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Heuristic search

Heuristic search. Reading: AIMA 2 nd ed., Ch. 4.1-4.3. (Blind) Tree search so far. function TreeSearch ( problem , fringe ) n = InitialState ( problem ); fringe = Insert ( n ); while (1) If Empty ( fringe ) return failure; n = RemoveFront ( fringe );

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Heuristic search

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  1. Heuristic search Reading: AIMA 2nd ed., Ch. 4.1-4.3 Rutgers CS440, Fall 2003

  2. (Blind) Tree search so far • functionTreeSearch(problem, fringe) • n = InitialState(problem); • fringe = Insert(n); • while (1) • If Empty(fringe) return failure; • n = RemoveFront(fringe); • If Goal(problem,n) returnn; • fringe = Insert( Expand(problem,n) ); • end • end • Strategies: • BFS: fringe = FIFO • DFS: fringe = LIFO • Strategies defined by the order of node expansion Rutgers CS440, Fall 2003

  3. Example of BFS and DFS in a maze G G s0 s0 Rutgers CS440, Fall 2003

  4. Informed searches: best-first search • Idea: Add domain-specific information to select the best path to continue searching along. • How to do it? • Use evaluation function, f(n), which selects the most “desirable” (best-first) node to expand. • Fringe is a queue sorted in decreasing order of desirability. • Ideal case: f(n) = true cost to goal state = t(n) • Of course, t(n) is expensive to compute (BFS!) • Use an estimate (heuristic) instead Rutgers CS440, Fall 2003

  5. maze example • f(n) = straight-line distance to goal 2 1 G 3 2 1 1 4 3 2 2 2 s0 3 3 3 4 4 4 Rutgers CS440, Fall 2003

  6. Romania example Rutgers CS440, Fall 2003

  7. Greedy search • Evaluation function f(n) = h(n) --- heuristic = estimate of cost from n to closest goal • Greedy search expands nodes that appear to be closest to goal Rutgers CS440, Fall 2003

  8. Sibiu h = 253 Timisoara h = 329 Zerind h = 374 Arad h = 366 Fagaras h = 176 Oradea h = 380 Rimnicu V. h = 193 Sibiu h = 253 Bucharest h = 0 Example of greedy search Arad h = 366 Arad h = 366 Sibiu h = 253 Fagaras h = 176 Bucharest h = 0 Is this the optimal solution? Rutgers CS440, Fall 2003

  9. Properties of greedy search • Completeness: • Not complete, can get stuck in loops, e.g., Iasi -> Neamt -> Iasi -> Neamt -> … • Can be made complete in finite spaces with repeated-state checking • Time complexity: • O(bm), but good heuristic can give excellent improvement • Space complexity: • O(bm), keeps all nodes in memory • Optimality: • No, can wander off to suboptimal goal states Rutgers CS440, Fall 2003

  10. A-search • Improve greedy search by discouraging wandering-off • f(n) = g(n) + h(n)g(n) = cost of path to reach node nh(n) = estimated cost from node n to goal nodef(n) = estimated total cost through node n • Search along most promising path, not node • Is A-search optimal? Rutgers CS440, Fall 2003

  11. Optimality of A-search • Theorem: Let h(n) be at most  higher than t(n), for all n. Than, the A-search will be at most  “steps” longer than the optimal search. • Proof: If n is a node on the path found by A-search, then f(n) = g(n) + h(n)  g(n) + t(n) +  = optimal +  Rutgers CS440, Fall 2003

  12. A* search • How to make A-search optimal? • Make  = 0. • This means heuristic must always underestimate the true cost to goal. h(n) has to be an Optimistic estimate. • 0  h(n)  t(n) is called an admissible heuristic. • Theorem: Tree A*-search is optimal if h(n) is admissible. Rutgers CS440, Fall 2003

  13. Sibiu 393 = 140+253 Timisoara 477 = 118+329 Zerind 449 = 75+374 Arad 646 = 280+366 Fagaras 415 = 239+176 Oradea 671 = 291+380 Rimnicu V. 413 = 220+193 Craiova 526 = 366+160 Pitesti 417 = 317+100 Sibiu 553 = 300+253 Sibiu 591 = 338+253 Bucharest 450 = 450+0 Bucharest f = 418+0 Craiova 615 = 455+160 Rimnicu V. 607 = 414+193 Example of A* search Arad Arad h = 366 Sibiu Fagaras Rimnicu V. Pitesti Bucharest f = 418+0 Rutgers CS440, Fall 2003

  14. s n G’ G A* optimality proof • Suppose there is a suboptimal goal G’ on the queue. Let n be an unexpanded node on a shortest path to optimal goal G. • f(G’) = g(G’) + h(G’) = g(G’) since h(G’)=0> g(G) since G’ is suboptimal  f(n) = g(n) + h(n) since h is admissible • Hence, f(G’) > f(n), and f(G’) will never be expanded • Remember example on the previous page? Rutgers CS440, Fall 2003

  15. Consistency of A* • Consistency condition:h(n) - c(n,n’)  h(n’) where n’ is any successor of n • Guarantees optimality of graph search (remember, the proof was for tree search!) • Also called monotonicity:f(n’) = g(n’) + h(n’) = g(n) + c(n,n’) + h(n’)  g(n) + h(n) = f(n)f(n’)  f(n) • f(n) is monotonically non-decreasing Rutgers CS440, Fall 2003

  16. Example of consistency/monotonicity • A* expands nodes in order of increasing f-value • Adds “f-contours” of nodes • BFS adds layers Rutgers CS440, Fall 2003

  17. Properties of A* • Completeness: • Yes (for finite graphs) • Time complexity: • Exponential in | h(n) - t(n) | x length of solution • Memory requirements: • Keeps all nodes in memory • Optimal: • Yes Rutgers CS440, Fall 2003

  18. Admissible heuristics h(n) • Straight-line distance for the maze and Romanian examples • 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = number of squares from desired location of each tile (Manhattan distance) • h1(S) = 7 • h2(S) = 4+0+3+3+1+0+2+1 = 14 Rutgers CS440, Fall 2003

  19. Dominance • If h1(n) and h2(n) are both admissible and h1(n)  h2(n), then h2(n) dominates over h1(n) • Which one is better for search? • h2(n), because it is “closer” to t(n) • Typical search costsd = 14, IDS = 3,473,941 nodes A*(h1) = 539 A*(h2) = 113d = 24, IDS ~ 54 billion nodes A*(h1) = 39135 A*(h2) = 1641 Rutgers CS440, Fall 2003

  20. Relaxed problems • How to derive admissible heuristics? • Can be derived from exact solutions to problems that are “simpler” (relaxed) versions of the problem one is trying to solve • Examples: • 8-puzzle: • Tile can move anywhere from initial position (h1) • Tiles can occupy same square but have to move one square at a time (h2) • maze: • Can move any distance and over any obstacles • Important:The cost of optimal solution to relaxed problems is no greater than the optimal solution to the real problem. Rutgers CS440, Fall 2003

  21. Local search • In many problems one does not care about the path, rather one wants to find the goal state (based on a goal condition). • Use local search / iterative improvement algorithms: • Keep a single “current” state, try to improve it. Rutgers CS440, Fall 2003

  22. Example • N-queens problem:from initial configuration move to other configurations such that the number of conflicts is reduced Rutgers CS440, Fall 2003

  23. Hill-climbingGradient ascent / descent • Goal:n* = arg max Value( n ) • functionHillClimbing(problem) • n = InitialState(problem); • while (1) • neighbors = Expand(problem,n); • n* = arg max Value(neighbors); • If Value(n*) < Value(n), return n; • n = n*; • end • end Rutgers CS440, Fall 2003

  24. Hill climbing (cont’d) • Problem:Depending on initial state, can get stuck in local maxima (minima), ridges, plateaus Rutgers CS440, Fall 2003

  25. Beam search • Problem of local minima (maxima) in hill-climbing can be alleviated by starting HC from multiple random starting points • Or make it stochastic (Stochastic HC) by choosing successors at random, based on how “good” they are • Local beam search: somewhat similar to random-restart HC: • Start from N initial states. • Expand all N states and keep N best successors. • Stochastic beam search: stochastic version of LBS, similar to SHC. Rutgers CS440, Fall 2003

  26. Simulated annealing • Idea: • Allow bad moves, initially more, later fewer • Analogy with annealing in metallurgy • functionSimulatedAnnealing(problem,schedule) • n = InitialState(problem); • t = 1; • while (1) • T = schedule(t); • neighbors = Expand(problem,n); • n’ = Random(neighbors); • V = Value(n’) - Value(n); • IfV > 0, n* = n; • Elsen* = n’ with probability exp(V/T); • t = t + 1; • end • end Rutgers CS440, Fall 2003

  27. Properties of simulated annealing • At fixed temperature T, state occupation probability reaches Boltzman distribution, exp( V(n)/T ) • Devised by Metropolis et al. in 1953 Rutgers CS440, Fall 2003

  28. Genetic algorithms Rutgers CS440, Fall 2003

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