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Heuristic Search Russell and Norvig: Chapter 4 CSMSC 421 – Fall 2005 Modified Search Algorithm 1. INSERT(initial-node,FRINGE) 2. Repeat: If FRINGE is empty then return failure n REMOVE(FRINGE) s STATE(n) If GOAL?(s) then return path or goal state For every state s’ in SUCCESSORS(s) Create a node n’ INSERT(n’,FRINGE) Search Algorithms Blind search – BFS, DFS, uniform cost no notion concept of the “right direction” can only recognize goal once it’s achieved Heuristic search – we have rough idea of how good various states are, and use this knowledge to guide our search Best-first search Idea: use an evaluation function f(n) for each node Estimate of desirability Expand most desirable unexpanded node Implementation: fringe is queue sorted by descreasing order of desirability Greedy search A* search Heuristic Webster's Revised Unabridged Dictionary (1913) (web1913) Heuristic \Heu*ris"tic\, a. [Greek. to discover.] Serving to discover or find out. The Free On-line Dictionary of Computing (15Feb98) heuristic 1. <programming> A rule of thumb, simplification or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood. Unlike algorithms, heuristics do not guarantee feasible solutions and are often used with no theoretical guarantee. 2. <algorithm> approximation algorithm. From WordNet (r) 1.6 heuristic adj 1: (computer science) relating to or using a heuristic rule 2: of or relating to a general formulation that serves to guide investigation [ant: algorithmic] n : a commonsense rule (or set of rules) intended to increase the probability of solving some problem [syn: heuristic rule, heuristic program] Informed Search Add domain-specific information to select the best path along which to continue searching Define a heuristic function, h(n), that estimates the “goodness” of a node n. Specifically, h(n) = estimated cost (or distance) of minimal cost path from n to a goal state. The heuristic function is an estimate, based on domain-specific information that is computable from the current state description, of how close we are to a goal Greedy Search f(N) = h(N) greedy best-first Robot Navigation Robot Navigation f(N) = h(N), with h(N) = Manhattan distance to the goal 8 7 6 5 4 3 2 3 4 5 6 7 5 4 3 5 6 3 2 1 0 1 2 4 7 6 5 8 7 6 5 4 3 2 3 4 5 6 Robot Navigation f(N) = h(N), with h(N) = Manhattan distance to the goal 8 7 6 5 4 3 2 3 4 5 6 7 5 4 3 5 6 3 2 1 0 0 1 2 4 What happened??? 7 7 6 5 8 7 6 5 4 3 2 3 4 5 6 Greedy Search f(N) = h(N) greedy best-first Is it complete? If we eliminate endless loops, yes Is it optimal? More informed search We kept looking at nodes closer and closer to the goal, but were accumulating costs as we got further from the initial state Our goal is not to minimize the distance from the current head of our path to the goal, we want to minimize the overall length of the path to the goal! Let g(N) be the cost of the best path found so far between the initial node and N f(N) = g(N) + h(N) Robot Navigation f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal 8 7 6+3 5 4 3 2 3 6 8+3 7+4 6+5 5+6 4+7 3+8 2+9 3+10 4 5 6 7 7+2 5 4 3 5+6 4+7 3+8 5 6 6+1 2 1 0 3 2+9 1+10 0+11 1 2 4 7 6 7+0 6+1 5 8 7 6 5 4 3 2 3 8+1 7+2 6+3 5+4 4+5 3+6 2+7 3+8 4 5 6 Can we Prove Anything? If the state space is finite and we avoid repeated states, the search is complete, but in general is not optimal If the state space is finite and we do not avoid repeated states, the search is in general not complete If the state space is infinite, the search is in general not complete Admissible heuristic Let h*(N) be the true cost of the optimal path from N to a goal node Heuristic h(N) is admissible if: 0 h(N) h*(N) An admissible heuristic is always optimistic A* Search Evaluation function: f(N) = g(N) + h(N) where: g(N) is the cost of the best path found so far to N h(N) is an admissible heuristic Then, best-first search with this evaluation function is called A* search Important AI algorithm developed by Fikes and Nilsson in early 70s. Originally used in Shakey robot. Completeness & Optimality of A* Claim 1: If there is a path from the initial to a goal node, A* (with no removal of repeated states) terminates by finding the best path, hence is: complete optimal requirements: 0 < c(N,N’) Each node has a finite number of successors Completeness of A* Theorem: If there is a finite path from the initial state to a goal node, A* will find it. Proof of Completeness Let g be the cost of a best path to a goal node No path in search tree can get longer than g/, before the goal node is expanded Optimality of A* Theorem: If h(n) is admissable, then A* is optimal. Proof of Optimality Cost of best path f(G1) = g(G1) to a goal thru N N f(N) = g(N) + h(N) g(N) + h*(N) G1 G2 f(G1) g(N) + h(N) g(N) + h*(N) Heuristic Function Function h(N) that estimates the cost of the cheapest path from node N to goal node. Example: 8-puzzle 5 8 1 2 3 h(N) = number of misplaced tiles 4 2 1 4 5 6 =6 7 3 6 7 8 N goal Heuristic Function Function h(N) that estimate the cost of the cheapest path from node N to goal node. Example: 8-puzzle 5 8 1 2 3 h(N) = sum of the distances of 4 2 1 4 5 6 every tile to its goal position 7 3 6 7 8 =2+3+0+1+3+0+3+1 N goal = 13 8-Puzzle f(N) = h(N) = number of misplaced tiles 3 3 4 5 3 4 2 4 2 1 3 3 0 4 5 4 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 3+3 1+5 2+3 3+4 5+2 0+4 3+2 4+1 1+3 2+3 5+0 3+4 1+5 2+4 8-Puzzle f(N) = h(N) = distances of tiles to goal 6 5 2 5 2 1 4 3 0 4 6 5 8-Puzzle 5 8 1 2 3 4 2 1 4 5 6 7 3 6 7 8 N goal • h1(N) = number of misplaced tiles = 6 is admissible • h2(N) = sum of distances of each tile to goal = 13 is admissible • h3(N) = (sum of distances of each tile to goal) + 3 x (sum of score functions for each tile) = 49 is not admissible Robot navigation f(N) = g(N) + h(N), with h(N) = straight-line distance from N to goal Cost of one horizontal/vertical step = 1 Cost of one diagonal step = 2 Consistent Heuristic The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N: N h(N) c(N,N’) + h(N’) c(N,N’) N’ h(N) (triangular inequality) h(N’) 8-Puzzle 5 8 1 2 3 4 2 1 4 5 6 7 3 6 7 8 N goal • h1(N) = number of misplaced tiles • h2(N) = sum of distances of each tile to goal are both consistent Robot navigation Cost of one horizontal/vertical step = 1 Cost of one diagonal step = 2 h(N) = straight-line distance to the goal is consistent Claims If h is consistent, then the function f along any path is non-decreasing: N c(N,N’) f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) N’ h(N) h(N’) Claims If h is consistent, then the function f along any path is non-decreasing: N c(N,N’) f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) N’ h(N) h(N) c(N,N’) + h(N’) h(N’) f(N) f(N’) Claims If h is consistent, then the function f along any path is non-decreasing: N c(N,N’) f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) N’ h(N) h(N) c(N,N’) + h(N’) h(N’) f(N) f(N’) If h is consistent, then whenever A* expands a node it has already found an optimal path to the state associated with this node Avoiding Repeated States in A* If the heuristic h is consistent, then: Let CLOSED be the list of states associated with expanded nodes When a new node N is generated: If its state is in CLOSED, then discard N If it has the same state as another node in the fringe, then discard the node with the largest f Heuristic Accuracy h(N) = 0 for all nodes is admissible and consistent. Hence, breadth-first and uniform- cost are particular A* !!! Let h1 and h2 be two admissible and consistent heuristics such that for all nodes N: h1(N) h2(N). Then, every node expanded by A* using h2 is also expanded by A* using h1. h2 is more informed than h1 Iterative Deepening A* (IDA*) Use f(N) = g(N) + h(N) with admissible and consistent h Each iteration is depth-first with cutoff on the value of f of expanded nodes 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=4 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=4 4 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=4 4 5 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 5 4 Cutoff=4 4 5 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 6 5 4 Cutoff=4 4 5 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=5 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=5 4 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=5 4 5 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 Cutoff=5 4 5 7 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 5 Cutoff=5 4 5 7 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 5 5 Cutoff=5 4 5 7 6 6 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 5 5 Cutoff=5 4 5 7 6 6 About Heuristics Heuristics are intended to orient the search along promising paths The time spent computing heuristics must be recovered by a better search After all, a heuristic function could consist of solving the problem; then it would perfectly guide the search Deciding which node to expand is sometimes called meta-reasoning Heuristics may not always look like numbers and may involve large amount of knowledge What’s the Issue? Search is an iterative local procedure Good heuristics should provide some global look-ahead (at low computational cost) Another approach… for optimization problems rather than constructing an optimal solution from scratch, start with a suboptimal solution and iteratively improve it Local Search Algorithms Hill-climbing or Gradient descent Potential Fields Simulated Annealing Genetic Algorithms, others… Hill-climbing search If there exists a successor s for the current state n such that h(s) < h(n) h(s) <= h(t) for all the successors t of n, then move from n to s. Otherwise, halt at n. Looks one step ahead to determine if any successor is better than the current state; if there is, move to the best successor. Similar to Greedy search in that it uses h, but does not allow backtracking or jumping to an alternative path since it doesn’t “remember” where it has been. Not complete since the search will terminate at "local minima," "plateaus," and "ridges." Hill climbing on a surface of states Height Defined by Evaluation Function Hill climbing Steepest descent (~ greedy best-first with no search) may get stuck into local minimum Robot Navigation Local-minimum problem f(N) = h(N) = straight distance to the goal Examples of problems with HC applet Drawbacks of hill climbing Problems: Local Maxima: peaks that aren’t the highest point in the space Plateaus: the space has a broad flat region that gives the search algorithm no direction (random walk) Ridges: flat like a plateau, but with dropoffs to the sides; steps to the North, East, South and West may go down, but a step to the NW may go up. Remedy: Introduce randomness Random restart. Some problem spaces are great for hill climbing and others are terrible. What’s the Issue? Search is an iterative local procedure Good heuristics should provide some global look-ahead (at low computational cost) Hill climbing example 2 8 3 1 2 3 start 1 6 4 h = -4 goal 8 4 h=0 7 5 7 6 5 -5 -5 -2 2 8 3 1 2 3 1 4 h = -3 8 4 h = -1 7 6 5 7 6 5 -3 -4 2 3 2 3 1 8 4 1 8 4 h = -2 7 6 5 7 6 5 h = -3 -4 f(n) = -(number of tiles out of place) Example of a local maximum 1 2 5 -4 7 4 start 8 6 3 goal 1 2 5 1 2 5 1 2 5 7 4 7 4 -4 7 4 0 8 6 3 8 6 3 8 6 3 -3 1 2 5 7 4 -4 8 6 3 Potential Fields Idea: modify the heuristic function Goal is gravity well, drawing the robot toward it Obstacles have repelling fields, pushing the robot away from them This causes robot to “slide” around obstacles Potential field defined as sum of attractor field which get higher as you get closer to the goal and the indivual obstacle repelling field (often fixed radius that increases exponentially closer to the obstacle) Does it always work? No. But, it often works very well in practice Advantage #1: can search a very large search space without maintaining fringe of possiblities Scales well to high dimensions, where no other methods work Example: motion planning Advantage #2: local method. Can be done online Example: RoboSoccer All robots have same field: attracted to the ball Repulsive potential to other players Kicking field: attractive potential to the ball and local repulsive potential if clase to the ball, but not facing the direction of the opponent’s goal. Result is tangent, player goes around the ball. Single team: kicking field + repulsive field to avoid hitting other players + player position fields (paraboilic if outside your area of the field, 0 inside). Player nearest to the ball has the largest attractive coefficient, avoids all players crowding the ball. Simulated annealing Simulated annealing (SA) exploits an analogy between the way in which a metal cools and freezes into a minimum-energy crystalline structure (the annealing process) and the search for a minimum [or maximum] in a more general system. SA can avoid becoming trapped at local minima. SA uses a random search that accepts changes that increase objective function f, as well as some that decrease it. SA uses a control parameter T, which by analogy with the original application is known as the system “temperature.” T starts out high and gradually decreases toward 0. Simulated annealing (cont.) A “bad” move from A to B is accepted with a probability (f(B)-f(A)/T) e The higher the temperature, the more likely it is that a bad move can be made. As T tends to zero, this probability tends to zero, and SA becomes more like hill climbing If T is lowered slowly enough, SA is complete and admissible. The simulated annealing algorithm Simulated Annealing applet Successful application: circuit routing, traveling sales person (TSP) Summary: Local Search Algorithms Steepest descent (~ greedy best-first with no search) may get stuck into local minimum Better Heuristics: Potential Fields Simulated annealing Genetic algorithms When to Use Search Techniques? The search space is small, and There is no other available techniques, or It is not worth the effort to develop a more efficient technique The search space is large, and There is no other available techniques, and There exist “good” heuristics Summary Heuristic function Best-first search Admissible heuristic and A* A* is complete and optimal Consistent heuristic and repeated states Heuristic accuracy IDA*