Informed Search Methods
Chapter 4
Fall 2009
Copyright, 1996 © Dale Carnegie & Associates, Inc.
What we’ll learn
Informed search algorithms are more
efficient in most cases
What are informed search methods
How to use problem-specific knowledge
How to optimize a solution
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Best-First Search
Evaluation functions

It gives a measure about which node to expand
Minimizing the path cost g(n) – a true cost

Expands the node based on the past
Minimizing estimated cost to reach a goal
Greedy search at node n



heuristic function h(n)
 an example is straight-line distance between cities (Fig 4.1)
The simple Romania map
Finding the route using greedy search – example (Fig 4.2)
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Best-first search (2)
h(n) is independent of the path cost g(n)
Minimizing the total path cost

f(n) = g(n) + h(n)
estimated cost of the cheapest solution via n
Admissible heuristic function h

never overestimates the cost
 What is the most useless h?

optimistic
 One never overestimates
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A* search
How it works (Fig 4.3)
Characteristics of A*

Monotonicity (Consistency) - h is nondescreasing
 How to check – using triangle inequality


Tree-search to ensure monotonicity
Contours (Fig 4.4) - from circle to oval (ellipse)
Proof of the optimality of A*
The completeness of A* (Fig 4.4 Contours)
Complexity of A* (time and space)

For most problems, the number of nodes within the
goal contour search space is still exponential in the
length of the solution
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Improving A* - memory-bounded
heuristic search
Iterative-deepening A* (IDA*)



Using f-cost(g+h) rather than the depth
Cutoff value is the smallest f-cost of any node that exceeded the
cutoff on the previous iteration; keep these nodes only
Space complexity O(bd)
Recursive best-first search (RBFS)



Best-first search using only linear space complexity (Fig 4.5)
It replaces the f-value of each node along the path with the best fvalue of its children (Fig 4.6)
Space complexity O(bd) with excessive node regeneration
Simplified memory bounded A* (SMA*)



IDA* and RBFS use too little memory – excessive node regeneration
Expanding the best leaf until memory is full
Dropping the worst leaf node (highest f-value) by backing up to its
parent
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Different Search Strategies
Uniform-cost search

minimize the path cost so far
Greedy search

minimize the estimated path cost
A*


minimize the total path cost
Time and space issues of A*
 Designing good heuristic functions
 A* usually runs out of space long before it runs
out of time
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Heuristic Functions
An example (the 8-puzzle, Fig 4.7)

How simple can a heuristic be?
 The distance to its correct position
 Using Manhattan distance
What is a good heuristic?


Effective branching factor - close to 1 (Why?)
Value of h
 not too large - must be admissible (Why?)
 not too small - ineffective (oval to circle)
(expanding all nodes with f (n) < f*)

Goodness measures - no. of nodes expanded and
branching factor (Fig 4.8)
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Domination translates directly into efficiency


Larger h means smaller branching factor
If h2 >= h1, is h2 always better than h1?
 Proof? (h1 <= h2 <= C* - g)
Inventing heuristic functions – An important
component of A*


How to invent
One way is to work on relaxed problems
 Simplify the problem
 Remove some constraints
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8-puzzle revisited
Definition: A tile can move from A to B if A is
horizontally or vertically adjacent to B and B
is blank
Relaxation by removing one or both the
conditions



A tile can move from A to B if A ~ B
A tile can move from A to B if B is blank
A tile can move from A to B
Deriving a heuristic from the solution cost of
a sub-problem

Fig 4.9
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If we have admissible h1 … hm and none
dominates, we can have for node n
h = max(h1, …, hm)
Feature selection and combination

use only relevant features
 “number of misplaced tiles” as a feature
The cost of heuristic function calculation


<= the cost of expanding a node
otherwise, we need to rethink.
Learning heuristics from experience

Each optimal solution to 8-puzzle provides a
learning example
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Local Search Algorithms and
Optimization Problems
Sometimes the path to the goal constitutes a
solution; sometimes, the path to the goal is
irrelevant (e.g., 8-queen)
Local search algorithms operate using a
single current state and generally move only
to neighbors of that state.


The paths followed by the search are not retained
Key advantages: little memory usage; can find
reasonable solutions in large or infinite state space
where systematic search is not suitable
Global and local optima

Fig 4.10, from current state to global maximum
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Some local-search algorithms
Hill-climbing (maximization)


Well know drawbacks (Fig 4.13)
 Local maxima, Plateaus, Ridges
Random-restart
Simulated annealing


Gradient descent (minimization)
Escaping the local minima by controlled bouncing
Local beam search


Keeping track of k states instead of just one
Is it similar to have k random-start of Hill-climbing?
Genetic algorithms

Selection, cross-over, and mutation
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Online Search
Offline search – computing a complete
solution before acting
Online search – interleaving computation and
action
Solving an exploration problem where the states
and actions are unknown to the agent
 Good for domains where there is a penalty for
computing too long, or for stochastic domains
 An example –
A robot is placed in a new building: explore it to
build a map that it can use for getting A to B
 Any additional examples? Please send it to me if
you find one.

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Online search problems
An agent knows: e.g., Fig 4.18


Actions(s) in state s
Step-cost function c(s,a,s’)
 c() cannot be used until the agent knows s’ is the outcome
 In order to know c(), a must be actually tried


Goal-Test(s)
Others: with memory of states visited, and
admissible heuristic from current state to the goal
state
Objective:

Reaching a goal state while minimizing cost
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Measuring its performance
Competitive ratio: the true path cost over the
path cost if it knew the search space in
advance
The best achievable competitive ratio can be 1

If some actions are irreversible, it may reach a
dead-end (Fig 4.19 (a))
An adversary argument – Fig 4.19 (b)

No bounded competitive ratio can be guaranteed if
there are paths of unbounded cost
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Online search agents
It can expand only a node that it physically occupies, so
it should expand nodes in a local order
Online Depth-First Search (Fig 4.20)

Backtracking requires that actions are reversible
Hill-climbing search keeps one current state in memory



It can get stuck in a local minimum
Random restart does not work here (Why?)
Random walk selects at random one of the available actions from
the current state
 It can be very slow, Fig 4.21

Augmenting hill climbing with memory rather than randomness is
more effective
 Learning real-time agent, Fig 4.22


H(s) is updated as the agent gains experience
Encourages to explore new paths
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Summary
Heuristics are the key to reducing research
costs


f(n) = g(n)+h(n)
Understand their variants
A* is complete, optimal, and optimally efficient
among all optimal search algorithms, but ...
Iterative improvement algorithms are memory
efficient, but ...
Local search

There is a cost associated with it
Online search is different from offline search

Mainly for exploration problems
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