Informed search algorithms Outline Best-first search Greedy - - PowerPoint PPT Presentation

Informed search algorithms

Outline

• Best-first search
• Greedy best-first search
• A* search
• Heuristics
• Local search algorithms
• Hill-climbing search

Best-first search

• Idea: use an evaluation function f(n) for each node

– estimate of "desirability« -istenme derecesi- Expand most desirable unexpanded node

• Implementation:

Order the nodes in fringe in decreasing order of desirability

• Special cases:

– greedy best-first search – A* search

Romania with step costs in km

Greedy best-first search

• Evaluation function f(n) = h(n) (heuristic)
• = estimate of cost from n to goal
• 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

Greedy best-first search example

Greedy best-first search example

Greedy best-first search example

Greedy best-first search example

Properties of greedy best-first search

• Complete? No – can get stuck in loops, e.g., Iasi

 Neamt  Iasi  Neamt 

• Time? O(bm), but a good heuristic can give

dramatic improvement

• Space? O(bm) -- keeps all nodes in memory
• Optimal? No

A* search

• Idea: avoid expanding paths that are already

expensive

• Evaluation function f(n) = g(n) + h(n)
• g(n) = cost so far to reach n
• h(n) = estimated cost from n to goal
• 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

Admissible heuristics

• A heuristic h(n) is admissible – kabul edilebilir - if for every node n,

h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.

• An admissible heuristic never overestimates – fazla tahmin etmez -

the cost to reach the goal, i.e., it is optimistic

• Example: hSLD(n) (never overestimates the actual road distance)
• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal

Properties of A*

• Complete? Yes (unless there are infinitely many nodes

with f ≤ f(G) )

• Time? Exponential
• Space? Keeps all nodes in memory
• Optimal? Yes

Admissible heuristics

E.g., for the 8-puzzle:

• h1(n) = number of misplaced tiles
• h2(n) = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

• h1(S) = ?
• h2(S) = ?

Admissible heuristics

E.g., for the 8-puzzle:

• h1(n) = number of misplaced tiles
• h2(n) = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

• h1(S) = ? 8
• h2(S) = ? 3+1+2+2+2+3+3+2 = 18

Relaxed problems

• A problem with fewer restrictions on the actions is called

a relaxed problem

• The cost of an optimal solution to a relaxed problem is

an admissible heuristic for the original problem

Local search algorithms

• In many optimization problems, the path to the goal is

irrelevant; the goal state itself is the solution

• State space = set of "complete" configurations
• Find configuration satisfying constraints, e.g., n-queens
• In such cases, we can use local search algorithms
• keep a single "current" state, try to improve it

Example: n-queens

• Put n queens on an n × n board with no

two queens on the same row, column, or diagonal

Hill-climbing search

• "Like climbing Everest in thick fog with

amnesia"

Hill-climbing search

• Problem: depending on initial state, can

get stuck in local maxima

Hill-climbing search: 8-queens problem

• h = number of pairs of queens that are attacking each other, either directly
• r indirectly
• h = 17 for the above state

Hill-climbing search: 8-queens problem

• A local minimum with h = 1

Constraint Satisfaction Problems

Outline

• Constraint Satisfaction Problems (CSP)
• Backtracking search for CSPs

Constraint satisfaction problems (CSPs)

• Standard search problem:
• –

state is a "black box“ – any data structure that supports successor function, heuristic function, and goal test

• CSP:

– state is defined by variables Xi with values from domain Di – – goal test is a set of constraints specifying allowable combinations of values for subsets of variables

• Allows useful general-purpose algorithms with more power than standard

search algorithms

Example: Map-Coloring

• Variables WA, NT, Q, NSW, V, SA, T
• Domains Di = {red,green,blue}
• Constraints: adjacent regions must have different colors
• e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red),

(green,blue),(blue,red),(blue,green)}

Example: Map-Coloring

• Solutions are complete and consistent

assignments, e.g., WA = red, NT = green,Q = red,NSW = green,V = red,SA = blue,T = green

Constraint graph

• Constraint graph: nodes are variables, arcs are constraints

Real-world CSPs

• Assignment problems

– e.g., who teaches what class

• Timetabling problems
• – e.g., which class is offered when and where?
• Transportation scheduling
• Factory scheduling
• Notice that many real-world problems involve real-valued

variables

Standard search formulation (incremental)

Let's start with the straightforward approach, then fix it States are defined by the values assigned so far

• Initial state: the empty assignment { }
• Successor function: assign a value to an unassigned variable that does not conflict

with current assignment

 fail if no legal assignments

• Goal test: the current assignment is complete

1. This is the same for all CSPs 2. Every solution appears at depth n with n variables  use depth-first search 3. Path is irrelevant, so can also use complete-state formulation

Backtracking search

• Variable assignments are commutative - değişmeli}, i.e.,

[ WA = red then NT = green ] same as [ NT = green then WA = red ]

• It repeatedly chooses an unassigned variable and then tries all

values in the domain of that variable in turn, trying to find a solution.

• Backtracking search is the basic uninformed algorithm for CSPs

Backtracking search

Backtracking example

Backtracking example

Backtracking example

Backtracking example

Improving backtracking efficiency

• General-purpose methods can give huge

gains in speed:

– Which variable should be assigned next? – In what order should its values be tried? – Can we detect inevitable failure early?

Most constrained variable

• Most constrained variable:

choose the variable with the fewest legal values

• a.k.a. minimum remaining values (MRV)

heuristic

Most constraining variable

• Most constraining variable:
• choose the variable with the most

constraints on remaining variables

Least constraining value

• Given a variable, choose the least constraining value:

– the one that rules out the fewest values in the remaining variables

• Combining these heuristics makes 1000 queens feasible

Summary

• CSPs are a special kind of problem:

– states defined by values of a fixed set of variables – goal test defined by constraints on variable values

• Backtracking = depth-first search with one variable assigned per node
• Variable ordering and value selection heuristics help significantly