Informed Search. Russell and Norvig Chap. 3

Transcription

1 Iformed Search Russell ad Norvig Chap. 3

2 Not all search directios are equally promisig

3 Outlie Iformed: use problem-specific kowledge Add a sese of directio to search: work toward the goal Heuristic fuctios: a way to provide iformatio to a search algorithm

4 What determies a search strategy fuctio TREE-SEARCH(problem) retur a solutio or failure Iitialize frotier usig the iitial state of problem do if the frotier is empty the retur failure choose leaf ode from the frotier if ode is a goal state the retur solutio else expad the ode ad add resultig odes to the frotier A search strategy is determied by the order i which odes i the frotier are processed

5 Best-first search Iformed search strategy: expad the ode that appears best Factors goig ito determiatio of best: q Curret cost of the solutio path q Estimated distace to the earest goal state Node is selected for expasio based o a evaluatio fuctio f() Implemetatio: q Frige is a queue sorted by value of f q Special cases: greedy search, A* search

6 Heuristics Heuristic: A rule of thumb, simplificatio, or educated guess that reduces or limits the search for solutios q The heuristic fuctio h() estimates cost of the cheapest path from ode to goal ode. q If is a goal ode h()=0

7 Greedy best-first search Expad ode o the frotier closest to goal Determiatio of closest based upo the heuristic fuctio h

8 Greedy search: A example Cosider path plaig betwee two cities Use the straight lie distace heuristic, h SLD C A B D The greedy solutio is (A, C, D) The least cost solutio is (A, B, D)

9 A* Search Order states by their total estimated cost Always select the ode with the lowest value Total estimated cost: f() = g() + h() g() the cost to reach h() the estimated cost to the goal Hart, P. E.; Nilsso, N. J.; Raphael, B. (1968). "A Formal Basis for the Heuristic Determiatio of Miimum Cost Paths". IEEE Trasactios o Systems Sciece ad Cyberetics SSC4 4 (2):

10 A* Search Order states by their total estimated cost Always select the ode with the lowest value Total estimated cost: f() = g() + h() g() the cost to reach h() the estimated cost to the goal Uiform cost search is a special case where h()=0. Hart, P. E.; Nilsso, N. J.; Raphael, B. (1968). "A Formal Basis for the Heuristic Determiatio of Miimum Cost Paths". IEEE Trasactios o Systems Sciece ad Cyberetics SSC4 4 (2):

11 Repeated states Uiformed search: q Add to frige oly if state ot already visited. A*: q If ode represets state already visited, update cost accordig to lower total estimated cost.

12 Heuristic fuctios Heuristics for the 8 puzzle: h 1 = the umber of misplaced tiles q h 1 (s)=8 h 2 = the sum of the distaces of the tiles from their goal positios (mahatta distace) q h 2 (s)= =18

13 Compariso of heuristics Eve very simple heuristics like h 1 ad h 2 ca sigificatly reduce the search cost: Algorithm Depth 10 Depth 14 Iterative Deepeig 47,127 3,473,941 A* with h A* with h

14 A* i Romaia Goal: shortest route from Arad to Bucharest

15 A* i Romaia Get to Bucharest startig at Arad q f(arad) = c(arad,arad)+h(arad)=0+366=366

16 A* i Romaia Expad Arrad ad determie f(): q f(sibiu)=c(arad,sibiu)+h(sibiu)= =393 q f(timisoara)=c(arad,timisoara)+h(timisoara)= =447 q f(zerid)=c(arad,zerid)+h(zerid)=75+374=449 Best choice is Sibiu

17 A* i Romaia Expad Sibiu ad determie f() q q q q f(arad)=c(sibiu,arad)+h(arad)= =646 f(fagaras)=c(sibiu,fagaras)+h(fagaras)= =415 f(oradea)=c(sibiu,oradea)+h(oradea)= =671 f(rimicu Vilcea)=c(Sibiu,Rimicu Vilcea)+ h(rimicu Vilcea)= =413 Best choice is Rimicu Vilcea

18 A* i Romaia Expad Rimicu Vilcea ad determie f() q q q f(craiova)=c(rimicu Vilcea, Craiova)+h(Craiova)= =526 f(pitesti)=c(rimicu Vilcea, Pitesti)+h(Pitesti)= =417 f(sibiu)=c(rimicu Vilcea,Sibiu)+h(Sibiu)= =553 Best choice is Fagaras

19 A* i Romaia Expad Fagaras ad determie f() q q f(sibiu)=c(fagaras, Sibiu)+h(Sibiu)= =591 f(bucharest)=c(fagaras,bucharest)+h(bucharest)=450+0=450 Best choice is Pitesti!

20 A* i Romaia Expad Pitesti ad determie f() q f(bucharest)=c(pitesti,bucharest)+h(bucharest)=418+0=418 Best choice is Bucharest Note values alog optimal path!! Is the solutio optimal?

21 A* i Romaia Whole subtrees of the search tree got prued!

22 Admissible heuristics A heuristic is admissible if it ever overestimates the cost to reach the goal (optimistic) Formally: 1. h() h*() where h*() is the true cost from 2. h() 0 so h(g)=0 for ay goal G. Examples: h SLD () ever overestimates the actual road distace Heuristics for 8 puzzle

23 Cosistecy A heuristic is cosistet if: h() c(, a, ) + h( ) Give a cosistet heuristic: f( ) = g( ) + h( ) g() + c(,a, ) + h( ) g() + h() = f() A cosequece of cosistecy: f() odecreasig alog a path c(, a, ): cost of gettig to from usig actio a

24 Cosistecy ad admissibility Cosistecy implies admissibility Hard to fid heuristics that are admissible but ot cosistet Focus o cosistet heuristics for provig optimality of A*

25 Cosistecy ad the optimality of A* Lemma: Wheever A* selects a ode for expasio the optimal path to that ode has bee foud (assumig cosistet heuristic). Suppose ot: The there is a uexpaded ode o the optimal path to. From mootoicity: f() f( ), so should have already bee expaded. Therefore wheever a goal ode is expaded, it is the lowest cost, i.e. optimal goal ode

26 A* expasio cotours Expasio represeted as cotours of states with equal f value A* expads all odes with f() < C* A* may expad odes o the goal cotour

27 Properties of A* A* expads all odes with f() < C* But there ca still be expoetially may such odes!

28 Evaluatio of A* Completeess: YES q Uless there are ifiitely may odes with f<f(g), ad regardless of the heuristic

29 Evaluatio of A* Completeess: YES Time complexity: q Number of odes with f() < C* ca be expoetial

30 Evaluatio of A* Completeess: YES Time complexity: q Number of odes with f() < C* ca be expoetial Space complexity: also expoetial.

31 Evaluatio of A* Completeess: YES Time complexity: q Number of odes with f() < C* ca be expoetial Space complexity: also expoetial. Optimality: YES q A* does ot expad ay ode with f() > C* Also optimally efficiet (o other optimal algorithm is guarateed to expad fewer odes)

32 Memory-bouded heuristic search Some solutios to the A* space problem (maitaiig completeess ad optimality) q Iterative-deepeig A* (IDA*) Like IDS, but cutoff iformatio is the f-cost (g+h) istead of depth Expads by cotour

33 Memory-bouded heuristic search Some solutios to A* space problems (maitaiig completeess ad optimality) q Iterative-deepeig A* (IDA*) q Recursive best-first search (RBFS) q (Simplified) Memory-bouded A* ((S)MA*) SMA*: Drop the worst-leaf ode whe memory is full (regeerate it later if ecessary; back up the value of the forgotte ode to its paret)

34 Comparig heuristics Heuristics for the 8 puzzle: h 1 = the umber of misplaced tiles h 2 = the sum of the distaces of the tiles from their goal positios (mahatta distace) For every state s, h 2 (s) h 1 (s) We say that h 2 domiates h 1 A domiatig heuristic is better for search. WHY?

35 Ivetig heuristics Admissible heuristics ca be derived from the exact solutio cost of a relaxed versio of the problem q q q Relaxed 8-puzzle for h 1 : a tile ca move aywhere. Relaxed 8-puzzle for h 2 : a tile ca move to ay adjacet square. Aother relaxatio: a tile ca move to ay blak square. Admissibility: The optimal solutio cost of a relaxed problem is o greater tha the optimal solutio cost of the real problem.

36 Ivetig heuristics Admissible heuristics ca also be derived from the solutio cost of a subproblem of a give problem. This cost is a lower boud o the cost of the real problem. Costruct a database of solutios for subproblems.

37 Ivetig heuristics Havig the best of all worlds: give admissible heuristics h 1,,h m h() = max(h 1 (),,h m ()) is a domiatig admissible heuristic. Useful i the cotext of the subproblems approach.

38 Ivetig heuristics Learig from experiece: q Experiece = solvig lots of 8-puzzles q A learig algorithm ca be used to predict costs for states that arise durig search.