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Informed Search Strategies Lecture # 8 & 9

Informed Search Strategies Lecture # 8 & 9. Outline. Best-first search Greedy best-first search A * search Heuristics. Introduction. So far, we have looked at different blind search strategies.

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Informed Search Strategies Lecture # 8 & 9

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  1. Informed Search StrategiesLecture # 8 & 9

  2. Outline • Best-first search • Greedy best-first search • A* search • Heuristics

  3. Introduction • So far, we have looked at different blind search strategies. • Uninformed search strategies look for solutions by systematically generating new states and checking each of them against the goal. • This approach is very inefficient in most cases. • Most successor states are “obviously” a bad choice. • Such strategies do not know that because they have minimal problem-specific knowledge. • Using problem-specific knowledge can dramatically improve the search speed. • Informed search strategies exploit problem-specific knowledge as much as possible to drive the search. • At the heart of such algorithms there is the concept of a heuristic function.

  4. Heuristics • “Heuristics are criteria, methods or principles for deciding which among several alternative courses of action promises to be the most effective in order to achieve some goal”. • In heuristic search or informed search, heuristics are used to identify the most promising search path. • Heuristics can estimate the “goodness” of a particular node (or state) n: How close is n to a goal node? What might be the minimal cost path from n to a goal node? h(n) = estimated cost of the cheapest path from node n to a goal node

  5. Example of Heuristic Function • 8-puzzle: (i) Misplaced Tiles Heuristics:- number of tiles out of place. The first picture shows the current state n, and the second picture the goal state. h(n) = 5 : because the tiles 2, 8, 1, 6 and 7 are out of place. (ii) Manhattan Distance Heuristic: This heuristic sums the distance that the tiles are out of place. The distance of a tile is measured by the sum of the differences in the x-positions and the y-positions. h(n) = 1 + 1 + 0 + 0 + 0 + 1 + 1 + 2 = 6

  6. Best First Search • The algorithm maintains a priority queue of nodes to be explored. • A cost function f(n) is applied to each node. • The nodes are put in OPEN in the order of their f values. • Nodes with smaller f(n) values are expanded earlier. • The generic best first search algorithm is outlined on next slide • Special cases: greedy search A∗ search

  7. Best Search Algorithm We will now consider different ways of defining the function f. This leads to different search algorithms.

  8. Greedy best-first Search • In greedy search, the idea is to expand the node with the smallest estimated cost to reach the goal. • We use a heuristic function f(n) = h(n) • h(n) estimates the distance remaining to a goal. • Greedy algorithms often perform very well. • They tend to find good solutions quickly, although not always optimal ones. • Considering an example of a route finding problem: S is the starting state, G is the goal state. • A good heuristic for the route-finding problem would be straight-line distance to the goal.

  9. Greedy best-first Search The straight line distance heuristic estimates for the nodes are shown in the following figure:

  10. Illustration of Greedy Search Example

  11. Greedy best-first Search • The algorithm is not optimal. • The algorithm is also incomplete, and it may fail to find a solution even if one exists. • This can be seen by running greedy search on the following example – using straight-line distance heuristic for the route-finding problem

  12. The nodes expanded are expanded in following order: A, B, E, G, H --- and the path obtained is A-B-E-G-H with its path cost 99 … Where as path A-B-C-F-H is optimal with cost 39.

  13. Example

  14. Properties of greedy best-first search • Complete? No – can get stuck in loops, • Time?O(bm), but a good heuristic can give dramatic improvement • Space?O(bm) -- keeps all nodes in memory • Optimal? No

  15. A* search • A best-first search algorithm by Nilsson & Rafael in 1968 • Idea: avoid expanding paths that are already expensive • 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

  16. Example

  17. A* Example

  18. 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

  19. Properties of A* • Complete? Yes • (unless there are infinitely many nodes with f ≤ f(G) ) • Time? Exponential • Space? Keeps all nodes in memory • Optimal? Yes

  20. Properties of Heuristics Dominance: • h2 is said to dominate h1 iff h2(n) ≥ h1(n) for any node n. • A* will expand fewer nodes on average using h2 than h1.

  21. Using multiple heuristics • Suppose you have identified a number of non-overestimating heuristics for a problem: h1(n), h2(n), … , hk(n) • Then max (h1(n), h2(n), … , hk(n)) is a more powerful non-overestimating heuristic. This follows from the property of dominance

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