Uninformed Search Complexity. Informed Search. Search Revisited. Day 2/3 of Search

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1 Informed Search ay 2/3 of Search hap. 4, Ruel & Norvig FS IFS US PFS MEM FS IS Uninformed Search omplexity N = Total number of tate = verage number of ucceor (branching factor) L = Length for tart to goal with mallet number of tep Q = verage ize of the priority queue Lmax = Length of longet path from to any tate lgorithm readth Firt Search i- irection. FS Uniform ot Search Path heck FS Memorizing FS Iterative eepening omplete, If cot > 0 Optimal, If all tran. have ame cot, If all tran. have ame cot, If cot > 0 N N, If all tran. have ame cot Time O(Min(N, L )) O(Min(N,2 L/2 )) O(log(Q)* /ε )) O( Lmax ) O(Min(N, Lmax )) O( L ) Space O(Min(N, L )) O(Min(N,2 L/2 )) O(Min(N, /ε )) O(L max ) O(Min(N, Lmax )) O(L) Material in part from Search Reviited State ready to be expanded (the fringe ) tate expanded o far f() f() f() F E Example: g() =0 g() =5 f() f() f() F E.Store a value f() at each tate 2.hooe the tate with lowet f to expand next 3.Inert it ucceor If f(.) i choen carefully, we will eventually find the lowet-cot equence US (Uniform ot Search): f() = g() = total cot of current hortet path from to Store tate awaiting expanion in a priority queue for efficient retrieval of minimum f Optimal Guaranteed to find lowet cot equence, but

2 Problem: No guidance a to how far any given tate i from the goal Solution: eign a function h(.) that give u an etimate of the ditance between a tate and the goal Our bet gue i that i cloer to than o maybe it i a more promiing tate to expand h() = 3 h() = 6 h() = 0 Heuritic Function h(.) i a heuritic function for the earch problem h() = etimate of the cot of the hortet path from to h(.) cannot be computed olely from the tate and tranition in the current problem If we could, we would already know the optimal path! h(.) i baed on external knowledge about the problem informed earch Quetion:. Typical example of h? 2. How to ue h? 3. What are deirable/neceary propertie of h? Heuritic Function Example X x The traight-line ditance i lower from than from o maybe ha a better chance to be on the bet path Heuritic Function Example X X How could we define h()? h() = Linear-geometric ditance to 2

3 Firt ttempt: Greedy et Firt Search Simplet ue of heuritic function: lway elect the node with mallet h(.) for expanion (i.e., f() = h()) Initialize PQ Inert with value h() in PQ While (PQ not empty and no goal tate i in PQ) Pop the tate with the minimum value of h from PQ For all in ucc() If i not already in PQ and ha not already been viited Inert in PQ with value h( ) Problem 2 2 h = 4 h = 3 h = 2 h = h = 0 What olution do we find in thi cae? 4 Greedy earch clearly not optimal, even though the heuritic function i non-tupid Trying to Fix the Problem g() = 0 g() = 5 f() = g() + h() = 3 h() = 3 h() = 6 f() = g() + h() = g() i the cot from to only h() etimate the cot from to Key inight: g() + h() etimate the total cot of the cheapet path from to going through * algorithm an * Fix the Problem? h = 4 h = 3 h = 2 h = h = 0 {(,4)} {(,5)} (f() = h() + g() = 3 + g() + cot(, ) = ) {(,5) (,)} (f() = h() + g() = + g() + cot(, ) = ) {(,5)} (f() = h() + g() = + g() + cot(, ) = ) {(,6)} 3

4 an * Fix the Problem? h = 4 h = 3 h = 2 h = h = 0 i placed in the queue with {(,4)} backpointer {,} {(,5)} (f() = h() + g() = 3 + g() + cot(, ) = ) {(,5) (,)} lower value of f() i found (f() = h() + g() = + g() + cot(, ) = + with 2 + backpointer 4) {,,} {(,5)} (f() = h() + g() = + g() + cot(, ) = ) {(,6)} * Termination ondition h = 8 S Queue: h = 3 {(,4) (,8)} h = h = 2 {(,4) (,8)} {(,4) (,8)} h = G {(,8) (G,0)} Stop when i popped from the queue! G h = 8 S h = * Termination ondition h = 2 h = h = 3 Queue: {(,4) (,8)} {(,4) (,8)} {(,4) (,8)} {(,8) (G,0)} We have encountered G before we have a chance to viit the branch going through. The problem i that at each tep we ue only an etimate of the path cot to the goal Stop when i popped from the queue! h = Reviiting State h = 8 /2 tate that wa already in the queue i re-viited. How i it priority updated? h = 3 h = 8 h = 4

5 h = Reviiting State h = 8 /2 tate that had been already expanded i re-viited. (areful: Thi i a different example.) h = 3 h = 2 h = Pop tate with lowet f() in queue If = return SUESS Ele expand : For all in ucc (): f = g( ) + h( ) = g() + cot(, ) + h( ) If ( not een before OR previouly expanded with f( ) > f OR in PQ with with f( ) > f ) Promote/Inert with new value f in PQ previou( ) Ele * lgorithm (inide loop) Ignore (becaue it ha been viited and it current path cot f( ) i till the lowet path cot from to ) Under what ondition i * Optimal? h = 6 h = 3 {(,6)} {(,3) (,8)} Final path: {, } with cot = 3 Problem: h(.) i a poor etimate of path cot to the goal tate dmiible Heuritic efine h*() = the true minimal cot to the goal from h i admiible if h() <= h*() for all tate In word: n admiible heuritic never overetimate the cot to the goal. Optimitic etimate of cot to goal. * i guaranteed to find the optimal path if h i admiible 5

6 X X x onitent (Monotonic) Heuritic h() h( ) ot(, ) h() <= h( ) + cot(, ) onitent (Monotonic) Heuritic h() ot(, ) Sort of triangular inequality implie that path cot alway increae + need to expand node only once h( ) h() <= h( ) + cot(, ) Pop tate with lowet f() in queue If = return SUESS Ele expand : For all in ucc (): f = g( ) + h( ) = g() + cot(, ) + h( ) If ( not een before OR previouly expanded with f( ) > f OR in PQ with with f( ) > f ) Promote/Inert with new value f in PQ previou( ) Ele Ignore (becaue it ha been viited and it current path cot f( ) i till the lowet path cot from to ) If h i conitent 8 h() Example 5 3 h()? For the navigation problem: The length of the hortet path i at leat the ditance between and Euclidean ditance i an admiible heuritic What about the puzzle? 6

7 h () = h 2 () = = 8 h = miplaced tile h 2 = Manhattan ditance omparing Heuritic Iterative eepening * with heuritic h * with heuritic h 2 L = 4 tep L = 8 tep 6,300 Overetimate * performance becaue of the tendency of IS to expand tate repeatedly Number of tate expanded doe not include log() time acce to queue L = 2 tep 3.6 x Example from Ruell&Norvig omparing Heuritic h () = h 2 () = = 8 h 2 i larger than h and, at ame time, * eem to be more efficient with h 2. I there a connection between thee two obervation? h 2 dominate h if h 2 () >= h () for all For any two heuritic h 2 and h : If h 2 dominate h then * i more efficient (expand fewer tate) with h 2 Limitation omputation: In the wort cae, we may have to explore all the tate O(N) The good new: * i optimally efficient For a given h(.), no other optimal algorithm will expand fewer node The bad new: Storage i alo potentially exponential O(N) Intuition: ince h <= h*, a larger h i a better approximation of the true path cot

8 I* (Iterative eepening *) Same idea a Iterative eepening FS except ue f() to control depth of earch intead of the number of tranition Example, auming integer cot:. Run FS, topping at tate uch that f() > 0 Stop if goal reached 2. Run FS, topping at tate uch that f() > Stop if goal reached 3. Run FS, topping at tate uch that f() > 2 Stop if goal reached..keep going by increaing the limit on f by every time omplete (auming we ue loop-avoiding FS) Optimal More expenive in computation cot than * Memory order L a in FS Summary Informed earch and heuritic Firt attempt: et-firt Greedy earch * algorithm Optimality ondition on heuritic function ompletene Limitation, pace complexity iue Extenion Nil Nilon. Problem Solving Method in rtificial Intelligence. McGraw Hill (9) Judea Pearl. Heuritic: Intelligent Search Strategie for omputer Problem Solving (984) hapter 3&4 Ruel & Norvig 8