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Notes

Notes. Dijstra’s Algorithm Corrected syllabus. Tree Search Implementation Strategies. Require data structure to model search tree Tree Node: State (e.g. Sibiu) Parent (e.g. Arad) Action (e.g. GoTo (Sibiu)) Path cost or depth (e.g. 140 )

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Notes

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  1. Notes • Dijstra’sAlgorithm • Corrected syllabus

  2. Tree Search Implementation Strategies • Require data structure to model search tree • Tree Node: • State (e.g. Sibiu) • Parent (e.g. Arad) • Action (e.g. GoTo(Sibiu)) • Path cost or depth (e.g. 140) • Children (e.g. Faragas, Oradea) (optional, helpful in debugging)

  3. Queue • Methods: • Empty(queue) • Returns true if there are no more elements • Pop(queue) • Remove and return the first element • Insert(queue, element) • Inserts element into the queue • InsertFIFO(queue, element) – inserts at the end • InsertLIFO(queue, element) – inserts at the front • InsertPriority(queue, element, value) – inserts sorted by value

  4. informed Search

  5. Search start goal Uninformed search Informed search

  6. Informed Search • What if we had an evaluation function h(n) that gave us an estimate of the cost associated with getting from n to the goal • h(n) is called a heuristic

  7. Romania with step costs in km h(n)

  8. Greedy best-first search • Evaluation function f(n) = h(n) (heuristic) • e.g., f(n) = hSLD(n) = straight-line distance from n to Bucharest • Greedy best-first search expands the node that is estimated to be closest to goal

  9. Romania with step costs in km

  10. Best-First Algorithm

  11. Performance of greedy best-first search • Complete? • Optimal?

  12. Failure case for best-first search

  13. Performance of greedy best-first search • Complete? • No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt • Optimal? • No

  14. Complexity of greedy best first search • Time? • O(bm), but a good heuristic can give dramatic improvement • Space? • O(bm) -- keeps all nodes in memory

  15. What can we do better?

  16. A* search • Ideas: • Avoid expanding paths that are already expensive • Consider • Cost to get here (known) – g(n) • Cost to get to goal (estimate from the heuristic) – h(n)

  17. A * Evaluation functions • Evaluation function 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 goal n start g(n) h(n) f(n)

  18. A* 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)

  19. What happens if heuristic is not admissible? • Will still find solution (complete) • But might not find best solution (not optimal)

  20. Properties of A* (w/ admissible heuristic) • Complete? • Yes (unless there are infinitely many nodes with f ≤ f(G) ) • Optimal? • Yes • Time? • Exponential, approximately O(bd)in the worst case • Space? • O(bm) Keeps all nodes in memory

  21. The heuristic h(x) guides the performance of A* • Let d(x) be the actual distance between S and G • h(x) = 0 : • A* is equivalent to Uniform-Cost Search • h(x) <= d (x) : • guarantee to compute the shortest path; the lower the value h(x), the more node A* expands • h(x) = d (x) : • follow the best path; never expand anything else; difficult to compute h(x) in this way! • h(x) > d(x) : • not guarantee to compute a best path; but very fast • h(x) >> g(x) : • h(n) dominates -> A* becomes the best first search

  22. Admissible heuristics

  23. Admissible heuristics E.g., for the 8-puzzle:

  24. Admissible heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = summed Manhattan distance for all tiles (i.e., no. of squares from desired location of each tile) • h1(S) = ? • h2(S) = ?

  25. 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 Which is better?

  26. Dominance • If h2(n) ≥ h1(n) for all n (both admissible) • then h2dominatesh1 •  h2is better for search • What does better mean? • All searches we’ve discussed are exponential in time

  27. Comparison of algorithms(number of nodes expanded)

  28. Visually

  29. Where do heuristics come from? • From people • Knowledge of the problem • From computers • By considering a simpler version of the problem • Called a relaxation

  30. Relaxed problems • 8-puzzle • 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 adjacent square, then h2(n) gives the shortest solution • Consider the example of straight line distance (Romania navigation) • Is that a relaxation?

  31. Iterative-Deepening A* (IDA*) • Further reduce memory requirements of A* • Regular Iterative-Deepening: regulated by depth • IDA*: regulated by f(n)=g(n)+h(n)

  32. Questions?

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