MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science Artificial Intelligence Fall, 2010

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1 MSSHUSETTS INSTITUTE OF TEHNOLOY epartment of Electrical Engineering and omputer Science 6.0 rtificial Intelligence Fall, 00 Search Me! Recitation, Thursday September Prof. ob erwick. ifference between Various Search Strategies: Uninformed search, and a bit more The most important data structure in the search algorithm is the queue Q, the list of partial paths. To make things efficient, we may also use a list of nodes already examined, via an extended list E. partial path is a path from the start node to some node X, in reverse order, e.g., (X S) The head of a partial path is the most recent node of the path, e.g., X. Let S be the start node and be the goal node. Here is a simple, uninformed search algorithm: : Initialize Q with partial path ((S)) as only entry: while Q is not empty do : Pick some path N from Q : if head(n)=, then : return N (we ve reached the goal) 6: end if 7: remove N from Q 8: for all children c in head(n) not in E do 9: extend the partial path N to c 0: end for : dd these extensions of N somewhere in Q : end while : Signal failure Two operations on Q primarily determine the search strategy: Step : Which path to extend? Strategy of picking element(s) N from Q. Step : How should the newly extended paths be added to Q? Strategy for adding path extensions from node(s) N. Search Strategy N dd extensions of N to Q epth-first First element Front of Q readth-first First element End of Q est-first est by heuristic value nywhere in Q Hill-limbing (no backup) First element Replace Q with sorted extensions of N Hill-limbing (w/backup) First element Front of Q, sorted by heuristic value eam (width k, no backup) est k by heuristic value nywhere in Q ll basic search methods, except for some so called informed/heuristic search methods (like best-first search and beam search), pick the first partial path in Q. s we will see, the most variation is in where the extended paths are inserted (and in some cases, also how). Note that in classical algorithms terminology, the paths added to Q, or the enqueued items, are called the open list ; the extended list is called the closed list. further note on the extended list. This is a lousy implementation. Most often space is allocated in the node itself to keep a mark, which makes adding to the extended list and checking whether a node is there a constant time operation. lternatively, a hash table may be used for efficient detection. In any case, the incremental space cost of an extended list will be proportional to the number of nodes in the worst case, which can be very high. To implement the searches w/o backtracking, we simply add

2 If, in addition, we use an extended list E, then the only difference is that () we add head(n) to E after N is extended; and () after we remove N from Q, we first check whether head(n) is in E. If head(n) is in E, then we continue at the top of the while loop without extending N. Otherwise, we continue with the remaining steps of the while loop (extending N and adding extensions to Q). To implement a search method without backtracking/backups (T), instead of adding all the extensions of N to Q, only one is added. Let s try out FS; FS; Hill-climb w/backup, ie, backtracking. FS: pick first element of Q; add path extensions to front of Q Step Q Extended list (S) ( S) ( S) S ( S) ( S) ( S), S ( S) ( S),, S ( S) ( S),,, S 6 Success- pop list,,,, S S w/ dded paths underlined. Why didn t we add ( S) in Step? Your turn FS: pick first element of Q; add extensions to end of Q. Step Q Extended list (S) ( S) ( S) S ( S)( S)( S), S [what happens now?] 6 Hill-climbing, with backup. (So this is a little bit informed!) Pick first element of Q. dd path extensions (sorted by heuristic value) to front of Q. Heuristic value is a measure of the goodness of the path, e.g., an estimate of how far to go, as the crow flies; or in some other terms if not a map. (We will see this a bit later how to work this into optimal search.) Note that hill-climbing only looks at next locally best step. (We tack the heuristic value to the front of the list, to keep track; note sorting. What if no backup?) Step Q Extended list (0 S) ( S) ( S) S ( S) ( S) ( S), S ( S) ( S),, S (0 S) ( S),,, S 6 Success,,,, S S Heuristic Values = = S=0 = = =0

3 est-first search: idea is to use an evaluation function for each node, an estimate of its desirability, and then expand most desirable unexpanded node along the entire fringe. Step Q Extended list (0 S) ( S) ( S) S ( S) ( S) ( S), S ( S) ( S),, S (0 S) ( S),,, S 6 Success,,,, S ost and Performance of Various Search Strategies (branching factor = b, depth = d) Worst case time = proportional to # nodes visited Worst case space= proportional to maximum length of Q Search Strategy Worst Time Worst Space Fewest Nodes? epth-first (with backup) b d+ bd No Yes readth-first b d+ b d Yes Yes Hill-limbing (no backup) d b No No Hill-limbing (with backup) b d+ bd No Yes est-first b d+ b d No Yes eam (beam width k, no backup) kd kb No No uaranteed to find path? How could we combine the space efficiency of FS with FS? (FS guaranteed to find path to goal with minimum number of nodes.) nswer: Iterative eepening Search (IS) search FS, level by level, until we run out of time. Let s see. ounting Nodes in a Tree Why is (b d+ )/(b-) the number of nodes in a tree? (branching factor = b, depth = d) If each node has b immediate descendents: o Then Level 0 (the root) has node. Level has b nodes. Level has b * b = b nodes. Level has b * b = b nodes. Level d has b d- * b = b d nodes. o o o b

4 So the total number of nodes is: N = + b + b + b + b + + b d bn = b + b + b + b + + b d + b d+ Subtracting: (b )N = b d+ N = b d+ b So we could do this to implement iterative deepening search (IS): : Initialize max =. (The goal node is of unknown depth d) : o : FS from S for fixed depth max : If found a goal node, depth d max then exit : max = max + ost is: O(b +b + +b L )=O(b L ) where L= length to goal. ut isn t IS wasteful because we repeat searches on the different iterations? No. For example, suppose b=0 and d=. Then the total # number of nodes N we look at for in each case is: N(IS) db+ (d )b + + b =,0, while for FS the # of nodes is appro: N(FS) b + b + + b =,0, or only about 0% less. Most of the time is spent at depth d. So, IS is asymptotically optimal; because most of the time is spent in the fringe of the search tree. It is the preferred method over FS, FS when the goal depth is unknown.. Searching smarter: optimal search (finding the best path to the goal) Heuristics, Patient rules of thumb, So often scorned: Sloppy, umb! Yet, Slowly, common sense come Ode to I There are functions that an agent can use to guide optimal search (smallest cost, path). g(n): function that measures cost from start node to the current node n (e.g., distance traveled so far, # tiles moved,. h(n): function that estimates future (forthcoming) costs from a node n to the goal node. This is usually called a heuristic function. (e.g., as crow flies distance, or even, h(n)=0 ) Note: the total estimated cost to the goal running through node n is thus given by: f(n)=g(n)+h(n) The total true cost to the goal will be denoted as: f * (n)=g(n)+h * (n) (where f * and h * indicate actual rather than estimated values. Note that in general we will set things up so that the following holds (why?) f * (n) f(n) a. ranch and bound (&) Pick best (measured by total path length) elt of Q; ie, Set h(n) = distance so far, always 0 (Q: what about ties?) dd path extensions anywhere in Q

5 & is like est-first except uses total length/cost of a path instead of the heuristic value of just the head node, pushing forward along the entire fringe of the search tree. lso called Uniform cost search in algorithms literature (we shall see why) Step Q E (0 S) ( S) ( S) S ( S) ( S)(6 S), S ( S)(6 S),,S (6 S)(6 S)(0 S),,,S 6 (6 S)(8 S)(9 S)(0 S),,,,S 7 (8 S)(9 S)(9 S)(0 S),,,,,S 8 Success! (8 S),,,,,S S What about ties, as in step? Note how the cost always increases monotonically in a branch-and-bound search. Q: Why don t we stop at step? We don t want to stop if the goal just appears in some path! Why not? Now we need to add two more things, to get optimal * search (ijkstra s algorithm). Note that & is really finding the optimal path to each node in the graph. It is not biased in the direction of the goal node. We use the heuristic function h to do that. * = & + dynamic programming + admissible heuristic. We know what & (branch and bound is). ynamic programming principle (Principle WP told you we d see it again!) (i) iven that path length is additive, the shortest path from S to via a node X must be made up of the shortest path from S to X and then the shortest path from X to. (ii) This means we only need to compute the single best path from S to any node X, because any other path will not be part of the final answer. (iii) Note that the first time & pulls a partial path of Q whose head node is X, this path is the shortest path from S to X. This follows from the fact that & enumerates paths in order of total length. (iv) So, once we find one path to N, we don t need to consider (extend) any other paths to N. This is where we can use the extended list again. If the head of the partial path we pull off of Q is in the extended list, we discard the partial path. We should also discard paths already in Q whose head is in the extended list. With all this, & is still correct, but inefficient. Why? (See answer above.). * and Heuristic functions. dmissible heuristics, consistency, and examples The main idea of * is to avoid expanding paths that are already expensive. We use the evaluation function f(n)=g(n)+h(n).

6 So, we sort the Q by this value, and pick the best f. heuristic is admissible if h(n) h * (n), i.e., the heuristic cost is always an underestimate (optimistic) estimate of the true cost to the goal from that node. Note that 0 will always be an admissible heuristic value, but then we are just doing & again, it won t prune the search space. The straight line distance is admissible if we re in a Euclidean space. ut we might be in other situations with costs, etc., where we can t assume this. We also require that h(n) be nonzero. If these conditions holds, then * search will provably find the optimal path. (If they don t hold, it may not, as an example will show. We may miss a better path.) Step Q E (0 S) ( S) (8 S) S ( S)(7 S)(8 S), S (7 S)(8 S),,S (8 S)(8 S)(0 S),,,S 6 Success! (8 S),,,,S Note the Q is shorter * is more efficient than &. Why didn t we expand (8 S) after step? S Heuristic Values = = S=0 = = =0 Note that if the heuristic values at S =0 and = 6, these would be inadmissible because, e.g., is an overestimate of the remaining distance to the goal,. So the h value at must be. dmissibility is a constraint that must hold between every node and the goal node. There is another constraint that is sometimes easier to check, that also implies admissibility, namely, consistency, which amounts to the triangle inequality, and ensures that f(n) is non-decreasing along any path. (However, admissibility does not imply consistency, so it s not a bi-conditional.) efinition: heuristic is consistent if, for every node n, every successor n' of n satisfies the following condition: h(n) c(n,a,n') + h(n') c(n,a,n ) n n h(n ) h(n) So if h is consistent, we have: f(n') = g(n') + h(n') [by dfn of f] = g(n) + c(n,a, n') + h(n') g(n) + h(n) = f(n) [substituting for dfn of consistent h & dfn of f] So f(n) is non-decreasing along any path. Question: is the search graph above consistent? (Hint: Look at node and calculate f values.)

7 Example. Is this search tree admissible? hange values so that it is admissible, but not consistent. g=0 g=80 h=0 80 f=0 h=00 f=00 g= h= f=80 0 g=90 g=0 g=8 h=0 E F h=80 h= f=0 f= f= 0 0 g=70 h=0 f=70 Properties of *. * is complete unless there are infinitely many nodes s.t. f f(). Time: Exponential in [relative error in h x length of solution path]. Space: Keeps all nodes in memory (the dark side of *, usually runs out of memory). Optimal: Yes, cannot expand f i+ until f i is finished * expands all nodes with f(n) < *, where * is the optimum cost/distance * expands some nodes with f(n) = * * expands no nodes with f(n) > * Optimality of * [optional] Start n Suppose the algorithm generates some suboptimal goal and is in the fringe, as in the picture. Let n be an unexpanded node in the fringe such that n is on a shortest path to the optimal goal. Then: () f( ) = g( ) since h( )=0 () g( ) > g() since is suboptimal () f() = g() since h() = 0 () f( ) > f() from (), (), () () h(n) h*(n) since h is admissible (6) g(n) +h(n) g(n) +h*(n) (7) f(n) f() by dfn of f() as g(n) +h*(n) ut then: (8) f( ) > f(n) by () and (7), so * will never select for expansion. QE.

8 nother picture, possibly more helpful (see properties of *). * expands in terms of increasing f values (like &), directed along contours pointing towards the goal. O Z N I T S R F V L P M 0 U H E. The value of good heuristics: the 8 puzzle Start State oal State What is a legal move;? What would be a good heuristic h for this puzzle? Note that even IS search is costly; if # tiles is, then IS typically searches,7,9 nodes. If # tiles is, then about,000,000,000 nodes. Two suggested heuristics, h =7; h =???? The first is called: The second is called: Question: can you guess what happens with efficiency of search if it s always the case that: h (n) h (n) for all n? (oth admissible). Why?