Chapter 4: Informed search & exploratIon

1. Chapter 4: Informedsearch & exploratIon Preparedby: Ece UYKUR

2. Outline • Evaluation of SearchStrategies • InformedSearchStrategies • Best-first search • Greedy best-first search • A*search • AdmissibleHeuristics • Memory-BoundedSearch • Iterative-Deepening A* Search • RecursiveBest-FirstSearch • SimplifiedMemory-Bounded A* Search • Summary

3. Evaluation of SearchStrategies • Search algorithms are commonly evaluated according to the following four criteria; • Completeness: does it always find a solution if one exists? • Time complexity: how long does it take as a function of number of nodes? • Space complexity: how much memory does it require? • Optimality: does it guarantee the least-cost solution? • Time and space complexity are measured in terms of: • b – max branching factor of the search tree • d – depth of the least-cost solution • m – max depth of the search tree (may be infinity)

4. BeforeStarting… • Solution is a sequence of operators that bring you from current state to the goal state. • The search strategy is determined by the order in which the nodes are expanded. • UninformedSearchStrategies; use only information available in the problem formulation. • Breadth-first • Uniform-cost • Depth-first • Depth-limited • Iterative deepening

5. Uninformed search strategies look for solutions bysystematically generating new states and checkingeach 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 haveminimal problem-specific knowledge.

6. InformedSearchStrategies • Informed search strategies exploit problem-specificknowledge as much as possible to drive the search. • They are almost always more efficient than uninformedsearches and often also optimal. • Also called heuristic search. • Usestheknowledge of the problem domain tobuild an evaluationfunction f. • The searchstrategy: For every node n in the search space, f(n) quantifiesthe desirability of expanding n,in order to reach thegoal.

7. Usesthe desirability value of the nodes in the fringeto decide which node to expand next. • f is typically an imperfect measure of the goodness ofthe node. The right choice of nodes is not always theone suggested by f. • It is possible to build a perfect evaluation function,which will always suggest the right choice. • The general approach is called “best-first search”.

8. Best-FirstSearch • Choose the most desirable (seemly-best) node forexpansion based on evaluation function. • Lowest cost/highest probability evaluation • Implementation; • Fringeis a priority queue in decreasing order ofdesirability. • Evaluation Function; f(n): select a node for expansion (usually the lowest cost node), desirability of node n. • Heuristic function; h(n): estimates cheapest path cost fromnode n to a goalnode. • g(n) = cost from the initial state to the current state n.

9. Best-FirstSearchStrategy; • Pick “best” element of Q. (measuredbyheuristicvalue of state) • Addpathextensionsanywhere in Q. (it may be moreefficienttokeepthe Q ordered in someway, so as tomake it easiertofindthe ‘best’ element) • Therearemanypossibleapproachestofindingthebestnode in Q; • Scanning Q tofindlowestvalue, • Sorting Q andpickingthefirst element, • Keepingthe Q sortedbydoing “sorted” insertions, • Keeping Q as apriorityqueue.

10. Several kinds of best-first search introduced • Greedy best-first search • A* search

11. GreedyBest-FirstSearch • Estimation function: • Expand the node that appears to be closest to the goal,based on the heuristic function only; • f(n) = h(n) = estimate of cost from n totheclosestgoal • Ex: thestraightlinedistanceheuristics • hSLD(n) = straight-line distance from n to Bucharest • Greedy search expands first the node that appears to be closest to the goal, according to h(n). • “greedy” – at each search step the algorithm alwaystries to get close to the goal as it can.

12. Romaniawithstep costs in km

13. hSLD(In(Arad)) = 366 • Notice that the values of hSLD cannot be computed from the problem itself. • It takes some experience to know that hSLD is correlated with actual road distances • Therefore a useful heuristic

18. Properties of GreedySearch • Complete? • No – can get stuck in loops / followssinglepathtogoal. Ex; Iasi > Neamt > Iasi > Neamt > … • Complete in finite space with repeated-state checking. • Time? • O(b^m) but a good heuristic can give dramatic improvement. • Space? • O(b^m) – keeps all nodes in memory • Optimal? • No.

19. A* Search • Idea;avoid expanding paths that are already expensive. • Evaluation function: f(n) = g(n) + h(n)with; g(n) – cost so far to reach n h(n) – estimated cost to goal from n f(n) – estimated total cost of path through n to goal • A* search uses an admissible heuristic, that is, h(n)  h*(n) where h*(n) is the true cost from n. Ex: hSLD(n) never overestimates actual road distance. • Theorem: A* search is optimal.

20. When h(n) = actual cost to goal • Only nodes in the correct path are expanded • Optimal solution is found • When h(n) < actual cost to goal • Additional nodes are expanded • Optimal solution is found • When h(n) > actual cost to goal • Optimal solution can be overlooked

27. 1 Optimality of A* (standardproof) • Suppose some suboptimal goal G2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G1.

28. ConsistentHeuristics • A heuristic is consistent if for every node n, every successor n' of ngenerated by any action a, • h(n) ≤ c(n,a,n') + h(n') • n' = successor of n generated by action a • The estimated cost of reaching the goal from n is no greater than the stepcost of getting to n' plus the estimated cost of reaching the goal from 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) • if h(n) is consistent then the values of f(n) along any path are nondecreasing • Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

29. Optimality of A* (moreusefulproof)

30. f-contours How do the contours look like when h(n) =0? Uniformed search; Bands circulate around the initial state A* search; Bands stretch toward the goal and is narrowly focused aroundthe optimal path if more accurate heuristicswere used

31. Properties of A* Search • Complete? • Yes – unless thereareinfinitely many nodes with f  f(G). • Time? • O(b^d ) – Exponential. [(relative error in h) x (length of solution)] • Space? • O(b^d) – Keeps allnodes in memory. • Optimal? • Yes – cannot expand fi+1 until fi is finished.

32. AdmissibleHeuristics

34. Relaxed Problem • Admissible heuristics can be derived from the exact solution cost of a relaxed version of the 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 adjacent square, then h2(n) gives the shortest solution. • Key point;the optimal solution cost of a relaxed problemis no greater than the optimal solution cost of the realproblem

35. Dominance • Definition: If h2(n) h1(n) for all n (both admissible)then h2 dominates h1. • For 8-puzzle, h2 indeed dominates h1. • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance • If h2 dominates h1, then h2 is better for search. • For 8-puzzle, search costs: d = 14 IDS = 3,473,941 nodes (IDS = Iterative Deepening Search) A(h1) = 539 nodesA(h2) = 113 nodes d = 24 IDS 54,000,000,000 nodes A(h1) = 39,135 nodesA(h2) = 1,641 nodes

36. Memory-BoundedSearch • Iterative-Deepening A* Search • RecursiveBest-FirstSearch • SimplifiedMemory-Bounded A* Search

37. Iterative-Deepening A* Search (IDA*) • The idea of iterative deepening was adapted to theheuristic search context to reduce memory requirements • At each iteration, DFS is performed by using the f-cost (g+h) as the cutoff rather than the depth • Ex;the smallest f-cost of any node that exceeded the cutoff onthe previous iteration

38. Properties of IDA* • IDA* is complete and optimal • Space complexity: O(bf(G)/δ) ≈ O(bd) • δ : the smallest step cost • f(G): the optimal solution cost • Time complexity: O(α b^d) • α: the number of distinct fvalues smaller than the optimalgoal • Between iterations, IDA* retains only a single number; the f-cost • IDA* has difficulties in implementation when dealing withreal-valued cost

39. Recursive Best-First Search (RBFS) • Attempt to mimic best-first search but use only linearspace • Can be implemented as a recursive algorithm. • Keep track of the f-value of the best alternative path from anyancestor of the current node. • If the current node exceeds the limit, then the recursion unwindsback to the alternative path. • As the recursion unwinds, the f-value of each node along thepath is replaced with the best f-value of its children.

40. Ex: TheRoutefinding Problem

43. Properties of RBFS • RBFS is complete and optimal • Space complexity: O(bd) • Time complexity : worse case O(b^d) • Depend on the heuristics and frequency of “mindchange” • The same states may be explored many times

44. Simplified Memory-Bounded A* Search (SMA*) • Make use of all available memory M to carry out A* • Expanding the best leaf like A* until memory is full • When full, drop the worst leaf node (with highest f-value) • Like RBFS, backup the value of the forgotten node to its parent ifit is the best among the sub-tree of its parent. • When all children nodes were deleted/dropped, put the parentnode to the fringe again for further expansion.

45. Properties of SMA* • Is complete if M ≥ d • Is optimal if M ≥ d • Space complexity: O(M) • Time complexity : worse case O(b^d)

46. Summary • This chapter has examined the application of heuristics to reduce search costs. • We have lookedat number of algorithms that use heuristics, and found that optimality comes at a stiff price interms of search cost, even with good heuristics. • Best-first search is just GENERAL-SEARCH where the minimum-cost nodes (according tosome measure) are expanded first. • If we minimize the estimated cost to reach the goal, h(n), we get greedy search. Thesearch time is usually decreased compared to an uninformed algorithm, but the algorithmis neither optimal nor complete. • Minimizing f(n) = g(n) + h(n) combines the advantages of uniform-cost search andgreedy search. If we handle repeated states and guarantee that h(n) never overestimates,we get A* search.

47. A* is complete, optimal, and optimally efficient among all optimal search algorithms. Itsspace complexity is still prohibitive. • The time complexity of heuristic algorithms depends on the quality of the heuristic function. • Good heuristics can sometimes be constructed by examining the problem definition or bygeneralizing from experience with the problem class. • We can reduce the space requirement of A* with memory-bounded algorithms such asIDA* (iterative deepening A*) and SMA* (simplified memory-bounded A*).

48. Thanksforyourpatience