded check / no check Ech ssignment % of grde, cp of 0%, so cn skip t lest one week, depends on how mny there re Solve in groups of ny size, write up lone A* Serch Heuristic Design Locl Serch Tody Recp: Serch Serch prolems: Sttes (configurtions of the world) Successor functions, costs, strt nd gol tests Serch trees: Nodes: represent pths / plns Pths hve costs (sum of ction costs) Strtegies differ (only) in fringe mngement enerl Tree Serch Uniform Cost Strtegy: expnd lowest pth cost The good: UCS is complete nd optiml! c c 2 c Expnding includes incrementing the pth cost! The d: Explores options in every direction No informtion out gol loction Strt ol

2 Best First Exmple: Heuristic Function Strtegy: expnd nodes which pper closest to gol Heuristic: function which mps sttes to distnce A common cse: Best-first tkes you stright to the (wrong) gol Worst- cse: like dly- guided DFS h(x) Comining UCS nd reedy Uniform-cost orders y pth cost, or ckwrd cost g(n) Best-first orders y gol proximity, or forwrd cost h(n) 2 S d h=5 2 h=6 h=2 h=0 c h=5 h=4 A* Serch orders y the sum: f(n) = g(n) + h(n) 5 e h= When should A* terminte? Should we stop when we enqueue gol? 2 A 2 S h = 2 h = B 2 h = No: only stop when we dequeue gol h = 0 Exmple: Teg renger Is A* Optiml? Admissile Heuristics A h = 6 A heuristic is dmissile (optimistic) if: S h = 7 h = 0 where is the true cost to nerest gol 5 Wht went wrong? Actul d gol cost > estimted good gol cost We need estimtes to e less thn ctul costs! E.g. Eucliden distnce on mp prolem Coming up with dmissile heuristics is most of wht s involved in using A* in prctice. 2

3 Optimlity of A*: Blocking UCS vs A* Contours Proof: Wht could go wrong? We d hve to hve to pop suoptiml gol off the fringe efore * This cn t hppen: Imgine suoptiml gol is on the queue Some node n which is supth of * must e on the fringe (why?) n will e popped efore Uniform-cost expnded in ll directions A* expnds minly towrd the gol, ut does hedge its ets to ensure optimlity Strt Strt ol ol Properties of A* Admissile Heuristics Uniform- Cost A* Most of the work is in coming up with dmissile heuristics Indmissile heuristics re often quite effective (especilly when you hve no choice) Very common hck: use α x h(n) for dmissile h, α > to generte fster ut less optiml indmissile h from dmissile h Exmple: 8 Puzzle 8 Puzzle I Numer of tiles misplced? Why is it dmissile? Wht re the sttes? Wht re the ctions? Wht sttes cn I rech from the strt stte? Wht should the costs e? h(strt) = 8 This is relxedprolem heuristic ID TILES Averge nodes expnded when optiml pth hs length 4 steps 8 steps 2 6, steps.6 x

4 8 Puzzle II 8 Puzzle III Wht if we hd n esier 8-puzzle where ny tile could slide ny direction t ny time, ignoring other tiles? Totl Mnhttn distnce Why dmissile? h(strt) = TILES = 8 MAN- HATTAN Averge nodes expnded when optiml pth hs length 4 steps 2 8 steps steps How out using the ctul cost s heuristic? Would it e dmissile? Would we sve on nodes? Wht s wrong with it? With A*: trde-off etween qulity of estimte nd work per node! Trivil Heuristics, Dominnce Dominnce: h h c if Heuristics form semi-lttice: Mx of dmissile heuristics is dmissile Trivil heuristics Bottom of lttice is the zero heuristic (wht does this give us?) Top of lttice is the exct heuristic Course Scheduling From the university s perspective: Set of courses {c, c 2, c n } Set of room / times {r, r 2, r n } Ech piring (c k, r m ) hs cost w km Wht s the est ssignment of courses to rooms? Sttes: list of pirings Actions: dd legl piring Costs: cost of the new piring Admissile heuristics? (Who cn think of cs70 nswer to this prolem?) Other A* Applictions Pthing / routing prolems Resource plnning prolems Root motion plnning Lnguge nlysis Mchine trnsltion Speech recognition Tree Serch: Extr Work? Filure to detect repeted sttes cn cuse exponentilly more work. Why? 4

5 rph Serch In BFS, for exmple, we shouldn t other expnding the circled nodes (why?) rph Serch Very simple fix: never expnd stte twice S d e p c e h r q h r p q f p q f q c q c Cn this wreck completeness? Optimlity? Optimlity of A* rph Serch Consider wht A* does: Expnds nodes in incresing totl f vlue (f-contours) Proof ide: optiml gols hve lower f vlue, so get expnded first We mde stronger ssumption thn in the lst proof Wht? Consistency Wit, how do we know we expnd in incresing f vlue? Couldn t we pop some node n, nd find its child n to hve lower f vlue? YES: h = 0 h = 8 B g = 0 Wht cn we ssume to prevent these inversions? Consistency: A h = 0 Rel cost lwys exceeds reduction in heuristic Optimlity Tree serch: A* optiml if heuristic is dmissile (nd nonnegtive) UCS is specil cse (h = 0) rph serch: A* optiml if heuristic is consistent UCS optiml (h = 0 is consistent) In generl, nturl dmissile heuristics tend to e consistent Summry: A* A* uses oth ckwrd costs nd (estimtes of) forwrd costs A* is optiml with dmissile heuristics Heuristic design is key: often use relxed prolems 5

6 Lrge Scle Prolems Limited Memory Options Wht sttes get expnded? All sttes with f-cost less thn optiml gol cost How fr in every direction will this e? Intuition: depth grows like the heuristic gp : h(strt) g(gol) p usully t lest liner in prolem size Work exponentil in depth In huge prolems, often A* isn t enough Stte spce just too ig Cn t visit ll sttes with f less thn optiml Often, cn t even store the entire fringe Solutions Better heuristics Bem serch (limited fringe size) reedy hill-climing (fringe size = ) Strt ol Hill-Climing Serch: Only est node kept round, no fringe! Usully prioritize successor choice y h (greedy hill climing) Compre to greedy cktrcking, which still hs fringe Bem Serch (Limited Memory Serch) In etween: keep K nodes in fringe Dump lowest priority nodes s needed Cn prioritize y h lone (greedy em serch), or h+g (limited memory A*) Why not pplied to UCS? We ll return to em serch lter No gurntees once you limit the fringe size! 6