What is Search? Intro to Search. Search problems. Finding goals in trees. Why is goal search not trivial? Blind Search Methods

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1 What is Search? lind Search Methods Search as an Tool S, DS, DS, US, ids Search is one of the most powerful approaches to problem solving in Search is a universal problem solving mechanism that Systematically explores the alternatives inds the sequence of steps towards a solution Problem Space ypothesis (llen Newell, SOR: n rchitecture for eneral ntelligence.) ll goal-oriented symbolic activities occur in a problem space Search in a problem space is claimed to be a completely general model of intelligence S 412, UMD, 2 ntro to Search ntroduce the notion of blind for goal nodes in trees Specifically, the general pseudo-code in Russell/Norvig (M) and in the course notes ut in a simplified context: without the notion of shortest path or minimum cost Search problems Search on graphs ind a goal node reachable from some given start node ind the shortest path between two given nodes (map problems) ind the shortest path from a given start node to any goal node (planning problems) Search on trees tree is a connected, acyclic graphs, that has precisely one path between any two nodes S 412, UMD, 3 S 412, UMD, 4 inding goals in trees Does the following tree contain a node? Yes. ow did you know that? read it? E D Why is goal not trivial? ecause the graph is not given in a nice picture on a piece of paper nstead the graph/tree is usually Explicitly known, but hidden. You need to discover it on the fly i.e. as you do the mplicitly known only. You are given a set of rules with which to create the graph on the fly Trivial! : so why the big deal about? S 412, UMD, S 412, UMD, 6 1

2 inding goals in - reality Does the tree under the following root contain a node? lind lind (aka uninformed) no information as to location of goal not giving you hotter or colder hints ll you get to see at first is the root and a guarantee that it is a tree The rest is up to you to discover during the process of S 412, UMD, 7 S 412, UMD, 8 ctions in ing a tree undamental actions (operators) that you can take: 1. Expand : sk a node for its children 2. Test : Test a node for whether it is a goal side: nternet Search Typically human will be incomplete, E.g. finding information on the internet before google, etc look at a few web pages, if no success then give up S 412, UMD, 9 S 412, UMD, 10 Properties of Search We will say a method is complete if it has both the following properties: if a goal exists then the will always find it if no goal exists then the will eventually finish and be able to say for sure that no goal exists We only look at complete methods inding oals in Trees: Reality Does the tree under the following root contain a node? llowed: Expand Test Whiteboard S 412, UMD, 11 S 412, UMD, 12 2

3 DDEN: Exercise irst step? Only choice is to ask for children of the root et people to select which node to expand Draw child nodes onto board Keep track of which nodes are eligible for expansion cross out visited ones with blue pen red-circle on the fringe Whiteboard: Results idden tree: ind a (oal) node The resulted in failure The nodes moved between three classes: E D K S 412, UMD, 13 S 412, UMD, 14 Undiscovered Nodes The set of nodes that have not yet been discovered as being reachable from the root ringe Nodes This is the set of nodes that have been discovered have not yet been processed : 1. have not yet discovered their children 2. (have not yet tested if they are a goal) lso called open nodes agenda S 412, UMD, 1 S 412, UMD, 16 Visited Nodes This is the set of nodes that have been discovered have been processed: 1. have discovered all their children 2. (have tested whether are a goal) lso called closed nodes ction on finding a oal irst match : Usually we just want one goal, or just to know whether or not one exists on discovering a goal, then return true ll Matches : Sometimes want all goals on discovering a goal, then record the fact that have found it, but continue with the S 412, UMD, 17 S 412, UMD, 18 3

4 eneral Method 1. fringe MKE-EMPTY-QUEUE() 2. fringe NSERT( root_node ) 3. found false // boolean flag 4. loop { 1. if fringe is empty then return found // finished the 2. node REMOVE-RST(fringe) 3. if node is a goal 1. print node // if want to list all matches to the goal 2. found true // so we remember we succeeded 3. (if only want first goal then return true) 4. L EXPND(node). fringe NSERT-LL( L, fringe ) } Selecting the Next Node The difference between es lies in the order in which nodes are selected for expansion The pseudo-code always uses the first node in the fringe queue The only way to control the ordering is to control the NSERT-LL S 412, UMD, 19 S 412, UMD, 20 Node Ordering: Why care? Later lectures: will see that using heuristic node orderings allow us to find goals quicker Even in blind, node orderings affect the memory usage different node orderings affect the shape of the fringe different shapes of the fringe can lead to very different memory usages Two Natural Node-Orderings Depth-irst Search (DS) readth-irst Search (S) S 412, UMD, 21 S 412, UMD, 22 Depth-irst Search Search Pattern: dive before spread ringe in red root Visited in blue Size of fringe O(bd) linear outline of tree readth-irst Search Search Pattern: spread before dive ringe in red Visited in blue root Size of fringe O(b d ) exponential outline of tree S 412, UMD, 23 S 412, UMD, 24 4

5 Two Natural Node-Orderings Depth-irst Search (DS) explores the depths first low memory usage readth-irst Search (S) explores all nodes at depth d before any at depth d+1 high memory usage (but forms basis for solving min-cost problems) riteria for evaluating strategies ompleteness: is the strategy guaranteed to find a solution when there is one? Time complexity: how long does it take to find a solution Space complexity: how much memory does it need to perform the? Optimality: does the strategy find the highestquality solution when there are several solutions? S 412, UMD, 2 S 412, UMD, 26 S ompleteness: Yes Time complexity: b d Space complexity: b d Optimality: Yes S vs DS S (b - branching factor, d - depth) DS ompleteness: No Time complexity: b m Space complexity: bm Optimality: No (b-branching factor, m-max depth of tree) S 412, UMD, 27 S 412, UMD, 28 S The example node set nitial state D E oal state K L M N O P Q R S T U V W X Y Z Press space to see a S of the example node set S 412, UMD, 29 S 412, UMD, 30

6 Node We then is backtrack expanded then to expand removed node from, the queue. and The the The process revealed then continues. moves nodes to are Press the added first space node to the END in the of the queue. queue. Press Press space space. to continue. This Node node is removed is then expanded from the to queue. reveal Each further revealed We begin (unexpanded) node with is our added initial nodes. to the state: Press END the space of node the queue. labeled Press. Press space space to continue to continue the. 8-tile puzzle S D E K L M N O P Q R S T U Node L is located and the returns a solution. Press space to end. Press space to continue begin the the Size of Queue: Queue:,, K,,,, L, E,,, D, Empty, K,, M,,, D, E,,, L, K,, L,,,, D, M, E,, N, K, L,, M,,,, E,, O, N, M, K, L,,,,, N, O, P, K, M,, L, N,, O, Q, P, K, L, M, O, N, P, Q, R, M, LP, O, N, Q, S, R, Q, NP, O, R, T, S, RQ PU ST Nodes expanded: urrent NSED ction: acktracking Expanding SER urrent level: n/a 0 12 REDT-RST SER PTTERN S 412, UMD, 31 S 412, UMD, 32 Weighted raphs Shortest path problems use weighted graphs These are graphs in which each edge is given a number called its weight The weight of an edge is intended to be its cost, or value, or length, etc Paths then have a weight which is just the sum of the weights of the edges in the path Desired Search Properties ompleteness: will find a goal if one is legally reachable Optimality: will find goal with shortest path to it and (usually) also find the shortest path itself Systematic: don t do the same work too many times but there is no generally agreed exact definition sometimes taken to mean to never repeat the same work S 412, UMD, 33 S 412, UMD, 34 Uniform ost Search in raphs Search Pattern: smaller cost nodes before larger ringe in red Visited in blue lind: still ignores goals outline of graph start goals Uniform cost breadth-first finds the shallowest goal state and will therefore be the cheapest solution provided the path cost is a function of the depth of the solution. ut, if this is not the case, then breadth-first is not guaranteed to find the best (i.e. cheapest solution). Uniform cost remedies this by expanding the lowest cost node on the fringe, where cost is the path cost, g(n). n the following slides those values that are attached to paths are the cost of using that path. increasing path cost from start S 412, UMD, 3 S 412, UMD, 36 6

7 onsider the following problem 1 10 S Once Node node is removed has been from expanded the queue it is and removed the revealed from the node queue (node and ) the is added revealed to the node queue. (node The We Node We queue now ) start S is expand is added. with removed again our the The sorted node initial from queue on at the state path the is queue again front and cost. expand of sorted Note, the queue, it on we revealed path have node cost. now nodes. Note, found Press are node added a space goal to state to now the continue. appears but queue. do not The in the recognise queue is it twice, then as it sorted is once not as at on 10 the path front and cost. once of Nodes the as queue. 11. with s Node cheaper 10 is is path the cost cheaper front have of node. the priority.n queue, Press we this space. now case proceed the queue to goal will state. be Node Press space. (1), node (), followed by node (1). Press space S 1 We wish to find the shortest route from node S to node ; that is, node S is the initial state and node is the goal state. n terms of path cost, we can clearly see that the route S is the cheapest route. owever, if we let breadth-first loose on the problem it will find the non-optimal path S, assuming that is the first node to be expanded at level 1. Press space to see a US of the same node set S 412, UMD, 37 Size of Queue: 13 0 Nodes expanded: Press space to begin the Queue: S,, Empty 10, 11,, 11, 1 The goal state is achieved and the path S-- is returned. n relation to path cost, US has found the optimal route. Press space to end. urrent NSED action: acktracking Expanding Waiting. SER urrent level: n/a 01 2 UNORM OST SER PTTERN S 412, UMD, 38 DS DS S 412, UMD, 39 S 412, UMD, 40 The example node set The Node process is expanded now continues and removed until the from goal The the queue. state is Revealed then achieved. moves nodes Press to the are space. first added node to in the the RONT queue. of Press the queue. space to Press continue space. Node is removed from the queue. Each We revealed This begin node node with is then is our added expanded initial to state: the to RONT reveal the node of labeled further queue.. (unexpanded) Press space nodes. to continue. Press space. nitial state D E D E oal state K L M N O P Q R S T U V W X Y Z Press space to see a DS of the example node set S 412, UMD, 41 K L Q R S T U Size of Queue: Press space to begin continue the the Queue: S,,, Q,,, K, L,, T, D, U,, Empty R,, E, D, L,,, D, E, D,, E, E, D, E, Node L is located and the returns a solution. Press space to end. Nodes expanded: urrent NSED ction: acktracking Expanding SER urrent level: n/a DEPT-RST SER PTTERN S 412, UMD, 42 7

8 8-tile puzzle DS Depth-Limited Search Practical DS DLS avoids the pitfalls of DS by imposing a cutoff on the maximum depth of a path. owever, if we choose a depth limit that is too small, then DLS is not even complete. The time and space complexity of DLS is similar to DS. ompleteness: Yes, if l >= d Time complexity: b l Space complexity: bl Optimality: No (b-branching factor, l-depth limit) S 412, UMD, 43 S 412, UMD, 44 Depth-Limited Search (cont) terative deepening The problem with depth-limited is deciding on a suitable depth parameter. To avoid this problem there is another called iterative deepening (DS). This method tries all possible depth limits; first 0, then 1, then 2 etc., until a solution is found. DS may seem wasteful as it is expanding nodes multiple times. ut the overhead is small in comparison to the growth of an exponential tree ig 4.16 Depth-first trees for binary tree. Same problem as ig 4.1 Depth limit, dl = 2 S 412, UMD, 4 or large spaces where is the depth of the solution is not known DS is normally the preferred method. The following slide illustrates an iterative deepening of 26 nodes (states) with an initial state of node and a goal state of node L. Press space to see the example node set. S 412, UMD, 46 The example node set We Node begin is with then our expanded initial state: and removed the node labeled from the. queue. This node Press is space. added to the queue. Press space to continue nitial state s this is the 0 th iteration of the, we cannot past any level greater than zero. This iteration now ends, and we begin the 1 st iteration. D E oal state K L M N O P Q R S T U V W X Y Z Press space to see a DS of the example node set S 412, UMD, 47 Press space to begin the Size of Queue: 10 Queue: Empty Nodes expanded: 01 urrent ction: Expanding urrent level: n/a 0 TERTVE DEEPENN SER PTTERN (0 th TERTON) S 412, UMD, 48 8

9 Node We The now is back expanded now track moves to and expand removed to level node one from, of and the queue. the the process node Press set. continues. space. Press space Press to space. continue Node We again is expanded, begin with then our initial removed state: from the the node queue, labeled and. the Note revealed that the nodes 1 st iteration are added carries to on the from front the. Press 0 th, and space. therefore the nodes expanded value is already set to 1. Press space to continue D E We The Node fter now expanding is move expanded then to moves level node and two to the we level of revealed backtrack the one node of set. the nodes to expand node Press added set. space node to Press the to. continue. space front The process of to the continue queue. then Press continues space until to continue. goal state. Press space gain, Node We again we is begin removed expand with node from our the initial to reveal queue state: the and level each the node revealed one labeled nodes. node Press. is Note added space. that to the the 2 nd front of iteration the queue. carries Press on space. from the 1 st, and therefore the nodes expanded value is already set to 7 (1+6). Press space to continue the D E s this is the 1 st iteration of the, we cannot past any level greater than level one. This iteration now ends, and we begin a 2 nd iteration. K L Node L is located on the second level and the returns a solution on its second iteration. Press space to end. Press space to continue begin the the Size of Queue: Queue:,, D, E, Empty, D, E, D, E, E, Nodes expanded: urrent ction: acktracking Expanding urrent level: n/a 01 TERTVE DEEPENN SER PTTERN (1 st TERTON) S 412, UMD, 49 Press space to continue the Size of Queue: Queue:,,,, K, L,, D, Empty,, D, E,,, L, D, E, D,, E, E, D, E, Nodes expanded: urrent SER ction: acktracking Expanding NSED urrent level: n/a 0 12 TERTVE DEEPENN SER PTTERN (2 nd TERTON) S 412, UMD, 0 S6. i-directional Search (cont) Search i-directional Search nitial State inal State d / 2 * Time complexity: O(b d/2 ) * Space complexity: O(b d/2 ) d ig 4.18 schematics view of a bi-directional S that is about to succeed, when a branch from the start node meets a branch from the goal node O(b d ) vs. O(b d/2 )? with b=10 and d=6 results in 1,111,111 vs. 2,222. S 412, UMD, 1 S 412, UMD, 2 omparing lind Search Strategies ig 4.19 omparison of 6 strategies in terms of the 4 evaluation criteria set forth in riteria for Evaluating Search Strategies b - branching factor; d is the depth of the solution; m is the maximum depth of the tree; l is the depth limit S 412, UMD, 3 9