1 / 39

Informed Search

EA C461 Artificial Intelligence. To discuss?. Search StrategiesBest First Search Greedy Best First SearchA* SearchMemory Bounded Heuristic Search Heuristic FunctionsHeuristic AccuracyFinding appropriate heuristics. EA C461 Artificial Intelligence. Introduction. SearchHeuristic / InformedUse

joie
Download Presentation

Informed Search

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    1. EA C461 Artificial Intelligence Informed Search S.P.Vimal BITS-Pilani

    2. EA C461 Artificial Intelligence To discuss… Search Strategies Best First Search Greedy Best First Search A* Search Memory Bounded Heuristic Search Heuristic Functions Heuristic Accuracy Finding appropriate heuristics

    3. EA C461 Artificial Intelligence Introduction Search Heuristic / Informed Uses additional information about nodes (heuristics) that have not yet been explored to decide which nodes to examine next Blind / Uninformed Assumes no additional information about the problem Heuristic function h (n) used in search provides an estimate of the distance from any given node to a goal node

    4. EA C461 Artificial Intelligence Best First Search A node is selected for expansion based on an evaluation function f(n) Choose the node which appears to be best while expanding Best First Search algorithms differs in the evaluation function Evaluation function incorporate the problem specific knowledge in the form of h(n) h(n) ? heuristic function , a component of f(n) ? Estimated cost of cheapest path to the goal node h(n) = 0, if n is the goal node Greedy Best First Search, A* Search

    5. EA C461 Artificial Intelligence Greedy best-first search Expands the node that is closest to the goal Consider route finding problem in Romania Use of hSLD, Straight Line Distance Heuristic Evaluation function f(n) = h(n) (heuristic), estimate of cost from n to goal For the problem of finding route from Arad to Burcharest…

    6. EA C461 Artificial Intelligence

    7. EA C461 Artificial Intelligence Greedy best-first search

    8. EA C461 Artificial Intelligence Greedy best-first search

    9. EA C461 Artificial Intelligence Greedy best-first search

    10. EA C461 Artificial Intelligence Greedy best-first search

    11. EA C461 Artificial Intelligence

    12. EA C461 Artificial Intelligence Critics Minimizing f(n) is susceptible to false start Same issues with DFS Not optimal, Incomplete (may search an infinite path) Performance depends problem and heuristic O(bm) ? Worst case space needed

    13. EA C461 Artificial Intelligence A* Search A better form of best-first search f(n)=g(n) + h(n) g(n) ?the cost to reach the node h(n) ?the estimated cost of the cheapest path from n to the goal f(n) ? the estimated cost of the cheapest solution through the node n h(n) ? Admissible Heuristic / Consistent (in case of graph search)

    14. EA C461 Artificial Intelligence A* search

    15. EA C461 Artificial Intelligence A* search

    16. EA C461 Artificial Intelligence A* search

    17. EA C461 Artificial Intelligence A* search

    18. EA C461 Artificial Intelligence A* search

    19. EA C461 Artificial Intelligence A* search

    20. EA C461 Artificial Intelligence A* Search : 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: If h(n) is admissible, A* using TREE-SEARCH is optimal

    21. EA C461 Artificial Intelligence A* Search : 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(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

    22. EA C461 Artificial Intelligence A* Search: Repeated states Sub optimal solutions can be returned, if we discard the optimal path to a repeated node Can be handled by keeping all nodes expanded in memory, and when repeated make a decision based on their costs Make Sure, repeated nodes are costlier Requires the heuristic to be consistent

    23. EA C461 Artificial Intelligence A* Search : 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: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

    24. EA C461 Artificial Intelligence A* Characteristics A* expands no nodes with f(n)>C* These nodes are said to be pruned This pruning still guarantees optimality Expands nodes in the increasing order of costs A* is optimally efficient For a given heuristic, A* finds optimal solution with the fewest number of nodes expansion Any algorithm that doesn’t expand nodes with f(n)<C* has the risk of missing an optimal solution For most problems the number of nodes with costs lesser than C* is exponential

    25. EA C461 Artificial Intelligence

    26. EA C461 Artificial Intelligence Recursive best-first search Keeps track of the f-value of the best-alternative path available. If current f-values exceeds this alternative f-value than backtrack to alternative path. Upon backtracking change f-value to best f-value of its children. Re-expansion of this result is thus still possible.

    27. EA C461 Artificial Intelligence Recursive best-first search, ex.

    28. EA C461 Artificial Intelligence Recursive best-first search, ex.

    29. EA C461 Artificial Intelligence Recursive best-first search, ex.

    30. EA C461 Artificial Intelligence RBFS evaluation RBFS is a bit more efficient than IDA* Still excessive node generation (mind changes) Like A*, optimal if h(n) is admissible Space complexity is O(bd). IDA* retains only one single number (the current f-cost limit) Time complexity difficult to characterize Depends on accuracy if h(n) and how often best path changes. IDA* en RBFS suffer from too little memory.

    31. EA C461 Artificial Intelligence (simplified) memory-bounded A* Use all available memory. I.e. expand best leafs until available memory is full When full, SMA* drops worst leaf node (highest f-value) Like RFBS backup forgotten node to its parent What if all leafs have the same f-value? Same node could be selected for expansion and deletion. SMA* solves this by expanding newest best leaf and deleting oldest worst leaf. SMA* is complete if solution is reachable, optimal if optimal solution is reachable.

    32. EA C461 Artificial Intelligence Heuristic functions E.g for the 8-puzzle Avg. solution cost is about 22 steps (branching factor +/- 3) Exhaustive search to depth 22: 3.1 x 1010 states. A good heuristic function can reduce the search process.

    33. EA C461 Artificial Intelligence Heuristic functions E.g for the 8-puzzle knows two commonly used heuristics h1 = the number of misplaced tiles h1(s)=8 h2 = the sum of the distances of the tiles from their goal positions (manhattan distance). h2(s)=3+1+2+2+2+3+3+2=18

    34. EA C461 Artificial Intelligence Heuristic quality Effective branching factor b* Is the branching factor that a uniform tree of depth d would have in order to contain N+1 nodes. Measure is fairly constant for sufficiently hard problems. Can thus provide a good guide to the heuristic’s overall usefulness. A good value of b* is 1.

    35. EA C461 Artificial Intelligence Heuristic quality and dominance 1200 random problems with solution lengths from 2 to 24. If h2(n) >= h1(n) for all n (both admissible) then h2 dominates h1 and is better for search

    36. EA C461 Artificial Intelligence Inventing admissible heuristics Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem: Relaxed 8-puzzle for h1 : a tile can move anywhere As a result, h1(n) gives the shortest solution Relaxed 8-puzzle for h2 : a tile can move to any adjacent square. As a result, h2(n) gives the shortest solution. The optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem. ABSolver found a usefull heuristic for the rubic cube.

    37. EA C461 Artificial Intelligence Inventing admissible heuristics Admissible heuristics can also be derived from the solution cost of a subproblem of a given problem. This cost is a lower bound on the cost of the real problem. Pattern databases store the exact solution to for every possible subproblem instance. The complete heuristic is constructed using the patterns in the DB

    38. EA C461 Artificial Intelligence Inventing admissible heuristics Another way to find an admissible heuristic is through learning from experience: Experience = solving lots of 8-puzzles An inductive learning algorithm can be used to predict costs for other states that arise during search.

    39. EA C461 Artificial Intelligence Reference : Chapter 4.1, 4.2

More Related