Chapter 3
This presentation is the property of its rightful owner.
Sponsored Links
1 / 40

Chapter 3 PowerPoint PPT Presentation


  • 108 Views
  • Uploaded on
  • Presentation posted in: General

King Saud University College of Computer and Information Sciences Information Technology Department IT422 - Intelligent systems . Chapter 3. PROBLEM SOLVING BY SEARCHING (2). Informed Search.

Download Presentation

Chapter 3

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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


King Saud University

College of Computer and Information Sciences

Information Technology Department

IT422 - Intelligent systems

Chapter 3

PROBLEM SOLVING BY SEARCHING (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.


G

L

K

J

I

H

F

N

E

C

B

A

M

D

O

Informed Search

  • Best-First Search

    • Greedy Best First Search

    • A* Search

  • Local search algorithms

  • Stochastic Search algorithms

Search only in this subtree!!


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:

    • frontier: maintain the frontier in ascending order of f-values

  • Special cases:

    • Greedy best-first search

    • A* search


Formal description of Best-First Search algorithm

Function Best-First Graph-Search(problem,frontier,f) returns a solution or a failure

// f: evaluation function

children an empty set;

explored← an empty set;

frontier← Insert (Make-Node(Initial-state[problem],NULL,NULL,d,c),frontier)

Loop do

If Empty?(frontier) then return failure

node ← POP(frontier)

If Goal-Test[ problem] applied to State[node] succeeds thenreturn Solution(node)

add State[node] to explored

childrenExpand (node,problem)

for each child in children

If state [child] is not in explored or frontierthen

frontierinsert(State [child], frontier) // sort frontier in ascending order of f-values

else if child[state] is in frontier with higher f-values then

replace that frontier node with child

End Loop


Greedy best-first 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 best-first 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.


Map of Romania with Step Costs in km


Greedy best-first search


Greedy Best-first search


Greedy Best-first search


Greedy best-first search Properties

  • Finds solution without ever expanding a node that is not on the solution path.

  • It is not optimal: the optimal path goes through 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

  • The graph search version is Complete in finite spaces with repeated state checking but not in infinite ones.


A* search

  • Most widely known form of 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 Best-First Search).

    • f(n) = estimated total cost of path through n to goal.


A* search example


A* search example


A* search example


A* search example


A* search example


A* search example


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 to the goal.

  • 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.


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


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.


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.

  • Example: the previous search: f(Bucharest)=450 was not chosen for expansion, even though Bucharest is the goal.


A* search

  • For Graph-Search, A* is optimal if h(n) is consistent.

  • 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 non decreasing.

h(n)

n

c(n,n’)

g

h(n’)

n’


A* search properties

  • Completeness: Yes, with f ≤ f(G).

  • Optimality: Yes. The tree-search version is optimal if h(n) is admissible, while the graph-search version is optimal if h(n) is consistent.

  • Time complexity: Exponential.

  • Space complexity: Keeps all nodes in memory.


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.


Local Search algorithms

  • Blind search and informed search strategies addressed a single category of problems: observable, deterministic, known environments where the solution is a sequence of actions.

  • The search algorithms we have seen so far keep track of the current state, the “frontier” 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.


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 does not need to 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.


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


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).

  • Hill-climbing search modifies the current state to try to improve it, as shown by the arrow in figure on slide 28.

  • 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.


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


Simulated Annealing

  • Basic inspiration: What is annealing?

  • In metallurgy, 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 crystalline state.

  • Heating then slowly cooling a substance to obtain a strong crystalline 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.


Simulated Annealing


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


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?


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. It is also called chromosome.


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 offspring themselves are created by crossing over the parent strings at the crossover point.

  • Mutation is performed randomly with a small independent probability.


Stochastic search: Genetic algorithms


Stochastic search: Genetic algorithms


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 best-first 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.


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.


  • Login