Intelligent Search Techniques in Problem Solving: Best-First and A* Algorithms

1. Shaqra University College of Computer and Information Sciences Information Technology Department Cs 401 - Intelligent systems Chapter 3 PROBLEM SOLVING BY SEARCHING (2)

2. Informed Search • One that uses problem specific knowledge beyond the definition of the problem itself to guide the search. • Why? • Without incorporating knowledge into searching, one is forced to look everywhere to find the answer. Hence, the complexity of uninformed search is intractable. • With knowledge, one can search the state space as if he was given “hints” when exploring a maze. • Heuristic information in search = Hints • Leads to dramatic speed up in efficiency.

3. Informed Search • Best-First Search • Greedy Best First Search • A* Search • Local search algorithms • Stochastic Search algorithms

4. Best first search • Key idea: • Use an evaluation function f(n) for each node: • estimate of “distance” to the goal. • Node with the lowest evaluation is chosen for expansion. • Implementation: • fringe: maintain the fringe in ascending order of f-values • Special cases: • Greedy search • A* search

5. Formal description of Best-First Search algorithm Function Best-First Search(problem,fringe,f) returns a solution or a failure // f: evaluation function fringe ← Insert (Make-Node(Initial-state[problem],NULL,NULL,d,c),fringe) Loop do If Empty?(fringe) then return failure node ← Remove-First (fringe) If Goal-Test[ problem] applied to State[node] succeeds then return Solution (node) fringe ← Insert-all( Expand (node,problem),fringe) sort fringe in ascending order of f-values

6. Greedy search • Let evaluation function f(n) be an estimate of cost from node n to goal • This function is often called a heuristic and is denoted by h(n). • f(n) = h(n) • e.g. hSLD(n) = straight-line distance from n to Bucharest • Greedy search expands the node that appears to be closest to goal. • Contrast with uniform-cost search in which lowest cost path from start is expanded. • Heuristic function is the way knowledge about the problem is used to guide the search process.

8. Greedy search

9. Greedy search

10. Greedy search

11. Greedy Search Properties • Finds solution without ever expanding a node that is not on the solution path. • It is not optimal: the optimal path goes trough Ptesti. • Minimizing h(n) is susceptible to false starts. • e.g getting from Iasi to Fagaras: according to h(n), we take Neamt node to expand but it is a dead end. • If repeated states are not detected, the solution will never be found. Search gets stuck in loops: • Iasi →Neamet → Iasi → Neamet • Complete in finite spaces with repeated state checking.

12. A* search • Most widely known for best-first search. • Key idea:avoid expanding paths that are already expensive. • Evaluation function:f(n) = g(n) + h(n) • g(n) = path cost so far to reach n. (used in Uniform Cost Search). • h(n) = estimated path cost to goal from n. (used in Greedy Search). • f(n) = estimated total cost of path through n to goal.

13. A* search

14. A* search

15. A* search

16. A* search

17. A* search

18. A* search • Definition: a heuristic h(n) is said to be admissible if it never overestimates the cost to reach the goal. h(n)  h*(n) • where h*(n) is the TRUE cost from n. • e.g: hsld straight line can not be an overestimate. • Consequently: if h(n) is an admissible heuristic, then f(n) never overestimates the true cost of a solution through n. WHY? • It is true because g(n) gives the exact cost to reach n.

19. h*(n): true minimum cost to goal A* search root g(n):cost of path n h(n): Heuristic (expected) minimum cost to goal. (estimation) Goal

20. A* search • Theorem: When Tree-Search is used, A* is optimal if h(n) is an admissible heuristic. • Proof: • Let G be the optimal goal state reached by a path with cost : C* =f(G) = g(G). • Let G2 be some other goal state or the same state, but reached by a more costly path.

21. A* search f(G2) = g(G2)+h(G2) = g(G2) since h(G2) = 0 g(G2) > C* since G2 is suboptimal • Let n be any unexpanded node on the shortest path to the optimal goal G. f(n) = g(n) + h(n) ≤ C* since h is admissible Therefore, f(n) ≤ C* ≤ f(G2) • As a consequence, G2 will not be expanded and A* must return an optimal solution.

22. A* search • Consistency (= Monotonicity): A heuristic is said to be consistent when for any node n, successor n’ of n, we have h(n) ≤ c(n,n’) + h(n’), where c(n,n’) is the (minimum) cost of a step from n to n’. • This is a form of triangular inequality: • Consistent heuristics are admissible. Not all admissible heuristics are consistent. • When a heuristic is consistent, the values of f(n) along any path are nondecreasing. • A* with a consistent heuristic is optimal. h(n) n c(n,n’) g h(n’) n’

23. A* search properties • Completeness: Yes, unless there are infinitely many nodes with f ≤ f(G). • Optimality: Yes. • Time: Exponential. • Space: Keeps all nodes in memory.

24. Some admissible heuristics • 8-Puzzle: • g(n):the path cost can be measured by the total number of horizontal and vertical moves. • h(n): two different heuristics • h1 (n): number of misplaced tiles. • h2 (n): the sum of the distances of the tiles from their goal positions.

25. Local Search algorithms • The search algorithms we have seen so far keep track of the current state, the “fringe” of the search space, and the path to the final state. • In some problems, one doesn’t care about a solution path but only the final goal state. The solution is the goal state. Example: 8-queen problem. • Local search algorithms are also useful for optimization problems where the goal is to find a state such that an objective function is optimized. • For the 8-queen algorithm, the objective function may be the number of attacks.

26. Local Search algorithms • Basic idea: • Local search algorithms operate on a single state – current state – and move to one of its neighboring states. • Therefore: Solution path needs not be maintained. • Hence, the search is “local”. • Two advantages: • Use little memory. • More applicable in searching large/infinite search space. They find reasonable solutions in this case.

27. Local Search algorithms • A state space landscape is a graph of states associated with their costs • Problem: local search can get stuck on a local maximum and not find the optimal solution

28. Hill Climbing • Hill climbing search algorithm (also known as greedy local search) uses a loop that continually moves in the direction of increasing values (that is uphill). • It terminates when it reaches a peak where no neighbor has a higher value. • A complete local search algorithm always find a goal if one exists. • An optimal algorithm always finds a global maximum/minimum.

29. Steepest ascent version Function Hill climbing (problem) return state that is a local maximum Inputs: problem, a problem Local variables: current, a node neighbor, a node Current ← Make-Node (initial-state [problem]) Loop do neighbor ← a highest-valued successor of current IfValue[neighbor] ≤ Value[current] then return state [current] Current ← neighbor

30. Simulated Annealing • Basic inspiration: What is annealing? • In mettallurgy, annealing is the physical process used to temper or harden metals or glass by heating them to a high temperature and then gradually cooling them, thus allowing the material to coalesce into a low energy cristalline state. • Heating then slowly cooling a substance to obtain a strong cristalline structure. • Key idea: Simulated Annealing combines Hill Climbing with a random walk in some way that yields both efficiency and completeness. • Used to solve VLSI layout problems in the early 1980.

31. Simulated Annealing

32. Simulated Annealing Function Simulated annealing (problem, schedule) return a solution state Inputs:problem, a problem schedule, a mapping from time to temperature Local variables: current, a node next, a node T, a temperature controlling the probability of downward steps Current ← Make-Node (initial-state [problem]) Fort←1 to ∞ do T ← schedule [t] If T=0 then return current Loopdo Next ← a randomly selected successor of current. ∆E ← Value [next] - Value [current] If∆E > 0 thencurrent ← next Elsecurrent ← next only with probability e -∆E/kT

33. Local Beam Search • Unlike Hill Climbing, Local Beam Search keeps track of k states rather than just one. • It starts with k randomly generated states. • At each step, all the successors of all the states are generated. • If any one is a goal, the algorithm halts, otherwise it selects the k best successors from the complete list and repeats. • LBS≠ running k random restarts in parallel instead of sequence. • Drawback: less diversity → Stochastic Beam Search

34. Stochastic search: Genetic algorithms • Formally introduced in the US in the 70s by John Holland. • GAs emulate ideas from genetics and natural selection and can search potentially large spaces. • Before we can apply Genetic Algorithm to a problem, we need to answer: • How is an individual represented • What is the fitness function? • How are individuals selected? • How do individuals reproduce?

35. Stochastic search: Genetic algorithms • Genetic algorithms is a variant of local beam search. • Successors in this case are generated by combining two parent states rather than modifying a single state. • Like local beam search, genetic algorithms starts with a set of k randomly generated states called Population. • Each state or individual is represented as a string over a finite alphabet. • Each state or individual is represented as a string over a finite alphabet. It is also called chromosome.

36. Stochastic search: Genetic algorithms • Each state is rated by the evaluation function called fitness function. • Fitness function should return higher values for better states. • For reproduction, individuals are selected with a probability which is directly proportional to the fitness score. • For each pair to be mated, a crossover point is randomly chosen from the positions in the string. • The offsprings themselves are created by crossing over the parent strings at the crossover point. • Mutation is performed randomly with a small independent probability.

37. Stochastic search: Genetic algorithms

38. Stochastic search: Genetic algorithms Function Genetic-Algorithm (population, fitness-Fn) returns individual Inputs:population, a set of individuals fitness-Fn, a function that measures the fitness of an individual Repeat New population ← empty set; fori from 1 toSizePopulationdo {x ← random-Selection (population, fitness-Fn); y ← random-Selection (population, fitness-Fn); Child ← Reproduce (x,y); If (small random probability) thenchild ← mutate(child) Add child to new population } Population ← new population Until some individual is fit enough or enough time has elapsed; Return the best individual in population, according to Fitness-Fn

39. Summary • Informed search uses knowledge about the problem to reduce search costs. • This knowledge is expressed in terms of heuristics. • Best first search is a class of methods that use a variant of graph-search where the minimum-cost unexpanded nodes are chosen for expansion. • Best first search methods use a heuristic function h(n) that estimates the cost of a solution from a node. • Greedy search is a best first search that expands nodes with minimal h(n). It is not optimal but often efficient. • A* search is a best first search that takes into account the total cost from the root node to goal node. It expands node with minimal f(n) = g(n) + h(n). It is complete and optimal provided that h(n) is admissible (for tree search) or consistent (for graph search). The space complexity is prohibitive.

40. Summary • Construction of heuristics can be done by relaxing the problem definition (in a sense simplifying the problem), by precomputing solution costs for subproblems or learning from experience with the problem class. • Local search methods keep small number of nodes in memory. They are suitable for problems where the solution is the goal state itself and not the path. • Hill climbing, simulated annealing and local beam search are examples of local search algorithms. • Stochastic algorithms represent another class of methods for informed search. Genetic algorithms are a kind of stochastic hill-climbing search in which a large population of states is maintained. New states are generated by mutation and by crossover which combines pairs of states from the population.