What are informed search strategies in Artificial intelligence

informed search strategies in artificial intelligence examples, informed search strategies in artificial intelligence examples,types of informed search strategies and differentiate informed and uninformed search strategies
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INFORMED SEARCH AND 4 EXPLORATION : In which we see how information about the state space can prevent algorithms from blundering about in the dark. Chapter 3 showed that uninformed search strategies can find solutions to problems by system- atically generating new states and testing them against the goal. Unfortunately, these strate- gies are incredibly inefficient in most cases. This chapter shows how an informed search strategy-one that uses problem-specific knowledge-can find solutions more efficiently. Section 4.1 describes informed versions of the algorithms in Chapter 3, and Section 4.2 ex- plains how the necessary problem-specific information can be obtained. Sections 4.3 and 4.4 cover algorithms that perform purely local search in the state space, evaluating and modify- ing one or more current states rather than systematically exploring paths from an initial state. These algorithms are suitable for problems in which the path cost is irrelevant and all that matters is the solution state itself. The family of local-search algorithms includes methods inspired by statistical physics (simulated annealing) and evolutionary biology (genetic al- gorithms). Finally, Section 4.5 investigates online search, in which the agent is faced with a state space that is completely unknown. INFORMED SEARCH This section shows how an informed search strategyone that uses problem-specific knowl- edge beyond the definition of the problem itself-can find solutions more efficiently than an uninformed strategy. The general approach we will consider is called best-first search. Best-first search is BEST-FIRST SEARCH an instance of the general TREE-SEARCH or GRAPH-SEARCH algorithm in which a node is EVALUATION selected for expansion based on an evaluation function, f (n) . Traditionally, the node with FUNCTION the lowest evaluation is selected for expansion, because the evaluation measures distance to the goal. Best-first search can be implemented within our general search framework via a priority queue, a data structure that will maintain the fringe in ascending order of f -values. 7 The name "best-first search ' is a venerable but inaccurate one. After all, if we could really expand the best node first, it would not be a search at all; it would be a straight march to Section 4.1. Informed (Heuristic) Search Strategies 95 the goal. All we can do is choose the node that appears to be best according to the evaluation function. If the evaluation function is exactly accurate, then this will indeed ble the best node; in reality, the evaluation function will sometimes be off, and can lead the search astray. Nevertheless, we will stick with the name "best-first search," because "seemingly-best-first search" is a little awkward. There is a whole family of BEST-FIRST-SEARCH algorithms with different evaluation HEURISTIC functions.' A key component of these algorithms is a heuristic fnction, denoted h(n): FUNCTION h(n) = estimated cost of the cheapest path from node n to a goal node. For example, in Romania, one might estimate the cost of the cheapest path from Arad to Bucharest via the straight-line distance from Arad to Bucharest. Heuristic functions are the most common form in which additional knowledge of the problem is imparted to the search algorithm. We will study heuristics in more depth in Sec- tion 4.2. For now, we will consider them to be arbitrary problem-specific functions, with one constraint: if n is a goal node, then h(n) = 0. The remainder of this section covers two ways to use heuristic information to guide search. Greedy best-first search 3 Greedy best-first search tries to expand the node that is closest to the goal, on the: grounds SEARCH that this is likely to lead to a solution quickly. Thus, it evaluates nodes by using just the heuristic function: f (n) = h(n). Let us see how this works for route-finding problems in Romania, using the straight- line distance heuristic, which we will call hsLD. If the goal is Bucharest, we will need to DlSTANCli know the straight-line distances to Bucharest, which are shown in Figure 4.1. For example, hsLD(In(Arad)) = 366. Notice that the values of hsLD cannot be computed from the prob- lem description itself. Moreover, it takes a certain amount of experience to know that hSLD is correlated with actual road distances and is, therefore, a useful heuristic. Arad 366 Mehadia 24 1 Bucharest 0 Neamt 234 Craiova 160 Oradea 380 Drobeta 242 Pitesti 100 Eforie 161 Rimnicu Vilcea 193 Fagaras 176 Sibiu 253 Giurgiu 77 Timisoara 329 Hirsova 151 Urziceni 80 Iasi 226 Vaslui 199 Lugoj 244 Zerind 374 Figure 4.1 Values of SLD-straight-line distances to Bucharest. Exercise 4.3 asks you to show that this family includes several familiar uninformed algorithms. "heuristic function h(n) takes a node as input, but it depends only on the state at that node. Our first edition called this greedy search; other authors have called it best-first search. Our mare general usage of the latter tern follows Pearl (1984). 96 Chapter 4. Informed Search and Exploration (a) The initial state (c) After expanding Sibiu d& && (d) After expanding Fagaras 253 0 Figure 4.2 Stages in a greedy best-first search for Bucharest using the straight-line dis- tance heuristic hsto. Nodes are labeled with their h-values. Figure 4.2 shows the progress of a greedy best-first search using hsLD to find a path from Arad to Bucharest. The first node to be expanded from Arad will be Sibiu, because it is closer to Bucharest than either Zerind or Timisoara. The next node to be expanded will be Fagaras, because it is closest. Fagaras in turn generates Bucharest, which is the goal. For this particular problem, greedy best-first search using hsLD finds a solution without ever expanding a node that is not on the solution path; hence, its search cost is minimal. It is not optimal, however: the path via Sibiu and Fagaras to Bucharest is 32 kilometers longer than the path through Rimnicu Vilcea and Pitesti. This shows why the algorithm is called 2 "greedy '-at each step it tries to get as close to the goal as it can. Minimizing h(n) is susceptible to false starts. Consider the problem of getting from Iasi to Fagaras. The heuristic suggests that Neamt be expanded first, because it is closest Section 4.1. Informed (Heuristic) Search Strategies 97 to Fagaras, but it is a dead end. The solution is to go first to Vaslui- a step that is actually farther from the goal according to the heuristic-and then to continue to Urziceni, Bucharest, and Fagaras. In this case, then, the heuristic causes unnecessary nodes to be expanded. Fur- thermore, if we are not careful to detect repeated states, the solution will never be found-the search will oscillate between Neamt and Iasi. Greedy best-first search resembles depth-first search in the way it prefers to follow a single path all the way to the goal, but will back up when it hits a dead end. It suffers from the same defects as depth-first search-it is not optimal, and it is incomplete (becixuse it can start down an infinite path and never return to try other possibilities). The worst-case time m and space complexity is O(b ), where m is the maximum depth of the search space. With a good heuristic function, however, the complexity can be reduced substantially. Tlie amount of the reduction depends on the particular problem and on the quality of the heuristic. A search: Minimizing the total estimated solution cost A SEARCH The most widely-known form of best-first search is called A search (pronounced "A-star search"). It evaluates nodes by combining g(n), the cost to reach the node, and h(n.), the cost to get from the node to the goal: Since g(n) gives the path cost from the start node to node n, and h(n) is the estirnated cost of the cheapest path from n to the goal, we have f (n) = estimated cost of the cheapest solution through n Thus, if we are trying to find the cheapest solution, a reasonable thing to try first is the node with the lowest value of g(n) + h(n). It turns out that this strategy is more than just reasonable: provided that the heuristic function h(n) satisfies certain conditions, K search is both complete and optimal. The optimality of A is straightforward to analyze if it is used with TREE-SEARCH. ADMISSIBLE HEURISTIC In this case, A is optimal if h(n) is an admissible heuristic-that is, provided that h(n) never overestimates the cost to reach the goal. Admissible heuristics are by nature optimistic, because they think the cost of solving the problem is less than it actually is. Since g(n) is the exact cost to reach n, we have as immediate consequence that f (n) never overestimates the true cost of a solution through n. An obvious example of an admissible heuristic is the straight-line distance hsLD that we used in getting to Bucharest. Straight-line distance is admissible because the shortest path between any two points is a straight line, so the straight line cannot be an overestimate. In Figure 4.3, we show the progress of an A tree search for Bucharest. The values of g are computed from the step costs in Figure 3.2, and the values of hsLD are given in Figure 4.1. Notice in particular that Bucharest first appears on the fringe at step (e), but it is noit selected for expansion because its f -cost (450) is higher than that of Pitesti (417). Another way to say this is that there might be a solution through Pitesti whose cost is as low as 417, so the algorithm will not settle for a solution that costs 450. From this example, we can extract a general proof that A using TREE-SEARCH is optimal if h(n) is admissible. Suppose a 98 Chapter 4. Informed Search and Exploration (a) The initial state 366=0+366 (b) After expanding Arad 393=140+253 (c) After expanding Sibiu 646=280+366 415=239+176 671=291+380 413=220+193 (d) After expanding Rimnicu Vilce 526=366+160 417=317+100 553=300+253 (e) After expanding Fagaras 591=338+253 450=450+0 526=366+160 417=317+100 553=300+253 (f) After expanding Pitesti 418=418+0 615=455+160 607=414+193 Stages in an A search for Bucharest. Nodes are labeled with f = g + h. The Figure 4.3 h values are the straight-line distances to Bucharest taken from Figure 4.1. Section 4.1. Informed (Heuristic) Search Strategies 99 suboptimal goal node G2 appears on the fringe, and let the cost of the optimal solution be C. Then, because G2 is suboptimal and because h(G2) = 0 (true for any goal node), vie know f (G2) = g(G2) + h(G2) = g(G2) C . Now consider a fringe node n that is on an optimal solution path-for example, Pitesti in the example of the preceding paragraph. (There must always be such a node if a solution exists.) If h(n) does not overestimate the cost of completing the solution path, then we know that f (n) = g(n) + h(n) 5 C . Now we have shown that f (n) 5 C f (G2), so G2 will not be expanded anti A must return an optimal solution. If we use the GRAPH-SEARCH algorithm of Figure 3.19 instead of TREE-SEARCH, then this proof breaks down. Suboptimal solutions can be returned because GRAPH-SEARCH can discard the optimal path to a repeated state if it is not the first one generated. (See There are two ways to fix this problem. The first solution is to extend Exercise 4.4.) GRAPH-SEARCH so that it discards the more expensive of any two paths found to the same node. (See the discussion in Section 3.5.) The extra bookkeeping is messy, but it does guar- antee optimality. The second solution is to ensure that the optimal path to any repeated state is always the first one followed-as is the case with uniform-cost search. This property holds if CONSISTENCY we impose an extra requirement on h(n), namely the requirement of consistency (also called MONOTONICIN monotonicity). A heuristic h(n) is consistent if, for every node n and every successor n' of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus the estimated cost of reaching the goal from n': h(n) 5 c(n, a, n') t h(nf) . TRlANGLli This is a form of the general triangle inequality, which stipulates that each side of ;I triangle INEQUALIN cannot be longer than the sum of the other two sides. Here, the triangle is formed by n, n', and the goal closest to n. It is fairly easy to show (Exercise 4.7) that every consistent heuristic is also admissible. The most important consequence of consistency is the following: A using GRAPH-SEARCH is optimal h(n) is consistent. Although consistency is a stricter requirement than admissibility, one has to wlork quite hard to concoct heuristics that are admissible but not consistent. All the admissible heuristics we discuss in this chapter are also consistent. Consider, for example, hsLD. We know that Ithe general triangle inequality is satisfied when each side is measured by the straight-line distance, and that the straight-line distance between n and n' is no greater than c(n, a, n'). IHence, hsLD is a consistent heuristic. Another important consequence of consistency is the following: Zf h(n) is consistent, then the values off (n) along any path are nondecueasing. The proof follows directly from the definition of consistency. Suppose n' is a successor of n; then g(n') = g(n) + c(n, a, n') for some a, and we have f(nf) = g(nf) + h(nf) = g(n) + c(n, a, n') + h(nf) 2 g(n) + h(n) = f (n) . It follows that the sequence of nodes expanded by A using GRAPH-SEARCH is in nonde- creasing order of f (n). Hence, the first goal node selected for expansion must be an optimal solution, since all later nodes will be at least as expensive. 100 Chapter 4. Informed Search and Exploration Figure 4.4 Map of Romania showing contours at f = 380, f = 400 and f = 420, with Arad as the start state. Nodes inside a given contour have f -costs less than or equal to the contour value. The fact that f -costs are nondecreasing along any path also means that we can draw contours in the state space, just like the contours in a topographic map. Figure 4.4 shows an CONTOURS example. Inside the contour labeled 400, all nodes have f (n) less than or equal to 400, and so on. Then, because A expands the fringe node of lowest f -cost, we can see that an A search fans out from the start node, adding nodes in concentric bands of increasing f -cost. With uniform-cost search (A search using h(n) = O), the bands will be "circular" around the start state. With more accurate heuristics, the bands will stretch toward the goal state and become more narrowly focused around the optimal path. If C is the cost of the optimal solution path, then we can say the following: A expands all nodes with f (n) C. A might then expand some of the nodes right on the "goal contour" (where f (n) = C) before selecting a goal node. Intuitively, it is obvious that the first solution found must be an optimal one, because goal nodes in all subsequent contours will have higher f -cost, and thus higher g-cost (because all goal nodes have h(n) = 0). Intuitively, it is also obvious that A search is complete. As we add bands of increasing f, we must eventually reach a band where f is equal to the cost of the path to a goal state.4 Notice that A expands no nodes with f (n) C-for example, Timisoara is not expanded in Figure 4.3 even though it is a child of the root. We say that the subtree below PRUNING Timisoara is pruned; because hsLD is admissible, the algorithm can safely ignore this subtree Completeness requires that there be only finitely many nodes with cost less than or equal to C, a condition that is true if all step costs exceed some finite t and if b is finite. Section 4.1. Informed (Heuristic) Search Strategies 101 while still guaranteeing optimality. The concept of pruning-eliminating possibilities from consideration without having to examine them-is important for many areas of AI. One final observation is that among optimal algorithms of this type-algorithms that extend search paths from the root- A is optimally efficient for any given heuristic function. That is, no other optimal algorithm is guaranteed to expand fewer nodes than A (except possibly through tie-breaking among nodes with f (n) = C). This is because any algorithm that does not expand all nodes with f (n) C runs the risk of missing the optimal solution. That A search is complete, optimal, and optimally efficient among all such algorithms is rather satisfying. Unfortunately, it does not mean that A is the answer to all our searching needs. The catch is that, for most problems, the number of nodes within the goal contour search space is still exponential in the length of the solution. Although the proof of the result is beyond the scope of this book, it has been shown that exponential growth will occur unless the error in the heuristic function grows no faster than the logarithm of the actual path cost. In mathematical notation, the condition for subexponential growth is that where h (n) is the true cost of getting from n to the goal. For almost all heuristics in practical use, the error is at least proportional to the path cost, and the resulting exponential growth eventually overtakes any computer. For this reason, it is often impractical to insist on finding an optimal solution. One can use variants of A that find suboptimal solutions quickly, or one can sometimes design heuristics that are more accurate, but not strictly admissible. In any case, the use of a good heuristic still provides enormolus savings compared to the use of an uninformed search. In Section 4.2, we will look at the question of designing good heuristics. Computation time is not, however, A's main drawback. Because it keeps all generated RAPH-SEARCH algorithms), A usually runs out of space long nodes in memory (as do all G before it runs out of time. For this reason, A is not practical for many large-scale prob- lems. Recently developed algorithms have overcome the space problem without sacrificing optimality or completeness, at a small cost in execution time. These are discussed next. Memory-bounded heuristic search The simples1 way to reduce memory requirements for A" is to adapt the idea of iterative deep- ening to the heuristic search context, resulting in the iterative-deepening A " (IDA) algorithm. The main difference between IDA and standard iterative deepening is that the cutoff used f -cost (g + h) rather than the depth; at each iteration, the cutoff value is the small- is the est f -cost of any node that exceeded the cutoff on the previous iteration. IDA is practical for many problems with unit step costs and avoids the substantial overhead associated with keeping a sorted queue of nodes. Unfortunately, it suffers from the same difficulties with real- valued costs as does the iterative version of uniform-cost search described in Exercise 3.11. This section briefly examines two more recent memory-bounded algorithms, called RBFS and MA. RECURSIVE BEST-FIRST SEARCH Recursive best-first search (RBFS) is a simple recursive algorithm that attempts to -first search, but using only linear space. The algorithm is mimic the operation of standard best shown in Figure 4.5. Its structure is similar to that of a recursive depth-first search, but rather 102 Chapter 4. Informed Search and Exploration function RECURSIVE-BET-FIRT-SEARCH(O) returns a solution, or failure RBFS(problem, MAKE-NoDE(INITIAL-STATEO), oo) function RBFS(problem, node, f-limit) returns a solution, or failure and a new f -cost limit if GoAL-TEsTo(STATEO) then return node successors +- EXPAND(O, problem) ' if successors is empty then return failure, oo for each s in successors do f sl +max(g(s) + h(s), f nodel) repeat best +the lowest f -value node in successors iff best f-limit then return failure, f best alternative +- the second-lowest f -value among successors result, f best + RBFS(problem, best,min( f-limit, alternative)) if result failure then return result Figure 4.5 The algorithm for recursive best-first search. 1 than continuing indefinitely down the current path, it keeps track of the f-value of the best alternative path available from any ancestor of the current node. If the current node exceeds this limit, the recursion unwinds back to the alternative path. As the recursion unwinds, RBFS replaces the f -value of each node along the path with the best f -value of its children. In this way, RBFS remembers the f -value of the best leaf in the forgotten subtree and can therefore decide whether it's worth reexpanding the subtree at some later time. Figure 4.6 shows how RBFS reaches Bucharest. RBFS is somewhat more efficient than IDA, but still suffers from excessive node re- generation. In the example in Figure 4.6, RBFS first follows the path via Rimnicu Vilcea, then "changes its mind" and tries Fagaras, and then changes its mind back again. These mind changes occur because every time the current best path is extended, there is a good chance that its f -value will increase-h is usually less optimistic for nodes closer to the goal. When this happens, particularly in large search spaces, the second-best path might become the best path, so the search has to backtrack to follow it. Each mind change corresponds to an iteration of IDA, and could require many reexpansions of forgotten nodes to recreate the best path and extend it one more node. Like A, RBFS is an optimal algorithm if the heuristic function h(n) is admissible. Its space complexity is linear in the depth of the deepest optimal solution, but its time complexity is rather difficult to characterize: it depends both on the accuracy of the heuristic function and on how often the best path changes as nodes are expanded. Both IDA and RBFS are subject to the potentially exponential increase in complexity associated with searching on graphs (see Section 3.5), because they cannot check for repeated states other than those on the current path. Thus, they may explore the same state many times. IDA and RBFS suffer from using too little memory. Between iterations, IDA retains only a single number: the current f -cost limit. RBFS retains more information in memory, Section 4.1. Informed (Heuristic) Search Strategies 103 (a) (a) After After expandi expanding Arad, Sibiu, and and Rimnicu Rimnicu Vilcea Vilcea (b) After unwinding back to Sibiu and expanding Fagaras (c) (c) After After switching switching back back to to Rimnicu Rimnicu Vilcea Vilcea and and expanding expanding Pitesti Pitesti 9 Figure 4.6 Stages in an RBFS search for the shortest route to Bucharest. The f-limit value for each recursive call is shown on top of each current node. (a) The path via Rimnicu Vilcea is followed until the current best leaf (Pitesti) has a value that is worse than the best alternative path (Fagaras). (b) The recursion unwinds and the best leaf value of the forgotten subtree (417) is backed up to Rimnicu Vilcea; then Fagaras is expanded, revealing a best leaf value of 450. (c) The recursion unwinds and the best leaf value of the forgotten subtree (450) is backed up to Fagaras; then Rirnnicu Vilcea is expanded. This time, because the best alternative path (through Timisoara) costs at least 447, the expansion continues to Bucharest. J but it uses only linear space: even if more memory were available, RBFS has no way to make use of it. It seems sensible, therefore, to use all available memory. Two algorithms that do this MA' are MA (memory-bounded A) and SMA (simplified MA). We will describe SMA, which SMA' 1 04 Chapter 4. Informed Search and Exploration is-well-simpler. SMA proceeds just like A, expanding the best leaf until memory is full. At this point, it cannot add a new node to the search tree without dropping an old one. SMA always drops the worst leaf node-the one with the highest f-value. Like RBFS, SMA then backs up the value of the forgotten node to its parent. In this way, the ancestor of a forgotten subtree knows the quality of the best path in that subtree. With this information, SMA regenerates the subtree only when all otherpaths have been shown to look worse than the path it has forgotten. Another way of saying this is that, if all the descendants of a node n are forgotten, then we will not know which way to go from n, but we will still have an idea of how worthwhile it is to go anywhere from n. The complete algorithm is too complicated to reproduce here,5 but there is one subtlety worth mentioning. We said that SMA expands the best leaf and deletes the worst leaf. What if all the leaf nodes have the same f -value? Then the algorithm might select the same node for deletion and expansion. SMA solves this problem by expanding the newest best leaf and deleting the oldest worst leaf. These can be the same node only if there is only one leaf; in that case, the current search tree must be a single path from root to leaf that fills all of memory. If the leaf is not a goal node, then even if it is on an optimal solution path, that solution is not reachable with the available memory. Therefore, the node can be discarded exactly as if it had no successors. SMA is complete if there is any reachable solution-that is, if d, the depth of the shallowest goal node, is less than the memory size (expressed in nodes). It is optimal if any optimal solution is reachable; otherwise it returns the best reachable solution. In practical terms, SMA might well be the best general-purpose algorithm for finding optimal solutions, particularly when the state space is a graph, step costs are not uniform, and node generation is expensive compared to the additional overhead of maintaining the open and closed lists. On very hard problems, however, it will often be the case that SMA is forced to switch back and forth continually between a set of candidate solution paths, only a small subset of THRASHING which can fit in memory. (This resembles the problem of thrashing in disk paging systems.) Then the extra time required for repeated regeneration of the same nodes means that problems that would be practically solvable by A, given unlimited memory, become intractable for SMA. That is to say, memory limitations can make a problem intractable from the point of view of computation time. Although there is no theory to explain the tradeoff between time and memory, it seems that this is an inescapable problem. The only way out is to drop the optimality requirement. Learning to search better We have presented several fixed strategies-breadth-first, greedy best-first, and so on-that have been designed by computer scientists. Could an agent learn how to search better? The answer is yes, and the method rests on an important concept called the metalevel state space. SPACE Each state in a metalevel state space captures the internal (computational) state of a program STATE that is searching in an object-level state space such as Romania. For example, the internal SPACE state of the A algorithm consists of the current search tree. Each action in the metalevel state A rough sketch appeared in the first edition of this book. Section 4.2. Heuristic Functions 105 space is a computation step that alters the internal state; for example, each computation step in A expands a leaf node and adds its successors to the tree. Thus, Figure 4.3, which shows a sequence of larger and larger search trees, can be seten as depicting a path in the metalevel state space where each state on the path is an object-level search tree. Now, the path in Figure 4.3 has five steps, including one step, the expansion of Fagaras, that is not especially helpful. For harder problems, there will be many such missteps, and a metalevel learning algorithm can learn from these experiences to avoid exploring unpromis- ing subtrees. The techniques used for this kind of learning are described in Chapter 21. The goal of leaning is to minimize the total cost of problem solving, trading off computational expense and path cost. In this section, we will look at heuristics for the 8-puzzle, in order to shed light on the nature of heuristics in general. The 8-puzzle was one of the earliest heuristic search problems. As mentioned in Sec- tion 3.2, the object of the puzzle is to slide the tiles horizontally or vertically into the empty space until the configuration matches the goal configuration (Figure 4.7). Start State Goal State Figure 4.7 A typical instance of the 8-puzzle. The solution p- is 26 steps long. The average solution cost for a randomly generated %-puzzle instance is about 22 steps. The branching factor is about 3. (When the empty tile is in the middle, there are four possible moves; when it is in a corner there are two; and when it is along an edge there are three.) This means that an exhaustive search to depth 22 would look at about 322 = 3.1 x lo1' states. By keeping track of repeated states, we could cut this down by a factor of about 170,000, because there are only 9/2 = 181,440 distinct states that are reachable. (See Exercise 3.4.) This is a manageable number, but the corresponding number for the 15-puzzle is roughly 1013, so the next order of business is to find a good heuristic function. If we want to find the shortest solutions by using , we need a heuristic function that never overestimates the number of steps to the goal. There is a long history of such heuristics for the 15-puzzle; here are two commonly-used candidates: 106 Chapter 4. Informed Search and Exploration hl = the number of misplaced tiles. For Figure 4.7, all of the eight tiles are out of position, so the start state would have hl = 8. hl is an admissible heuristic, because it is clear that any tile that is out of place must be moved at least once. h2 = the sum of the distances of the tiles from their goal positions. Because tiles cannot move along diagonals, the distance we will count is the sum of the horizontal and vertical distances. This is sometimes called the city block distance or Manhattan MANHATTAN distance. h2 is also admissible, because all any move can do is move one tile one step DISTANCE closer to the goal. Tiles 1 to 8 in the start state give a Manhattan distance of h2=3+l+2+2+2+3+3+2=18. As we would hope, neither of these overestimates the true solution cost, which is 26. The effect of heuristic accuracy on performance EFFECTIVE FACTOR One way to characterize the quality of a heuristic is the effective branching factor b. If the total number of nodes generated by A for a particular problem is N, and the solution depth is d, then b is the branching factor that a uniform tree of depth d would have to have in order to contain N + 1 nodes. Thus, N + 1 = 1 + b + (b)2 + . . . + (b)d. For example, if A finds a solution at depth 5 using 52 nodes, then the effective branching factor is 1.92. The effective branching factor can vary across problem instances, but usually it is fairly constant for sufficiently hard problems. Therefore, experimental measurements of b on a small set of problems can provide a good guide to the heuristic's overall usefulness. A well-designed heuristic would have a value of b close to 1, allowing fairly large problems to be solved. To test the heuristic functions hl and ha, we generated 1200 random problems with solution lengths from 2 to 24 (100 for each even number) and solved them with iterative deepening search and with A tree search using both hl and h2. Figure 4.8 gives the average number of nodes generated by each strategy and the effective branching factor. The results suggest that h2 is better than hl, and is far better than using iterative deepening search. On our solutions with length 14, A with hz is 30,000 times more efficient than uninformed iterative deepening search. One might ask whether h2 is always better than hl. The answer is yes. It is easy to see from the definitions of the two heuristics that, for any node n, h2(n) 2 hl (n). We thus say DOMINATION that hZ dominates hl. Domination translates directly into efficiency: A using h2 will never expand more nodes than A using hl (except possibly for some nodes with f (n) = C). The argument is simple. Recall the observation on page 100 that every node with f (n) C will surely be expanded. This is the same as saying that every node with h(n) C - g(n) will surely be expanded. But because ha is at least as big as hl for all nodes, every node that is surely expanded by A search with ha will also surely be expanded with hl, and hl might also cause other nodes to be expanded as well. Hence, it is always better to use a heuristic function with higher values, provided it does not overestimate and that the computation time for the heuristic is not too large. Section 4.2. Heuristic Functions 107 Search Cost Effective Branching Factor IDS IDS d A(hl) A(ha) A(hl) A(h2) 1.79 10 6 6 2.45 1.79 2 1.45 112 13 12 2.87 1.48 4 1.30 680 20 18 ;.73 1.34 6 1.24 6384 39 25 2.80 1.33 8 1.22 47127 93 39 2+.79 1.38 10 12 227 73 2.78 1.42 1.24 3644035 - - 539 113 1.44 1.23 14 1.45 1.25 1.46 1.26 1 ;;:: 1 ;:: 1 - 7276 676 1.47 1.27 18094 1219 - 1.48 1.28 - 39135 1641 - 1.48 1.26 ,I Figure 4.8 Comparison of the search costs and effective branching factors for the ITERATI VE-DEEPENING-SEARCH and A algorithms ,with hl , h2. Data are averaged over 100 instances of the &puzzle, for various solution lengths. Inventing admissible heuristic functions We have seen that both hl (misplaced tiles) and 1 (Manhattan distance) are fairly good heuristics for the 8-puzzle and that hz is better. How might one have come up with h2? Is it possible for a computer to invent such a heuristic mechanically? hl and hz are estimates of the remaining path length for the 8-puzzle, but they are also perfectly accurate path lengths for simpl.ed versions of the puzzle. If the rules of the puzzle were changed so that a tile could move anywhere, instead of just to the adjacent empty square, then hl would give the exact number of steps in the shortest solution. Similarly, if a tile could move one square in any direction, even onto an occupied square, then h2 would give the exact number of steps in the shortest solution. A problem with fewer restrictions on RELAXED PROBLEM the actions is called a relaxed problem. The cost of an optimal solution to a relaxedproblem is an admissible heuristic for the original problem. The heuristic is admissible because the optimal solution in the original problem is, by defjnition, also a solution in the relaxed problem and therefore must be at least as expensive as the optimal solution in the relaxed problem. Because the derived heuristic is an exact cost for the relaxed problem, it must obey the triangle inequality and is therefore consistent (see page 99). If a problem definition is written down in a formal1 language, it is possible to construct relaxed problems atornaticall. For example, if the 8-puzzle actions are described as A tile can move from square A to square B if A is horizontally or vertically adjacent to B an(d B is blank, In Chapters 8 and 11, we will describe formal languages suitable for this task; with formal descriptions that can be manipulated, the construction of relaxed problems can be automated. For now, we will use English. 108 Chapter 4. Informed Search and Exploration we can generate three relaxed problems by removing one or both of the conditions: (a) A tile can move from square A to square B if A is adjacent to B. (b) A tile can move from square A to square B if B is blank. (c) A tile can move from square A to square B. From (a), we can derive hz (Manhattan distance). The reasoning is that h2 would be the proper score if we moved each tile in turn to its destination. The heuristic derived from (b) is discussed in Exercise 4.9. From (c), we can derive hl (misplaced tiles), because it would be the proper score if tiles could move to their intended destination in one step. Notice that it is crucial that the relaxed problems generated by this technique can be solved essentially without search, because the relaxed rules allow the problem to be decomposed into eight independent subproblems. If the relaxed problem is hard to solve, then the values of the corresponding heuristic will be expensive to btain. A program called ABSOLVER can generate heuristics automatically from problem def- initions, using the "relaxed problem" method and various other techniques (Prieditis, 1993). ABSOLVER generated a new heuristic for the 8-puzzle better than any preexisting heuristic and found the first useful heuristic for the famous Rubik's cube puzzle. One problem with generating new heuristic functions is that one often fails to get one "clearly best" heuristic. If a collection of admissible heuristics hl . . . h, is available for a problem, and none of them dominates any of the others, which should we choose? As it turns out, we need not make a choice. We can have the best of all worlds, by defining h(n) = maxhl (n), . . . , h,(n) ) . This composite heuristic uses whichever function is most accurate on the node in question. Because the component heuristics are admissible, h is admissible; it is also easy to prove that h is consistent. Furthermore, h dominates all of its component heuristics. SUBPROBLEM Admissible heuristics can also be derived from the solution cost of a subproblem of a given problem. For example, Figure 4.9 shows a subproblem of the 8-puzzle instance in Figure 4.7. The subproblem involves getting tiles 1, 2, 3, 4 into their correct positions. Clearly, the cost of the optimal solution of this subproblem is a lower bound on the cost of the complete problem. It turns out to be substantially more accurate than Manhattan distance in some cases. PAUERN DATABASES The idea behind pattern databases is to store these exact solution costs for every pos- sible subproblem instance-in our example, every possible configuration of the four tiles and the blank. (Notice that the locations of the other four tiles are irrelevant for the purposes of solving the subproblem, but moves of those tiles do count towards the cost.) Then, we com- pute an admissible heuristic hDB for each complete state encountered during a search simply by looking up the corresponding subproblem configuration in the database. The database itself is constructed by searching backwards froin the goal state and recording the cost of each new pattern encountered; the expense of this search is amortized over many subsequent problem instances. Note that a perfect heuristic can be obtained simply by allowing h to run a full breadth-first search "on the sly." Thus, there is a tradeoff between accuracy and computation time for heuristic functions. Section 4.2. Heuristic Functions 109 Start State Goal State Figure 4.9 A subproblem of the 8-puzzle instance given in Figure 4.7. The task is to get tiles 1, 2, 3, and 4 into their correct positions, without worrying about what happens to the other tiles. The choice of 1-2-3-4 is fairly arbitrary; we could also construct databases for 5-6-7-8, and for 2-4-64, and so on. Each database yields an admissible heuristic, and these heuristics can be combined, as explained earlier, by taking the maximum value. A combined heuristic of this lund is much more accurate than the Manhattan distance; the number of nodes generated when solving random 15-puzzles can be reduced by a factor of 1000. One might wonder whether the heuristics obtained from the 1-2-3-4 database and the 5-6-7-8 could be added, since the two subproblems seem not to overlap. Would this still give an admissible heuristic? The answer is no, because the solutions of the 1-2-3-4 subproblem and the 5-6-7-8 subproblem for a given state will almost certainly share some moves-it is unlikely that 1-2-3-4 can be moved into place without touching 5-6-7-8, and vice versa. But what if we don't count those moves? That is, we record niot the total cost of solving the 1-2-3-4 subproblem, but just the number of moves involving 1-2-3-4. Then it is easy to see that the sum of the two costs is still a lower bound on the cost of solving the entire problem. This is the idea behind disjoint pattern databases. Using such databases, it is possible to DATABASES solve random 15-puzzles in a few milliseconds-the numbes of nodes generated is reduced by a factor of 10,000 compared with using Manhattan distance. For 24-puzzles, a speedup of roughly a million can be obtained. Disjoint pattern databases work for sliding-tile puzzles because the problem can be divided up in such a way that each move affects only onie subproblem-because only one tile is moved at a time. For a problem such as Rubik's cube, this kind of subdivision cannot be done because each move affects 8 or 9 of the 26 cubies. Currently, it is not clear how to define disjoint databases for such problems. Learning heuristics from experience A heuristic function h(n) is supposed to estimate the cost of a solution beginning from the state at node n. How could an agent construct such a function? One solution was given in the preceding section-namely, to devise relaxed problems for which an optimal solution can be found easily. Another solution is to learn from experience. "E.xperience" here means solving lots of 8-puzzles, for instance. Each optimal solution to an 8-puzzle problem provides ex- 110 Chapter 4. Informed Search and Exploration amples from which h(n) can be learned. Each example consists of a state from the solution path and the actual cost of the solution from that point. From these examples, an inductive learning algorithm can be used to construct a function h(n) that can (with luck) predict solu- tion costs for other states that arise during search. Techniques for doing just this using neural nets, decision trees, and other methods are demonstrated in Chapter 18. (The reinforcement learning methods described in Chapter 21 are also applicable.) FEATURES Inductive learning methods work best when supplied with features of a state that are relevant to its evaluation, rather than with just the raw state description. For example, the feature "number of misplaced tiles" might be helpful in predicting the actual distance of a state from the goal. Let's call this feature XI (n). We could take 100 randomly generated 8-puzzle configurations and gather statistics on their actual solution costs. We might find that when XI (n) is 5, the average solution cost is around 14, and so on. Given these data, the value of XI can be used to predict h(n). Of course, we can use several features. A second feature xz(n) might be "number of pairs of adjacent tiles that are also adjacent in the goal state." How should XI (n) and xa(n) be combined to predict h(n)? A common approach is to use a linear combination: h(n) = clx(n) + c2x(n) . The constants cl and ca are adjusted to give the best fit to the actual data on solution costs. Presumably, cl should be positive and c2 should be negative. 4.3 LOCAL SEARCH ALGORITHMS AND OPTIMIZATION PROBLEMS The search algorithms that we have seen so far are designed to explore search spaces sys- tematically. This systematicity is achieved by keeping one or more paths in memory and by recording which alternatives have been explored at each point along the path and which have not. When a goal is found, the path to that goal also constitutes a solution to the problem. In many problems, however, the path to the goal is irrelevant. For example, in the 8- queens problem (see page 66), what matters is the final configuration of queens, not the order in which they are added. This class of problems includes many important applications such as integrated-circuit design, factory-floor layout, job-shop scheduling, automatic programming, telecommunications network optimization, vehicle routing, and portfolio management. If the path to the goal does not matter, we might consider a different class of algo- LOCALSEARCH rithms, ones that do not worry about paths at all. Local search algorithms operate using CURRENTSTATE a single current state (rather than multiple paths) and generally move only to neighbors of that state. Typically, the paths followed by the search are not retained. Although local search algorithms are not systematic, they have two key advantages: (1) they use very little memory-usually a constant amount; and (2) they can often find reasonable solutions in large or infinite (continuous) state spaces for which systematic algorithms are unsuitable. In addition to finding goals, local search algorithms are useful for solving pure op- OPTIMIZATION timization problems, in which the aim is to find the best state according to an objective PROBLEMS OBJECTIVE function. Many optimization problems do not fit the "standard" search mode1 introduced in FUNCTION Section 4.3. Local Search Algorithms and Optimization Problems 111 Chapter 3. For example, nature provides an objective function-reproductive fitness-that Darwinian evolution could be seen as attempting to optimize, but there is no "goal test" and no "path cost" for this problem. To understand local search, we will find it very useful to consider the state space land- STATE SPACE scape (as in Figure 4.10). A landscape has both "localtion" (defined by the state) and "eleva- LANDSCAPE tion" (defined by the value of the heuristic cost function or objective function). If elevation GLOBALMINIMUM corresponds to cost, then the aim is to find the lowest valley-a global minimum; if eleva- tion corresponds to an objective function, then the aim is to find the highest peak- a global GLOBALMAXIMUM maximum,. (You can convert from one to the other just by inserting a minus sign.) Local search algorithms explore this landscape. A complete, local search algorithm always finds a goal if one exists; an optimal algorithm always finds a, global minimum/maximum. maximum I I b state space current state Figure 4.10 A one-dimensional state space landscape in which elevation corresponds to the objective function. The aim is to find the global maximum. Hill-climbing search modifies the current state to try to improve it, as shown by the an-ow. The various topographic features are defined in the text. I Hill-climbing search HILL-CLIMBING The hill-climbing search algorithm is shown in Figure 4.1 1. It is simply a loop that continu- ally moves in the direction of increasing value-that is, uphill. It terminates when it reaches a "peak" where no neighbor has a higher value. The algorithm does not maintain a search tree, so the current node data structure need only record the state and its objective function value. Hill-climbing does not look ahead beyond the immediate neighbors of the current state. This resembles trying to find the top of Mount Everest in a thick fog while suffering from amnesia. To illustrate hill-climbing, we will use the 8-queens problem introduced on page 66. Local-search algorithms typically use a complete-state formulation, where each state has 8 queens on the board, one per column. The successor function returns all possible states generated by moving a single queen to another square in the same column (so each state has 112 Chapter 4. Informed Search and Exploration function HILL-CLIMBING() returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current +- MAKE-NODE(INITIAL-STATEO) loop do neighbor +- a highest-valued successor of curren.t if VAuEneighbor VALUEcurrent then return Scurrent current t neighbor Figure 4.11 The hill-climbing search algorithm (steepest ascent version), which is the most basic local search technique. At each step the current node is replaced by the best neighbor; in this version, that means the neighbor with the highest VALUE, but if a heuristic cost estimate h is used, we would find the neighbor with the lowest h. (a) (b) (a) An 8-queens state with heuristic cost estimate h = 17, showing the value Figure 4.12 of h for each possible successor obtained by moving a queen within its column. The best moves are marked. (b) A local minimum in the 8-queens state space; the state has h = 1 but every successor has a higher cost. 8 x 7 = 56 successors). The heuristic cost function h is the number of pairs of queens that are attacking each other, either directly or indirectly. The global minimum of this function is zero, which occurs only at perfect solutions. Figure 4.12(a) shows a state with h = 17. The figure also shows the values of all its successors, with the best successors having h = 12. Hill-climbing algorithms typically choose randomly among the set of best successors, if there is more than one. Section 4.3. Local Search Algorithms and Optimization Problems 113 GREEDY LOCAL Hill climbing is sometimes called greedy local s'earch because it grabs a good neighbor SEARCH state without thinking ahead about where to go next. Although greed is considered one of the seven deadly sins, it turns out that greedy algorithms often perform quite well. Hill climbing often makes very rapid progress towards a solution, be'cause it is usually quite easy to improve a bad state. For example, from the state in Figure 4.12(a), it takes just five steps to reach the state in Figure 4.12(b), which has h = 1 and is very nearly a solution. Unfortunately, hill climbing often gets stuck for the following reasons: 0 Local maxima: a local maximum is a peak that is higher than each of its neighboring states, but lower than the global maximum. Hill-climbing algorithms that reach the vicinity of a local maximum will be drawn upwards towards the peak, but will then be stuck with nowhere else to go. Figure 4.10 illusfrates the problem schematically. More concretely, the state in Figure 4.12(b) is in fact a local maximum (i.e., a local minimum for the cost h); every move of a single queen makes the situation worse. 0 Ridges: a ridge is shown in Figure 4.13. Ridges result in a sequence of local maxima that is very difficult for greedy algorithms to navigate. 0 Plateaux: a plateau is an area of the state space landscape where the evaluation function SHOULDER is flat. It can be a flat local maximum, from which no uphill exit exists, or a shoulder, from which it is possible to make progress. (See Figure 4.10.) A hill-climbing search might be unable to find its way off the plateau. In each case, the algorithm reaches a point at which no progress is being made. Starting from a randomly generated %queens state, steepest-ascent hill climbing gets stuck 86% of the time, solving only 14% of problem instances. It works quickly, taking just 4 steps on average when it succeeds and 3 when it gets stuck-not bad for a state space with 88 "N 17 million states. The algorithm in Figure 4.11 halts if it reaches a plateau where the best successor has the same value as the current state. Might it not be a good idea to keep going-to allow a SIDEWAYSMOVE sideways move in the hope that the plateau is really a shoulder, as shown in Figure 4. lo? The answer is usually yes, but we must take care. If we always allow sideways moves when there are no uphill moves, an infinite loop will occur whenlever the algorithm reaches a flat local maximum that is not a shoulder. One common solution is to put a limit on the number of con- secutive sidleways moves allowed. For example, we could allow up to, say, 100 consecutive sideways moves in the %queens problem. This raises the percentage of problem instances solved by hill-climbing from 14% to 94%. Success comes at a cost: the algorithm averages roughly 21 steps for each successful instance and 64 for each failure. STOCHASTIC HILL Many variants of hill-climbing have been invented. Stochastic hill climbing chooses at CLIMBING random from among the uphill moves; the probability of selection can vary with the steepness of the uphK1 move. This usually converges more slowly than steepest ascent, but in some C FIRSCHOIcEH'LL LIMBING state landscapes it finds better solutions. First-choice hill climbing implements stochastic hill climbing by generating successors randomly until one is generated that is better than the current state. This is a good strategy when a state has many (e.g., thousands) of successors. Exercise 4.16 asks you to investigate. The hill-climbing algorithms described so far are incomplete-they often fail to find a goal when one exists because they can get stuck on local maxima. Random-restart hill

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