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- 1. Jarrar © 2014 1 Chapter 4 Informed Searching Dr. Mustafa Jarrar Sina Institute, University of Birzeit mjarrar@birzeit.edu www.jarrar.info Lecture Notes on Informed Searching University of Birzeit, Palestine 1st Semester, 2014 Artificial Intelligence
- 2. Jarrar © 2014 2 Watch this lecture and download the slides from http://jarrar-courses.blogspot.com/2011/11/artificial-intelligence-fall-2011.html Most information based on Chapter 4 of [1]
- 3. Jarrar © 2014 3 Discussion and Motivation How to determine the minimum number of coins to give while making change? The coin of the highest value first ?
- 4. Jarrar © 2014 4 Discussion and Motivation Travel Salesperson Problem • 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)? 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.
- 5. Jarrar © 2014 5 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. A heuristic function ranks alternatives at each branching step based on the available information (heuristically) in order to make a decision about which branch to follow during a search. • Implementation: Order the nodes in fringe in decreasing order of desirability. • Special cases: – greedy best-first search – A* search
- 6. Jarrar © 2014 6 Romania with step costs in km Suppose we can have this info (SLD) How can we use it to improve our search?
- 7. Jarrar © 2014 7 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., hSLD(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).
- 8. Jarrar © 2014 8 Greedy best-first search example Example from [1]
- 9. Jarrar © 2014 9 Greedy best-first search example
- 10. Jarrar © 2014 10 Greedy best-first search example
- 11. Jarrar © 2014 11 Greedy best-first search example
- 12. Jarrar © 2014 12 Properties of greedy best-first search • Complete: No – can get stuck in loops (e.g., Iasi Neamt Iasi Neamt ….) • Time: O(bm), but a good heuristic can give significant improvement • Space: O(bm) -- keeps all nodes in memory • Optimal: No b branching factor m maximum depth of the search tree
- 13. Jarrar © 2014 13 Discussion • Do you think hSLD(n) is admissible? • Would you use hSLD(n) in Palestine? How? Why? • Did you find the Greedy idea useful? Ideas to improve it?
- 14. Jarrar © 2014 14 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
- 15. Jarrar © 2014 15 A* search example Example from [1]
- 16. Jarrar © 2014 16 A* search example
- 17. Jarrar © 2014 17 A* search example
- 18. Jarrar © 2014 18 A* search example
- 19. Jarrar © 2014 19 A* search example
- 20. Jarrar © 2014 20 A* search example
- 21. Jarrar © 2014 21 A* Exercise How will A* get from Iasi to Fagaras?
- 22. Jarrar © 2014 22 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
- 23. Jarrar © 2014 23 Solution to 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
- 24. Jarrar © 2014 24 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
- 25. Jarrar © 2014 25 Solution to 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
- 26. Jarrar © 2014 26 Another Exercise Node C g(n) h(n) A (5,10) 0.0 8.0 B (3,8) 2.8 6.3 C (7,8) 2.8 6.3 D (2,6) 5.0 5.0 E (5,6) 5.6 4.0 F (6,7) 4.2 5.1 G (8,6) 5.0 5.0 H (1,4) 7.2 4.5 I (3,4) 7.2 2.8 J (7,3) 8.1 2.2 K (8,4) 7.0 3.6 L (5,2) 9.6 0.0 Do 1) A* Search and 2) Greedy Best-Fit Search
- 27. Jarrar © 2014 27 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: hSLD(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?) Based on [4]
- 28. Jarrar © 2014 28 Optimality of A* (proof) • Recall that f(n) = g(n) + h(n • Now, 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(G2) = g(G2) since h(G2) = 0 g(G2) > g(G) since G2 is suboptimal f(G) = g(G) since h(G) = 0 f(G2) > 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(G2) > f(n), and n is expanded contradiction! thus, A* will never select G2 for expansion We want to prove: f(n) < f(G2) (then A* will prefer n over G2) Based on [4]
- 29. Jarrar © 2014 29 Optimality of A* (proof) • Recall that f(n) = g(n) + h(n • Now, 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. We want to prove: f(n) < f(G2) (then A* will prefer n over G2) • In other words: f(G2) = g(G2) + h(G2) = g(G2) > C*, since G2 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(G2), so G2 will never be expanded A* will not expand goals on sub-optimal paths
- 30. Jarrar © 2014 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 Based on [5]
- 31. Jarrar © 2014 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 = fi, where fi < fi+1 Based on [6]
- 32. Jarrar © 2014 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 Based on [2]
- 33. Jarrar © 2014 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
- 34. Jarrar © 2014 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 SMA*
- 35. Jarrar © 2014 35 Recursive Best First Search (RBFS) 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. Ay idea to improve this? best alternative over fringe nodes, which are not children: do I want to back up?
- 36. Jarrar © 2014 36 Simple Memory Bounded A* (SMA*) • 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. • SMA* finds the optimal reachable solution given the memory constraint. • But time can still be exponential.
- 37. Jarrar © 2014 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 Based on [2]
- 38. Jarrar © 2014 38 Simple Memory-bounded A* (SMA*) SMA* is a shortest path algorithm based on the A* algorithm. The advantage of SMA* is that it uses a bounded memory, while the A* algorithm might need exponential memory. All other characteristics of SMA* are inherited from A*. How it works: • Like A*, it expands the best leaf until memory is full. • 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.
- 39. Jarrar © 2014 39 Simple Memory-bounded A* (SMA*) 24+0=24 A B G C D E F H J I K 0+12=12 10+5=15 20+5=25 30+5=35 20+0=20 30+0=30 8+5=13 16+2=18 24+0=24 24+5=29 10 8 10 10 10 10 8 16 8 8 A 12 A B 12 15 A B G 13 15 13 H 13 A G 18 13[15] A G 24[] I 15[15] 24 A B G 15 15 24 A B C 15[24] 15 25 f = g+h (Example with 3-node memory) A B D 8 20 20[24] 20[] Progress of SMA*. Each node is labeled with its current f-cost. Values in parentheses show the value of the best forgotten descendant. Is given to nodes that the path up to it uses all available memory. Can tell you when best solution found within memory constraint is optimal or not. = goal Search space
- 40. Jarrar © 2014 40 The Algorithm proceeds as follows [3]
- 41. Jarrar © 2014 41 SMA* Properties [2] • It works with a heuristic, just as A* • It is complete if the allowed memory is high enough to store the shallowest solution. • It is optimal if the allowed memory is high enough to store the shallowest optimal solution, otherwise it will return the best solution that fits in the allowed memory • It avoids repeated states as long as the memory bound allows it • It will use all memory available • Enlarging the memory bound of the algorithm will only speed up the calculation • When enough memory is available to contain the entire search tree, then calculation has an optimal speed
- 42. Jarrar © 2014 42 Admissible Heuristics How can you invent a good admissible heuristic function? E.g., for the 8-puzzle
- 43. Jarrar © 2014 43 Admissible heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) • h1(S) = ? • h2(S) = ?
- 44. Jarrar © 2014 44 Admissible Heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) • h1(S) = ? 8 • h2(S) = ? 3+1+2+2+2+3+3+2 = 18
- 45. Jarrar © 2014 45 Dominance • If h2(n) ≥ h1(n) for all n, and both are admissible. • then h2 dominates h1 • h2 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*(h1) = 227 nodes A*(h2) = 73 nodes • d=24 IDS = too many nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes • What to do If we have h1…hm, but none dominates the other? h(n) = max{h1(n), . . .hm(n)}
- 46. Jarrar © 2014 46 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 h1(n) gives the shortest solution • If the rules are relaxed so that a tile can move to any near square, then h2(n) gives the shortest solution
- 47. Jarrar © 2014 47 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 automatically?
- 48. Jarrar © 2014 48 Local Search Algorithms
- 49. Jarrar © 2014 49 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
- 50. Jarrar © 2014 50 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.
- 51. Jarrar © 2014 51 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.
- 52. Jarrar © 2014 52 Hill-Climbing Search • continually moves in the direction of increasing value (i.e., uphill). • Terminates when it reaches a “peak”, no neighbor has a higher value. • Only records the state and its objective function value. • Does not look ahead beyond the immediate • Problem: can get stuck in local maxima, • Its success depends very much on the shape of the state-space land- scape: if there are few local maxima, random-restart hill climbing will find a good solution very quickly. Sometimes called Greedy Local Search
- 53. Jarrar © 2014 53 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.
- 54. Jarrar © 2014 54 Hill-climbing search: 8-queens problem • h = number of pairs of queens that are attacking each other, either directly or indirectly (h = 17 for the above state) Each number indicates h if we move a queen in its corresponding column
- 55. Jarrar © 2014 55 Hill-climbing search: 8-queens problem A local minimum with h = 1
- 56. Jarrar © 2014 56 Simulated Annealing Search • To avoid being stuck in a local maxima, It tries randomly (using a probability function) to move to another state, if this new state is better it moves into it, otherwise try another move… and so on. • In other word, 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 sit moves. • Terminates when finding 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. • Widely used in VLSI layout, airline scheduling, etc. Based on [1]
- 57. Jarrar © 2014 57 Properties of Simulated Annealing Search • The problem with this approach is that the neighbors of a state are not guaranteed to contain any of the existing better solutions which means that failure to find a better solution among them does not guarantee that no better solution exists. • It will not get stuck to a local optimum • If it runs for an infinite amount of time, the global optimum will be found.
- 58. Jarrar © 2014 58 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).
- 59. Jarrar © 2014 59 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.
- 60. Jarrar © 2014 60 • Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28) • 24/(24+23+20+11) = 31% • 23/(24+23+20+11) = 29% etc fitness: #non-attacking queens probability of being regenerated in next generation Genetic Algorithms
- 61. Jarrar © 2014 61 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]
- 62. Jarrar © 2014 62 References • [1] S. Russell and P. Norvig: Artificial Intelligence: A Modern Approach Prentice Hall, 2003, Second Edition • [2] http://en.wikipedia.org/wiki/SMA* • [3] Moonis Ali: Lecture Notes on Artificial Intelligence http://cs.txstate.edu/~ma04/files/CS5346/SMA%20search.pdf • [4] Max Welling: Lecture Notes on Artificial Intelligence https://www.ics.uci.edu/~welling/teaching/ICS175winter12/A-starSearch.pdf • [5] Kathleen McKeown: Lecture Notes on Artificial Intelligence http://www.cs.columbia.edu/~kathy/cs4701/documents/InformedSearch-AR-print.ppt • [6] Franz Kurfess: Lecture Notes on Artificial Intelligence http://users.csc.calpoly.edu/~fkurfess/Courses/Artificial-Intelligence/F09/Slides/3- Search.ppt

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