CMU-Q Lecture 3: Search algorithms: Informed. Teacher: Gianni A. Di Caro

Transcription

1 CMU-Q 5-38 Lecure 3: Search algorihms: Informed Teacher: Gianni A. Di Caro

2 UNINFORMED VS. INFORMED SEARCH Sraegy How desirable is o be in a cerain inermediae sae for he sake of (effecively) reaching a goal sae from sar Uninformed: Generaion order, esimaed cos-o-come Informed: Generaion order, esimaed cos-o-come, esimaed cos-o-go Heurisic informaion Sar ~ Cos-o-come (Pessimisic) ~ Cos-o-go (Opimisic) Goal 2

3 INFORMED SEARCH Informed Sraegies ha know wheher one non-goal is more promising han anoher based on heurisics 3

4 ESTIMATE OF COST-TO-GO 4

5 EVALUATION FUNCTION The uiliy (oward he goal) of a node n in he fronier se is measured by an evaluaion funcion f(n) cos (o-go) esimae (o reach a goal) Fronier se 5

6 GENERAL INFORMED STRATEGY: BEST-FIRST SEARCH f(n) can combine Backward (g(n), cos-o-come) and Forward (h(n), cos-o-go) cos esimaes: f n = g n + h(n) Bes-firs: The node in he fronier se wih he lowes cos esimae is expanded firs Daa srucure: Prioriy queue using f(n) for ordering 6

7 H(X) = HEURISTIC FUNCTION The funcion h(x) ha provides an esimae of he cos-o-go(al) and direcs sae expansion by looking forward is commonly referred o as a heurisic funcion Informed search heurisic search Heurisics are rules of humb, educaed guesses, inuiive judgmens or, simply, common sense The erm derives from he ancien Greek keuriskein, meaning serving o find ou, or discover. Archimedes Eureka! means I have found i! 7

8 EXAMPLE: HEURISTIC FOR PATH FINDING Ciy Arad 366 Sibiu 253 Rimnicu Vilcea Aerial disance 93 Fagaras 76 Piesi 00 No necessarily a good heurisic Monevideo 8

9 UNIFORM COST SEARCH AS INFORMED SEARCH Sraegy: Expand by f x = g x = cos-o-come This is where he erm uniform comes from! 9

10 UNIFORM COST SEARCH Sraegy: Expand by f x = g x = cos-o-come s c 3 2 a d 5 b e s a b d e c d g = 0 g = g = 2 g = 4 g = 6 g = 3 g = 6 g = 7 # #2 #3 #5 #6 #4 #7 0

11 GREEDY BEST-FIRST SEARCH (G-BFS) Sraegy: Expand by h x = heurisic evaluaion of cos-o-go(al) from x I would rivially work opimally wih exac h values (= o he acual pah coss), wih general h values won be opimal!

12 GREEDY BEST-FIRST SEARCH Sraegy: Expand by h x = heurisic evaluaion of cos-o-go(al) from x h = 6 h = 5 h = 2 h = 0 s c 3 2 a d 5 b e h = 7 h = 6 h = s a b d e d h = 6 h = 5 h = 6 h = 2 h = h = 2 h = 0 # #2 #3 #4 #5 A few wrong h esimaes resul in a srongly sub-opimal soluion (g() = 9) 2

13 A* SEARCH (968) Sraegy: Combine cos-o-come (pas) and heurisic esimae of cos-o-go (fuure), expand by f x = h x + g x UCS + G-BFS Robo moion planning, characer moving in games. 3

14 A* SEARCH Sraegy: Combine cos-o-come (pas) and heurisic esimae of cos-o-go (fuure), expand by f x = h x + g x Which node is expanded fourh?. d 2. e 3. s 4. c c h = 6 h = 5 h = a d 5 b e h = 7 h = 6 h = h = 0 4

15 A* SEARCH Should we sop when we discover a goal? (when we add i o fronier se) h = 3 s 2 2 h = 2 a b 2 3 h = 0 F = {s} à s F = {a,b}, f(a) = 2+2 = 4, f(b) = 2+ = 3 à b F = {a,}, f() = (2+3) + 0 = 5 à a F = {}, f() = 2+4 = 4 à h = No: Only sop when expand a goal! The fac ha anoher node in fronier has a beer evaluaion value migh indicae ha a beer pah o he goal can be found hrough i Same as in UCS, also he updaing for g(x) is he same 5

16 IS A* OPTIMAL? h = 7 h = 0 s 5 a 3 h = 6 F = a,, f a = + 6 = 7, f = Expand, goal check = rue à end Opimal pah no discovered! Wha s he problem? The good pah has a pessimisic esimae (h a > 3) Circumven his issue by being opimisic! 6

17 ADMISSIBLE HEURISTICS h is admissible, if for all nodes x, h x h x where h is he cos of he opimal pah from x o a goal An admissible heurisic is a lower bound on real cos, i s always opimisic abou he cos-o-go(al) (Max problems: An upper bound on real profis / rewards) Example: Aerial disance in he pah finding example Example: h 0 The igher he bound, he beer 7

18 OPTIMALITY OF A* Theorem: A* ree search wih an admissible heurisic reurns an opimal soluion s Proof (by conradicion): o Assume ha a subopimal goal is expanded before he opimal goal, such ha A* would no reurn he opimal soluion x Fronier 8

19 OPTIMALITY OF A* Proof (con.): o There is a node x in he fronier, on he opimal pah o ha has been discovered bu no expanded ye: f x mus be greaer han f() s o By admissibiliy: f x = g x + h x g x + h x x o Since x is on he opimal pah o, g x + h x = g o Bu g < g = f() (h = 0) Fronier o Therefore, f x < g, x should have been expanded before! 9

20 8-PUZZLE HEURISTICS Defining a good heurisic is no a rivial ask 5 2 h : #iles in wrong posiion [h (s) = 5] h 2 : sum of Manhaan disances of iles from goal [h 2 (s) = =0] Example sae 2 3 Which heurisic is admissible?. Only 2. Only h A 3. Boh and h A 4. Neiher one Goal sae 20

21 HOW TO DESIGN AN ADMISSIBLE HEURISTIC? Heurisic for designing admissible heurisics: relax he problem! Relaxaion: Remove funcional / domain consrains Add forbidden moves (e.g., eleporaion) 2

22 DOMINANCE h dominaes h iff x, h x h E x h is consisenly a igher bound compared o h 5 2 h : #iles in wrong posiion h 2 : sum of Manhaan disances of iles from goal Example sae Wha is he dominance relaion beween and h A? 2 3. dominaes h A 2. h A dominaes 3. and h A are incomparable Goal sae 22

23 8-PUZZLE HEURISTICS The following able gives he search cos (expanded nodes) of A* wih he wo heurisics, averaged over random 8-puzzles, for various soluion lenghs Lengh A (h ) A (h 2 ) Moral: Good heurisics are crucial! 23

24 GRAPH SEARCH AND OPTIMALITY Is opimaliy of A* under admissible heurisics preserved in he case of graph search? No! h = 2 s h = 4 a b h = 2 h = c 3 h = 0 (+4) a Search ree s f = (0+2) {a,b} b (+) (2+) c {a,c} c (3+) We need more han admissibiliy. (5+0) No! {a,} {} (6+0) 24

25 CONSISTENT HEURISTICS c x, y = real cos of cheapes pah beween x and y h is consisen if for every wo nodes x, y: h x c(x, y) + h(y) c x, y h y h(x) Triangle inequaliy Necessary for graph search opimaliy x y x y 25

26 CONSISTENT HEURISTICS Consisency : The esimaed disance h(x) o he goal from x canno be reduced by moving o a differen sae y and adding he esimae of he disance o he goal from y o he cos of reaching y from x c x, y h x h(y) The real cos mus be higher han, or he same of he cos implied by he values of he heurisic h = 2 s h = 4 a h = c 3 h = 0 a c x y b 2 h = 26

27 CONSISTENCY MONOTONICITY Lemma (Monooniciy of cos funcion f(x)) : If h(x) is consisen, hen in graph search he values of he cos funcion f x = g x + h(x) along any search pah are non-decreasing Proof: (you) ü In moving from a sae o is neighbor, (a consisen) h mus no decrease more han he cos of he edge ha connecs hem. ü Consisency is a propery of h(x), monooniciy is a propery of f(x) ha derives from monooniciy 27

28 CONTOURS IN THE STATE SPACE f-coss are non-decreasing along any pah from he sar Conours (isolines) in he sae-space, like in opographic maps During search, A* search fans ou adding nodes placed in near ellipic bands of increasing f-cos conained one in anoher The igh he lower bounds are, he more he bands will srech oward he goal sae Band shapes when using UCS? Z O N A T 380 S 400 R L F P I V D M C 420 G B U H E 28

29 CONSISTENT HEURISTICS h is consisen if for every wo nodes x, y, h x c(x, y) + h(y) h is admissible, if for all nodes x, h x h x Wha is he relaion beween admissibiliy and consisency? x y. Admissible consisen 2. Consisen admissible 3. They are equivalen 4. They are incomparable Assuming h() = 0 a goals 29

30 8-PUZZLE HEURISTICS, REVISITED h : #iles in wrong posiion h 2 : sum of Manhaan disances of iles from goal Which heurisic is consisen?. Only 2. Only h A 3. Boh and h A 4. Neiher one Example sae Goal sae 30

31 ADMISSIBLE BUT INCONSISTENT HEURISTICS? Keep he LB propery, bu violae monooniciy Inconsisen for a leas one pair of saes Manhaan disance for se {,2,3,4} Manhaan disance for se {5,6,7,8} A each sep, choose a random which se Wha is he relaion beween heurisic esimaes?

32 DESIGNING A CONSISTENT HEURISTIC? Heurisic for designing consisen heurisics: design an admissible heurisic! Will no work all he ime. 32

33 OPTIMALITY OF A*, REVISITED Theorem: A* graph search wih a consisen heurisic reurns an opimal soluion Proof: owhenever A* selecs a sae x for expansion, he opimal pah o x has been found. Oherwise, here would be a fronier node y (separaion propery) on he opimal pah from sar o x ha should be expanded firs because f is non decreasing along any pah (monooniciy) othe firs goal sae x seleced for expansion mus be opimal, because f(x ) is he rue (opimal) cos for goal nodes (f x = 0), and any oher laer goal node would be a leas as expensive because of f monooniciy 33

34 PRUNING, COMPLETENESS, COMPLEXITY O If C is he cos of he opimal pah, A* expands all nodes wih f x < C, and no nodes wih f x > C (auomaic pruning of hese nodes) A T Z 380 D S 400 R L M C F P 420 G N B I U V H E There migh be an exponenial number of saes wih f x < C! Difficuly increases when here are many near-opimal saes A* migh expand some of he nodes on he goal conour, where f x = C, before selecing he goal node Compleeness? Yes, if only a finiely many nodes wih cos less or equal o C are presen (b is finie and all sep coss are ε > 0) Time/Space complexiy srongly depend on problem srucure, bu are usually bad 34

35 A* SEARCH TREE FOR PATH FINDING (a) The iniial sae (b) Afer expanding Arad Arad 366=0+366 Arad Sibiu 393= Timisoara 447=8+329 Zerind 449= (c) Afer expanding Sibiu Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = = = (d) Afer expanding Rimnicu Vilcea Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea 646= = = (e) Afer expanding Fagaras Rimnicu Vilcea Craiova Piesi Sibiu 526= = = Arad Check conour lines in previous slide! Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = Sibiu Buchares Craiova Piesi Sibiu 59= = = = = (f) Afer expanding Piesi Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = Sibiu Buchares Craiova Piesi Sibiu 59= = = = Buchares Craiova Rimnicu Vilcea 48= = =

36 A* IS OPTIMALLY EFFICIENT Theorem: No oher opimal informed algorihm ha uses ha same heurisic h is guaraneed o expand fewer nodes ha A* given ha h is a consisen heurisic (excep for ie-breaking among nodes wih f x = C ) Inuiion behind he proof: any oher algorihm ha does no expand all nodes wih f x < C migh miss he opimal soluion The heorem doesn hold for he case when he heurisic is only admissible 36