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

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chapter 3

King Saud University

College of Computer and Information Sciences

Information Technology Department

IT422 - Intelligent systems

Chapter 3

PROBLEM SOLVING BY SEARCHING (2)

informed search
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.
informed search1

G

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H

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C

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A

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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
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
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
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.
greedy best first search properties
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
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 search1
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.
a search2

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 search3
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 search4
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 search5
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
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
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
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 algorithms1
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 algorithms2
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
  • 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
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
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.
local beam search
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
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 algorithms1
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 algorithms2
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.
summary
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.
summary1
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.
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