Out line Using knowledge Heurist ics I nf ormed Search Best -f - PDF document

Out line • Using knowledge – Heurist ics I nf ormed Search • Best -f irst search – Greedy best -f irst search – A* search CS 486/ 686 – Ot her variat ions of A* Univer sit y of Wat erloo • Back t o heurist ics May 10 1 2 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart Recall f rom last lect ure I nf ormed Search • Uninf ormed search met hods expand nodes based on “dist ance” f rom st art node • Our knowledge is of t en on t he merit of nodes – Never look ahead t o t he goal – Value of being at a node – E.g. in unif orm cost search expand t he cheapest • Dif f erent not ions of merit pat h. We never consider t he cost of get t ing t o t he – I f we are concerned about t he cost of t he goal solut ion, we might want a not ion of how expensive – Advant age is t hat we have t his inf ormat ion it is t o get f rom a st at e t o a goal – I f we are concerned wit h minimizing comput at ion, • But , we of t en have some addit ional knowledge we might want a not ion of how easy it is t o get a about t he problem st at e t o a goal – E.g. in t raveling around Romania we know t he dist ances bet ween cit ies so can measure t he – We will f ocus on cost of solut ion overhead of going in t he wrong direct ion 3 4 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart I nf ormed search I nf ormed search • We need t o develop a domain specif ic • I f h(n1)< h(n2) t hen we guess t hat it is heurist ic f unct ion, h(n) cheaper t o reach t he goal f rom n1 t han • h(n) guesses t he cost of reaching t he it is f rom n2 goal f rom node n – The heur ist ic f unct ion must be domain • We require specif ic – h(n)=0 when n is a goal node – We of t en have some inf ormat ion about t he problem t hat can be used in f or ming a – h(n)> = 0 f or all ot her nodes heurist ic f unct ion (i.e. heur ist ics are domain specif ic) 5 6 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart 1

Gr eedy best -f ir st sear ch: Greedy best -f irst search Example Heur ist ic • Use t he heur ist ic f unct ion, h(n), t o rank f unct ion t he nodes in t he f ringe h=4 h=3 h=2 h=1 h=0 • Search st r at egy S A B C G – Expand node wit h lowest h-value 2 1 1 2 P at h cost • Greedily t rying t o f ind t he least -cost 4 solut ion 7 8 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart Example cont … Example cont … h=4 h=3 h=2 h=1 h=0 h=4 h=3 h=2 h=1 h=0 S A B C G S A B C G 2 1 1 2 2 1 1 2 4 4 9 10 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart Example cont … Example cont … h=4 h=3 h=2 h=1 h=0 h=4 h=3 h=2 h=1 h=0 G S A B C S A B C G 2 1 1 2 2 1 1 2 4 4 11 12 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart 2

Example cont … Anot her Example h=0 h=4 h=3 h=4 h=4 h=3 h=2 h=1 h=0 S A B G S A B C G 2 1 2 1 1 2 2 1 1 Found t he goal 4 C But cheaper pat h is S, A, B, C, G Pat h is S, A, C, G Wit h cost 2+1+1+2=6 h=1 Cost of t he pat h is 2+4+2=8 Greedy best-first is not optimal 13 14 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart Anot her Example Propert ies of greedy search • Not opt imal! • Not complet e! h=0 – I f we check f or repeat ed st at es t hen we are ok h=4 h=3 h=4 • Exponent ial space in worst case since need t o S A B G keep all nodes in memory 2 1 2 • Exponent ial worst case t ime O(b m ) where m is 1 1 t he maximum dept h of t he t ree C – I f we choose a good heurist ic t hen we can do much h=1 bet t er Greedy best -f irst can get st uck in loops Not complet e 15 16 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart A* Search A* Example • Greedy best -f ir st search is t oo greedy – I t does not t ake int o account t he cost of h=4 h=3 h=2 h=1 h=0 t he pat h so f ar! S A B C G • Def ine 2 1 1 2 – f (n)=g(n)+h(n) – g(n) is t he cost of t he pat h t o node n 4 – h(n) is t he heurist ic est imat e of t he cost of 1. Expand S reaching t he goal f rom node n 2. Expand A • A* search 3. Choose bet ween B (f (B)=3+2=5) and C (f (C)=6+1=7) ) expand B – Expand node in f r inge (queue) wit h lowest f 4. Expand C value 5. Expand G – recognize it is t he goal 17 18 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart 3

When should A* t erminat e? When should A* t erminat e? • As soon as we f ind a goal st at e? • As soon as we f ind a goal st at e? 1 1 1 1 S S B B h=3 h=3 A A h=7 h=7 1 1 C C 7 h=2 7 h=2 1 1 G G D D h=1 h=1 7 7 A* Terminat es only when goal st at e is popped f rom t he queue 19 20 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart A* and revisit ing st at es A* and revisit ing st at es What if we revisit a st at e t hat was already expanded? What if we revisit a st at e t hat was already expanded? S S 1 1 1 1 h=7 h=7 A A B h=3 B h=3 2 2 1 1 h=2 h=2 C C G 7 G 7 I f we allow st at es t o be expanded again, we might get a bet t er solut ion! 21 22 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart I s A* Opt imal? Admissible heurist ics • Let h*(n) denot e t he t rue minimal cost h=6 t o t he goal f rom node n A 1 1 • A heurist ic, h, is admissible if S – h(n) ≤ h*(n) f or all n G 3 • Admissible heurist ics never overest imat e t he cost t o t he goal – Opt imist ic No. This example shows why not . 23 24 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart 4

Opt imalit y of A* Opt imalit y of A* • I f t he heurist ic is admissible t hen A* wit h t ree-search is opt imal • For searching graphs we require Let G be an opt imal goal st at e, and f (G) = f * = g(G). somet hing st ronger t han admissibilit y Let G 2 be a subopt imal goal st at e, i.e. f (G 2 ) = g(G 2 ) > f *. Assume f or cont radict ion t hat A* has select ed G 2 f rom t he queue. – Consist ency (monot onicit y): (This would t erminat e A* wit h a subopt imal solut ion) • h(n) ≤ cost (n,n’)+h(n’) Let n be a node t hat is current ly a leaf node on an opt imal pat h t o G. – Almost any admissible heurist ic f unct ion will also be consist ent • A* graph-search wit h a consist ent heurist ic is opt imal Because h is admissible, f * ≥ f (n). 2 , we must have f (n) ≥ f (G I f n is not chosen f or expansion over G 2 ) So, f * ≥ f (G 2 )=0, we have f * ≥ g(G 2 ). Because h(G 2 ), cont radict ion. 25 26 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart Memor y-bounded heur ist ic search Propert ies of A* • A* keeps most generat ed nodes in memory – On many problems A* will run out of memory • Complet e if t he heurist ic is consist ent • I t erat ive deepening A* (I DA*) – Along any pat h, f always increases ) if a – Like I DS but change f -cost rat her t han dept h at solut ion exist s somewhere t he f value will each it erat ion event ually get t o it s cost • SMA* (Simplif ied Memory-Bounded A*) • Exponent ial t ime complexit y in wor st case – Uses all available memory – A good heurist ic will help a lot here – Proceeds like A* but when it runs out of memory it drops t he worst leaf node (one wit h highest f - – O(bm) if t he heurist ic is per f ect value) • Exponent ial space complexit y – I f all leaf nodes have t he same f -value t hen it drops oldest and expands t he newest – Opt imal and complet e if dept h of shallowest goal node is less t han memory size 27 28 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart Heurist ic Funct ions 8-puzzle • A good heurist ic f unct ion can make all 7 2 4 1 2 t he dif f er ence! 5 6 3 4 5 • How do we get heurist ics? 8 3 1 6 7 8 – One appr oach is t o t hink of an easier Start State Goal State problem and let h(n) be t he cost of • Relax t he game reaching t he goal in t he easier problem 1. Can move t ile f rom posit ion A t o posit ion B if A is next t o B (ignore whet her or not posit ion is blank) 2. Can move t ile f rom posit ion A t o posit ion B if B is blank (ignore adj acency) 3. Can move t ile f rom posit ion A t o posit ion B 29 30 cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart cs486/686 Lecture Slides 2005 (c) K. Larson and P. Poupart 5