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Informed Search

Informed Search. Chapter 4. Search and Control Strategies. Word " Search " refers to the search for a solution in a problem space . Search proceeds with different types of " Search Control strategies ". A strategy is defined by selection of the order in which the nodes expand .

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Informed Search

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  1. Informed Search Chapter 4

  2. Search and Control Strategies • Word "Search" refers to the search for a solution in a problemspace. • Search proceeds with different types of "SearchControlstrategies". • A strategy is defined by selection of the order in which the nodesexpand. • The Searchstrategies are evaluated in the following dimensions: • Completeness, • Time complexity, • Space complexity, • Optimality.

  3. Search related terms Algorithm’s Performance and Complexity: A common measure to compareapproaches in order to select the most appropriatealgorithm for a given situation. ◊ Performance of an algorithm depends on internal and externalfactors. Internalfactors ‡ Time required, to run ‡ Space (memory) required to run Externalfactors ‡ Size of input to the algorithm ‡ Speed of the computer ‡ Quality of the compiler ◊ Complexity is a measure of the performance of an algorithm. It measures the internalfactors, usually in time than space.

  4. Computational Complexity: A measure of resources in terms of Time and Space. ◊ IfA is an algorithm that solves a decision problemf then runtime T of A is the number of steps taken on the input of lengthn. ◊ Time Complexity T(n) of a decision problemf is the runtime of the 'best' algorithmA forf . ◊ Space Complexity S(n) of a decision problemf is the amount of memory used by the `best' algorithmA forf .

  5. “Big - O” notation The "Big-O“ is theoretical measure of the execution of an algorithmto give an approximation to the run-time-efficiency of an algorithm . • the letter “O” is for orderofmagnitudeof operations or spaceatrun-time. ◊ The Big-O of an Algorithm A • If algorithmA requires timeproportional ton, then order of the algorithm is said to beO(n). • If algorithm A requires timeproportional ton2, then order of the algorithm is said to beO(n2). • If an algorithm A requires timeproportional tof(n), then the algorithm A is said to be of order f(n), and it is denoted asO(f(n)). • The functionf(n) is called the algorithm’sgrowth-ratefunction. • Similarly, for algorithms having performancecomplexity : O(log N), O(N log N) , O(2N) and so on. Note: O(logn) < O(n ) < O(nlogn) < O(n2 ) < O(n3 )

  6. ◊ Example 1 : 1-D array • determine the Big-O of an algorithm ; Calculate the sum of then elements in an integer array : a[0 . . . . n-1]. Line no Instructions No of execution steps line 1 sum = 0 1 line 2 for (i = 0; i < n; i++) n+1 line 3 sum += a[i] n line 4 print sum 1 Total = 2n+3 To determine the Big-O: • Ignoringconstants such as2 and3, the algorithm is of the ordern. So theBig-O of the algorithm isO(n). • In other words the run-time of this algorithm increases roughly as the size of the input datan

  7. ◊ Example 2 :2-D array determine the Big-O of an algorithm ; In a square 2-D array a[0 . . . . . n-1] [0 . . . . . n-1] , find the largestelement . Line no Instructions No of execution steps line 1 max = a[0][0] 1 line 2 for (row = 0; row < n; row++) n+1 line 3 for (col = 0; col < n; col++) (n+1)(n+1) line 4 if (a[row][col] > max) max = a[row][col]. n*(n) line 5 print max 1 Total = 2n2 +3n+4 To determine the Big-O: • Ignoring the constants such as2, 3and4, the algorithm is of order n2. So theBig-O of the algorithm isO(n2). • In other words, run-time of this algorithm increases roughly as the square of the size of the inputdata which isn2

  8. Search • Search is the systematicexamination of states to findpath from the start/root state to the goal state. − searchexploresknowledgealternatives to arrive at the bestanswer. − searchalgorithmoutputis a solution (a path)from the initial state to a state that satisfies the goal test. − search deals with findingnodes having certainproperties in a graph that represents searchspace. − searchmethodsexplore thesearchspace "intelligently", evaluating possibilities.

  9. Example :Search tree • The search trees are multilevelindexes used to guide the search for data items, given some searchcriteria.

  10. Search Algorithms : There are twotypes of search: Uninformed Search : Also calledblind, exhaustive or brute-forceالقوة الغاشمة search, uses noinformation about the problem to guide the search and therefore may not be very efficient. Informed Search : Also called heuristic orintelligentsearch, uses information about the problem to guide the search, usually guesses the distance to a goalstate, and therefore efficient, but the searchmaynot be alwayspossible.

  11. Search Space A set of allstates , which can be reached, constitute a searchspace. Example : Find route from Start to Goal state. Consider the vertices as city and the edges as distances.

  12. Search notations Search is the systematicexamination of states to findpath from the start or root state to the goal state. The notations used for defining search are: −h(n) is heuristicfunction that estimates leastcostpath from noden to goal node. −g(n) is costfunction that estimates leastcostpath from start node to noden.  −f(n) is evaluationfunction that estimates leastcostPath (solution) through noden. Where, f(n) = g(n) + h(n)

  13. Estimate Cost Function g* ◊ An estimatedleast cost path from start node to noden, is written asg*(n). ◊ g* is known by summingallpathcosts fromstart tocurrent state. ◊ If search space is atree, theng* = g, because there is onlyonepath from start node to current node. ◊ If search space is agraph, theng* ≥ g,

  14. Estimate Heuristic Function h* ◊ An estimatedleast cost path fromnoden togoal node , is written ash*(n) ◊ h* is a heuristicinformation, it represents a guess; ◊ h* may be estimated using an evaluationfunction f(n) that measures "goodness" of a node. ◊ h* may have different values (based on experience) the values lie between0 ≤ h*(n) ≤ h(n); they mean a different search algorithm. ◊ Ifh* = h, it is a "perfectheuristic"; it means nounnecessarynodes are ever expanded.

  15. Control Strategies Search for a solution in a problem space, requires "ControlStrategies" to control the searchprocesses. Strategies for Search Some widely used control strategies for search are: 1. Forward search : • Here, the control strategies for exploringsearch proceeds forward from initial state towards a solution; • This strategy is calleddata-directed. 2. Backward search : • Here, the control strategies for exploringsearch proceeds backward from a goal or final state towards either a solvable sub problem or the initial state; • This strategy is calledgoal-directed.

  16. 3. Both forward and backward search : Here, the control strategies for exploringsearch is amixture of bothforward and backward strategies . 4. Systematic search : Where searchspace is small, a systematic (but blind) method can be used to explore the wholesearchspace. • One such search method isdepth-first search • the other isbreath-first search. 5. Heuristic search : Many search depend on the knowledge of the problemdomain. They have some measure of relative merits to guide the search. The search so guided are calledheuristicsearchand the methods used are calledheuristics. Note : A heuristicsearch might notalwaysfind the bestsolution but it is guaranteed to find a goodsolution in reasonabletime.

  17. Condition-action rules − one way of encodingKnowledge is condition-actionrules − the rules are written as: if < condition> then < conclusion > ◊ Chaining −Chaining refers to sharingconditions between rules, so that the same condition is evaluated once for allrules. − When one or moreconditions are shared between rules, they are considered "chained." − Chaining are of two types : −Forwardchaining is called data-driven And −Backwardchaining is called query-driven

  18. Forward Chaining Algorithm Forwardchaining is a technique for drawinginferences from Rulebase. Forward-chaininginference is often calleddatadriven. ◊ The algorithm Proceeds from a givensituation to a desiredgoal, adding new assertions (facts) found. ◊ A forward-chaining, system comparesdata in theworkingmemoryagainst the conditions in the IFparts of the rules and determineswhichrule to fire. ◊ Data Driven

  19. ◊ Example : Forward Channing

  20. Backward Chaining Algorithm • Backwardchaining is a techniques for drawinginferences from Rulebase. • Backward-chaininginference is often calledgoaldriven. • The algorithmproceeds from desiredgoal, addingnewassertions found. • A backward-chaining, system looks for the action in the THEN clause of the rules that matches the specified goal.

  21. Heuristic Search Techniques • For complexproblems, the traditionalBlind Searching algorithms, are unable to find the solution within some practicaltime and spacelimits. • Consequently, many specialtechniques are developed, usingheuristicfunctions. قواعد تعتمد على الخبرة • − Blindsearch is not always possible, because they require toomuchtime or Space (memory). • − Heuristics are rules of thumb; they do notguarantee for a solution to a problem. • − HeuristicSearchis a weaktechnique but can be effective if applied correctly; they require domainspecificinformation

  22. Characteristics of Heuristic Search ◊Heuristics, are knowledge about domain, which helpsearch and reasoning in its domain. ◊Heuristicsearchincorporatesdomainknowledge to improveefficiency over blind search. ◊Heuristic is a function that, when applied to a state, returns value as estimated merit of state, with respect to goal. ◊Heuristicevaluationfunctionestimateslikelihood of given stateleading to goal state.

  23. Heuristic Search compared with other search •  The Heuristic search is compared with Brute force or Blind search techniques • Compare Algorithms

  24. ◊ Solution: optimal sequence of operators ◊ Action: “blank moves” • Condition: the move is within the board • Directions: Left, Right, Up,Dn ◊ Problem • - which 8-puzzle move is best? • - what heuristic(s) can decide? • - which move is “best” (worth considering first) ?

  25. ◊ Apply the Heuristic : Three different approaches -Countcorrectposition of each tile, compare to goal state -Countincorrectposition of each tile, compare to goal state - Count how faraway each tile is from it is correctposition. ApproachesLeftRightUp 1. Count correct position 6 4 5 2. Count incorrect position 2 4 3 3. Count how faraway 2 4 4

  26. ◊ Heuristic :  Three different approaches  ■ 1st approach : Count correct position of each tile, compare to goal state. ‡Higher the number the better it is. ‡Easy to compute (fast and takes little memory). ‡ Probably the simplest possible heuristic.  ■ 2nd approach Count incorrect position of each tile, compare to goal state ‡Lower the number the better it is. ‡ The “best” move is where lowest number returned by heuristic.  ■ 3rd approach Count how faraway each tile is from it’s correct position ‡ Count how far away (how many tile movements) each tile is from it’s correct position. ‡ Sum up these count over all the tiles. ‡ The “best” move is where lowest number returned by heuristic.

  27. Eight Queens Puzzle Problem : How can one put 8 queens on a (8 x 8) chess board such that no queen can attack any other queen ? • the puzzle has 92 distinct solutions. • Humans would find it hard to solve N-Queens puzzle while N becomes more. 

  28. The possiblenumber of configurations are :  − For 4-Queens there are 256 different configurations. − For 8-Queens there are 16,777,216 configurations. − For 16-Queens there are 18,446,744,073,709,551,616 configurations. - in general for N configurations, there NN Configurations. − For N = 16, this would take about 12,000 years on a fast machine. • How do we solve such problems ? Three computer based approaches or models are stated below. They are ‡GenerateandTest (GT) , ‡Backtracking (BT) and ‡ConstrainSatisfactionProblems (CSPs)

  29. ◊ Generate and Test (GT) : n = 4 Queens puzzle  One possible solution is to systematically try every placement of queens until we find a solution. The process is known as "Generate and Test". Examples of Generate and Test conditions for solutions :

  30. ◊ Backtracking (BT) : n = 4 Queens puzzle The Backtracking method is based on systematic examination of the possiblesolutions. − The algorithmstryeachpossibility until they find the rightone. − It differs from simple bruteforce, which generatesallsolutions, even those arising from infeasiblepartialsolutions. Backtracking is similar to a depth-first search but uses lessspace, keeping just one current solution state and updating it. ‡ during search, if an alternative does notwork, the searchbacktracks to the choicepoint, the place which presented different alternatives, and tries the next alternative. ‡when the alternatives are exhausted, the searchreturns to the previouschoicepoint and tries the nextalternative there. ‡if there are no more choicepoints, the searchfails.

  31. Example : Backtracking to solve N = 4 Queens problem.

  32. Backtracking to solve N Queens problem. • The problemproceeds either by rows or by columns. • for no particularly good reason, select columns to proceed. • for eachcolumn, select a row to place the queen. Algorithm 1. Move “left to right” processing onecolumn at a time. 2. For column J, select a row position for the queen. Check for feasibility. a. If there are one or more attacks possible from queens in columns 1 through (J – 1), discard the solution. b. For each feasibleplacement in columnJ, make the placement and try placement in column (J + 1). c. If there are no more feasible placements in columnJ, return to column (J – 1) and try another placement. 3. Continue until allNcolumns are assigned or until no feasible solution is found.

  33. Best-first search • Idea: use an evaluation functionf(n)for each node • The evaluation measures distance to Goal.Expandlowestevaluationunexpanded node • Implementation: use priority Queue that maintain the nodes in fringe in ascendingorder of f-values. • Special cases: • Greedybest-first search • A*search

  34. Romania with step costs in km

  35. Greedy best-first search • Greedy best-first search expands the node that appears to be closestto the goal • It evaluates nodes by using just the heuristic function h • Evaluation function f(n) = h(n) (heuristic) = estimate of cost from n to goal • Ex: hSLD(n) = straight-line distance from n to Bucharest Note: hSLD cannot be computed from the problem description. (it is Heuristic data gain from experience)

  36. Greedy best-first search example

  37. Greedy best-first search example

  38. Greedy best-first search example

  39. Greedy best-first search example

  40. Properties of greedy best-first search • Complete?No – can get stuck in loops, (Iasi Neamt Iasi Neamt) , or dead end (start node = lasi, goal node = faragas, heuristic suggests that be expanded first) Neamt • Time?O(bm), but a good heuristic can give dramatic improvement ( m = maximum depth of search)Space?O(bm)-- keeps all nodes in memory Optimal?No the path (Arad, RimnicuVilcea, Pitesti, Bucharest) is Shorter than that selected by Greedy (Arad, Sibiu, Faragas, Bucharest). This shows why the algorithm is called “Greedy” طماع

  41. A* search • Idea: evaluates nodes by combining the cost to reach the nodeg(n) and the cost to get from node to goalh(n) • Evaluation function f(n) = g(n) + h(n) • g(n) = cost so far toreachn • h(n) = estimatedcost from n to goal • f(n)= estimated totalcost of paththroughn to goal

  42. A* search example

  43. A* search example

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