Dr. Mustafa Jarrar. Chapter 4 Informed Searching. Sina Institute, University of Birzeit

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1 Lecture Notes, Advanced Artificial Intelligence (SCOM7341) Sina Institute, University of Birzeit 2 nd Semester, 2012 Advanced Artificial Intelligence (SCOM7341) Chapter 4 Informed Searching Dr. Mustafa Jarrar Sina Institute, University of Birzeit mjarrar@birzeit.edu Jarrar

2 Discussion and Motivation How to determine the minimum number of coins to give while making change? The coin of the highest value first? Jarrar

3 Discussion and Motivation Travel Salesperson Problem Given a list of cities and their pair wise distances, the task is to find a shortest possible tour that visits each city exactly once. Any idea how to improve this type of search? What type of information we may use to improve our search? Do you think this idea is useful: (At each stage visit the unvisited city nearest to the current city)? Jarrar

4 Best-first search Idea: use an evaluation function f(n) for each node family of search methods with various evaluation functions (estimate of "desirability ) usually gives an estimate of the distance to the goal often referred to as heuristics in this context Expand most desirable unexpanded node. Implementation: Order the nodes in fringe in decreasing order of desirability. Special cases: greedy best-first search A * search Jarrar

5 Romania with step costs in km Suppose we can have this info (SLD) How can we use it to improve our search? Jarrar

6 Greedy best-first search Greedy best-first search expands the node that appears to be closest to goal. Estimate of cost from n to goal,e.g., h SLD (n) = straight-line distance from n to Bucharest. Utilizes a heuristic function as evaluation function f(n) = h(n) = estimated cost from the current node to a goal. Heuristic functions are problem-specific. Often straight-line distance for route-finding and similar problems. Often better than depth-first, although worst-time complexities are equal or worse (space). Jarrar

7 Greedy best-first search example Jarrar

8 Greedy best-first search example Jarrar

9 Greedy best-first search example Jarrar

10 Greedy best-first search example Jarrar

11 Properties of greedy best-first search Complete: No can get stuck in loops (e.g., Iasi Neamt Iasi Neamt.) Time: O(b m ), but a good heuristic can give significant improvement Space: O(b m ) -- keeps all nodes in memory Optimal: No b m branching factor maximum depth of the search tree Jarrar

12 Discussion Do you think h SLD (n) is admissible? Would you use h SLD (n) in Palestine? How? Why? Did you find the Greedy idea useful? Ideas to improve it? Jarrar

13 A * search Idea: avoid expanding paths that are already expensive. Evaluation function = path cost + estimated cost to the goal f(n) = g(n) + h(n) -g(n) = cost so far to reach n -h(n) = estimated cost from n to goal -f(n) = estimated total cost of path through n to goal Combines greedy and uniform-cost search to find the (estimated) cheapest path through the current node Heuristics must be admissible Never overestimate the cost to reach the goal Very good search method, but with complexity problems Jarrar

14 A * search example Jarrar

15 A * search example Jarrar

16 A * search example Jarrar

17 A * search example Jarrar

18 A * search example Jarrar

19 A * search example Jarrar

20 A* Exercise How will A* get from Iasi to Fagaras? Jarrar

21 A* Exercise Node Coordinates SL Distance A (5,9) 8.0 B (3,8) 7.3 C (8,8) 7.6 D (5,7) 6.0 E (7,6) 5.4 F (4,5) 4.1 G (6,5) 4.1 H (3,3) 2.8 I (5,3) 2.0 J (7,2) 2.2 K (5,1) 0.0 Jarrar

22 Solution to A* Exercise Jarrar

23 Greedy Best-First Exercise Node Coordinates Distance A (5,9) 8.0 B (3,8) 7.3 C (8,8) 7.6 D (5,7) 6.0 E (7,6) 5.4 F (4,5) 4.1 G (6,5) 4.1 H (3,3) 2.8 I (5,3) 2.0 J (7,2) 2.2 K (5,1) 0.0 Jarrar

24 Solution to Greedy Best-First Exercise Jarrar

25 Another Exercise Do 1) A* Search and 2) Greedy Best-Fit Search Node C g(n) h(n) A (5,10) B (3,8) C (7,8) D (2,6) E (5,6) F (6,7) G (8,6) H (1,4) I (3,4) J (7,3) K (8,4) L (5,2) Jarrar

26 Admissible Heuristics A heuristic h(n) is admissible if for every node n, h(n) h * (n), where h * (n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic Example: h SLD (n) (never overestimates the actual road distance) Theorem-1: If h(n) is admissible, A * using TREE-SEARCH is optimal. (Ideas to prove this theorem?) Jarrar

27 Optimality of A * (proof) Suppose some suboptimal goal G 2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. We want to prove: f(n) < f(g2) (then A* will prefer n over G f(g 2 ) = g(g 2 ) since h(g 2 ) = 0 g(g 2 ) > g(g) since G 2 is suboptimal f(g) = g(g) since h(g) = 0 f(g 2 ) > f(g) from above Jarrar

28 Optimality of A * (proof) Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. f(g 2 ) > f(g) from above h(n) h^*(n) since h is admissible g(n) + h(n) g(n) + h * (n) f(n) f(g) Hence f(g 2 ) > f(n), and n is expanded contradiction! thus, A * will never select G 2 for expansion Jarrar

29 Optimality of A * (proof) Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. In other words: f(g 2 ) = g(g 2 ) + h(g 2 ) = g(g 2 ) > C*, since G 2 is a goal on a non-optimal path (C* is the optimal cost) f(n) = g(n) + h(n) <= C*, since h is admissible f(n) <= C* < f(g 2 ), so G 2 will never be expanded A* will not expand goals on sub-optimal paths Jarrar

30 Consistent Heuristics A heuristic is consistent if for every node n, every successor n' of n generated by any action a, h(n) c(n,a,n') + h(n') If h is consistent, we have f(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') g(n) + h(n) = f(n) i.e., f(n) is non-decreasing along any path. Theorem-2: If h(n) is consistent, A* using GRAPH-SEARCH is optimal. consistency is also called monotonicity Jarrar

31 Optimality of A * A * expands nodes in order of increasing f value Gradually adds "f-contours" of nodes Contour i has all nodes with f=f i, where f i < f i+1 Jarrar

32 Complexity of A* The number of nodes within the goal contour search space is still exponential with respect to the length of the solution better than other algorithms, but still problematic Frequently, space complexity is more important than time complexity A* keeps all generated nodes in memory Jarrar

33 Properties of A* Complete: Yes (unless there are infinitely many nodes with f f(g) ) Time: Exponential Because all nodes such that f(n) <= C* are expanded! Space: Keeps all nodes in memory, Fringe is exponentially large Optimal: Yes who can propose an idea to improve the time/space complexity Jarrar

34 Memory Bounded Heuristic Search How can we solve the memory problem for A* search? Idea: Try something like iterative deeping search, but the cutoff is f-cost (g+h) at each iteration, rather than depth first. Two types of memory bounded heuristic searches: Recursive BFS MA* Jarrar

35 Recursive Best First Search (RBFS) best alternative over fringe nodes, which are not children: do I want to back up? RBFS changes its mind very often in practice. This is because the f=g+h become more accurate (less optimistic) as we approach the goal. Hence, higher level nodes have smaller f-values and will be explored first. Problem? If we have more memory we cannot make use of it. Jarrar

36 Simple Memory Bounded A* (MA*) This is like A*, but when memory is full we delete the worst node (largest f-value). Like RBFS, we remember the best descendent in the branch we delete. If there is a tie (equal f-values) we first delete the oldest nodes first. Simple-MBA* finds the optimal reachable solution given the memory constraint. But time can still be exponential. Jarrar

37 SMA* pseudocode function SMA*(problem) returns a solution sequence inputs: problem, a problem static: Queue, a queue of nodes ordered by f-cost Queue MAKE-QUEUE({MAKE-NODE(INITIAL-STATE[problem])}) loop do if Queue is empty then return failure n deepest least-f-cost node in Queue if GOAL-TEST(n) then return success s NEXT-SUCCESSOR(n) if s is not a goal and is at maximum depth then f(s) else f(s) MAX(f(n),g(s)+h(s)) if all of n s successors have been generated then update n s f-cost and those of its ancestors if necessary if SUCCESSORS(n) all in memory then remove n from Queue if memory is full then delete shallowest, highest-f-cost node in Queue remove it from its parent s successor list insert its parent on Queue if necessary insert s in Queue end Jarrar

38 Simple Memory-bounded A* (MA*) (Example with 3-node memory) Search space Progress of MA*. Each node is labeled with its current f-cost. Values in parentheses show the value of the best forgotten descendant. f = g+h B 10+5= = goal A 0+12= G 8+5= A 15 B 12 A A 13 B G A 13[15] G 13 H C D 20+5= = E F 30+5= =30 H I 16+2= = J K 24+0= =29 A 15[15] G 24[ ] 15 B A G A 15[24] B 15 A 20[24] 8 B 20[ ] 24 I C 25 D 20 Can tell you when best solution found within memory constraint is optimal or not. Jarrar

39 Admissible Heuristics How can you invent a good admissible heuristic function? E.g., for the 8-puzzle Jarrar

40 Admissible heuristics E.g., for the 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) h 1 (S) =? h 2 (S) =? Jarrar

41 Admissible Heuristics E.g., for the 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) h 1 (S) =? 8 h 2 (S) =? = 18 Jarrar

42 Dominance If h 2 (n) h 1 (n) for all n (both admissible) then h 2 dominates h 1 h 2 is better for search: it is guaranteed to expand less nodes. Typical search costs (average number of nodes expanded): d=12 IDS = 3,644,035 nodes A * (h 1 ) = 227 nodes A * (h 2 ) = 73 nodes d=24 IDS = too many nodes A * (h 1 ) = 39,135 nodes A * (h 2 ) = 1,641 nodes What to do If we have h 1 h m, but none dominates the other? h(n) = max{h 1 (n),...h m (n)} Jarrar

43 Relaxed Problems A problem with fewer restrictions on the actions is called a relaxed problem The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h 1 (n) gives the shortest solution If the rules are relaxed so that a tile can move to any near square, then h 2 (n) gives the shortest solution Jarrar

44 Admissible Heuristics How can you invent a good admissible heuristic function? Try to relax the problem, from which an optimal solution can be found easily. Learn from experience. Can machines invite an admissible heuristic automaticlly? Jarrar

45 Local Search Algorithms In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution. State space = set of "complete" configurations. Find configuration satisfying constraints, e.g., n-queens. In such cases, we can use local search algorithms. keep a single "current" state, try to improve it according to an objective function Advantages: 1. Uses little memory 2. Finds reasonable solutions in large infinite spaces Jarrar

46 Local Search Algorithms Local search can be used on problems that can be formulated as finding a solution maximizing a criterion among a number of candidate solutions. Local search algorithms move from solution to solution in the space of candidate solutions (the search space) until a solution deemed optimal is found or a time bound is elapsed. For example, the travelling salesman problem, in which a solution is a cycle containing all nodes of the graph and the target is to minimize the total length of the cycle. i.e. a solution can be a cycle and the criterion to maximize is a combination of the number of nodes and the length of the cycle. A local search algorithm starts from a candidate solution and then iteratively moves to a neighbor solution. Jarrar

47 Local Search Algorithms If every candidate solution has more than one neighbor solution; the choice of which one to move to is taken using only information about the solutions in the neighborhood of the current one. hence the name local search. Terminate on a time bound or if the situation is not improved after number of steps. Local search algorithms are typically incomplete algorithms, as the search may stop even if the best solution found by the algorithm is not optimal. Jarrar

48 Example: n-queens Put n queens on an n n board with no two queens on the same row, column, or diagonal. Move a queen to reduce number of conflicts. Jarrar

49 Hill-Climbing Search Technique which belongs to the family of local search. starts with a random (potentially poor) solution, and iteratively makes small changes to the solution, each time improving it a little. When the algorithm cannot see any improvement anymore, it terminates. Problem: depending on initial state, can get stuck in local maxima. Hill climbing can be used to solve problems that have many solutions, some of which are better than others. (e.g. Bisimilarity ) Jarrar

50 Hill-climbing search: 8-queens problem Each number indicates h if we move a queen in its corresponding column h = number of pairs of queens that are attacking each other, either directly or indirectly (h = 17 for the above state) Jarrar

51 Hill-climbing search: 8-queens problem A local minimum with h = 1 Jarrar

52 Simulated Annealing Search Find an acceptably good solution in a fixed amount of time, rather than the best possible solution. Locating a good approximation to the global minimum of a given function in a large search space. At each step, the SA heuristic considers some neighbour s' of the current state s, and probabilistically decides between moving the system to state s' or staying in state s. Widely used in VLSI layout, airline scheduling, etc. Jarrar

53 Properties of Simulated Annealing Search One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1 (however, this may take VERY long) Widely used in VLSI layout, airline scheduling, etc. Jarrar

54 Genetic Algorithms Inspired by evolutionary biology such as inheritance. Evolves toward better solutions. A successor state is generated by combining two parent states. Start with k randomly generated states (population). Jarrar

55 Genetic Algorithms A state is represented as a string over a finite alphabet (often a string of 0s and 1s) Evaluation function (fitness function). Higher values for better states. Produce the next generation of states by selection, crossover, and mutation. Commonly, the algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population. Jarrar

56 Genetic Algorithms fitness: #non-attacking queens probability of being regenerated in next generation Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 7/2 = 28) 24/( ) = 31% 23/( ) = 29% etc Jarrar

57 Genetic Algorithms [2,6,9,3,5 0,4,1,7,8] [3,6,9,7,3 8,0,4,7,1] [2,6,9,3,5,8,0,4,7,1] [3,6,9,7,3,0,4,1,7,8] Jarrar

58 Homework-1 Draw the map of about 20 towns (including Jerusalem), Illustrate the greedy, A*, and RBSF algorithms to go between a town and Birzeit University. Estimate the straight line distance between these towns (use Google earth). Prove (theoretically, and by example) that your estimation is consistent and admissible; and that the obtained path is optimal. Overestimate the distances, and proof (theoretically, and by example) that the obtained path to Birzeit is not optimal. Upload this homework to Ritaj, and don t send it to me directly. Your solution should be clear, and with animation (use ppt). Each student should select different towns (whole Palestine!). Don t send me (Only to Ritaj). File name: AAI10.Search.RamiHodrob.v52.ppt Spilling of Town should correct (as found on the map or Wikipedia) Deadline (9/2/2011) Jarrar