Heuristic Search
Russell and Norvig: Chapter 4
CSMSC 421 – Fall 2005
Modified Search Algorithm
1. INSERT(initial-node,FRINGE)
2. Repeat:
If FRINGE is empty then return failure
n  REMOVE(FRINGE)
s  STATE(n)
If GOAL?(s) then return path or goal state
For every state s’ in SUCCESSORS(s)
 Create a node n’
 INSERT(n’,FRINGE)
Search Algorithms
Blind search – BFS, DFS, uniform cost


no notion concept of the “right direction”
can only recognize goal once it’s achieved
Heuristic search – we have rough idea of how
good various states are, and use this
knowledge to guide our search
Best-first search
Idea: use an evaluation function f(n) for
each node

Estimate of desirability
Expand most desirable unexpanded
node
Implementation: fringe is queue sorted
by descreasing order of desirability


Greedy search
A* search
Heuristic
Webster's Revised Unabridged Dictionary (1913) (web1913)
Heuristic \Heu*ris"tic\, a. [Greek. to discover.] Serving to discover
or find out.
The Free On-line Dictionary of Computing (15Feb98)
heuristic 1. <programming> A rule of thumb, simplification or
educated guess that reduces or limits the search for solutions
in domains that are difficult and poorly understood. Unlike
algorithms, heuristics do not guarantee feasible solutions and
are often used with no theoretical guarantee. 2. <algorithm>
approximation algorithm.
From WordNet (r) 1.6
heuristic adj 1: (computer science) relating to or using a heuristic
rule 2: of or relating to a general formulation that serves to
guide investigation [ant: algorithmic] n : a commonsense rule
(or set of rules) intended to increase the probability of solving
some problem [syn: heuristic rule, heuristic program]
Informed Search
Add domain-specific information to select the
best path along which to continue searching
Define a heuristic function, h(n), that
estimates the “goodness” of a node n.
Specifically, h(n) = estimated cost (or
distance) of minimal cost path from n to a
goal state.
The heuristic function is an estimate, based
on domain-specific information that is
computable from the current state
description, of how close we are to a goal
Greedy Search
f(N) = h(N)  greedy best-first
Robot Navigation
Robot Navigation
f(N) = h(N), with h(N) = Manhattan distance to the goal
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Robot Navigation
f(N) = h(N), with h(N) = Manhattan distance to the goal
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What happened???5
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Greedy Search
f(N) = h(N)  greedy best-first
Is it complete?
If we eliminate endless loops, yes
Is it optimal?
More informed search
We kept looking at nodes closer and closer to
the goal, but were accumulating costs as we
got further from the initial state
Our goal is not to minimize the distance from
the current head of our path to the goal, we
want to minimize the overall length of the
path to the goal!
Let g(N) be the cost of the best
path found so far between the initial
node and N
f(N) = g(N) + h(N)
Robot Navigation
f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal
8+3
8 7+4
7 6+3
6+5
6 5+6
5 4+7
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3 4
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Can we Prove Anything?
If the state space is finite and we avoid
repeated states, the search is complete,
but in general is not optimal
If the state space is finite and we do
not avoid repeated states, the search is
in general not complete
If the state space is infinite, the search
is in general not complete
Admissible heuristic
Let h*(N) be the true cost of the
optimal path from N to a goal node
Heuristic h(N) is admissible if:
0  h(N)  h*(N)
An admissible heuristic is always
optimistic
A* Search
Evaluation function:
f(N) = g(N) + h(N)
where:


g(N) is the cost of the best path found so far to N
h(N) is an admissible heuristic
Then, best-first search with this evaluation function is
called A* search
Important AI algorithm developed by Fikes and
Nilsson in early 70s. Originally used in Shakey robot.
Completeness & Optimality of A*
Claim 1: If there is a path from the
initial to a goal node, A* (with no
removal of repeated states) terminates
by finding the best path, hence is:
 complete
 optimal
requirements:


0 <   c(N,N’)
Each node has a finite number of successors
Completeness of A*
Theorem: If there is a finite path from
the initial state to a goal node, A* will
find it.
Proof of Completeness
Let g be the cost of a best path to a
goal node
No path in search tree can get longer
than g/, before the goal node is
expanded
Optimality of A*
Theorem: If h(n) is admissable, then
A* is optimal.
Proof of Optimality
f(G1) = g(G1)
N
G1
G2
Cost of best path
to a goal thru N
f(N) = g(N) + h(N)  g(N) + h*(N)
f(G1)  g(N) + h(N)  g(N) + h*(N)
Heuristic Function
Function h(N) that estimates the cost
of the cheapest path from node N to
goal node.
Example: 8-puzzle
5
8
4 2 1
7 3 6
N
1 2 3 h(N) = number of misplaced tiles
=6
4 5 6
7 8
goal
Heuristic Function
Function h(N) that estimate the cost of
the cheapest path from node N to goal
node.
Example: 8-puzzle
5
8
4 2 1
7 3 6
N
1 2 3 h(N) = sum of the distances of
every tile to its goal position
4 5 6
=2+3+0+1+3+0+3+1
7 8
= 13
goal
8-Puzzle
f(N) = h(N) = number of misplaced tiles
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5
3
4
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2
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0
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1
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
3+3
1+5
2+3
3+4
5+2
0+4
3+2
1+3
2+3
5+0
3+4
1+5
2+4
4+1
8-Puzzle
f(N) = h(N) =  distances of tiles to goal
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8-Puzzle
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N
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goal
• h1(N) = number of misplaced tiles = 6 is admissible
• h2(N) = sum of distances of each tile to goal = 13
is admissible
• h3(N) = (sum of distances of each tile to goal)
+ 3 x (sum of score functions for each tile) = 49
is not admissible
Robot navigation
f(N) = g(N) + h(N), with h(N) = straight-line distance from N to goal
Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2
Consistent Heuristic
The admissible heuristic h is consistent
(or satisfies the monotone restriction) if
for every node N and every successor N’
of N:
h(N)  c(N,N’) + h(N’)
(triangular inequality)
N
c(N,N’)
N’
h(N’)
h(N)
8-Puzzle
5
8
1
2
3
6
4
2
1
4
5
7
3
N
6
7
8
goal
• h1(N) = number of misplaced tiles
• h2(N) = sum of distances of each tile to goal
are both consistent
Robot navigation
Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2
h(N) = straight-line distance to the goal is consistent
Claims
If h is consistent, then the function f along
any path is non-decreasing:
N
f(N) = g(N) + h(N)
f(N’) = g(N) +c(N,N’) + h(N’)
c(N,N’)
N’
h(N’)
h(N)
Claims
If h is consistent, then the function f along
any path is non-decreasing:
N
f(N) = g(N) + h(N)
f(N’) = g(N) +c(N,N’) + h(N’)
h(N)  c(N,N’) + h(N’)
f(N)  f(N’)
c(N,N’)
N’
h(N’)
h(N)
Claims
If h is consistent, then the function f along
any path is non-decreasing:
N
f(N) = g(N) + h(N)
f(N’) = g(N) +c(N,N’) + h(N’)
h(N)  c(N,N’) + h(N’)
f(N)  f(N’)
c(N,N’)
N’
h(N)
h(N’)
If h is consistent, then whenever A* expands
a node it has already found an optimal path
to the state associated with this node
Avoiding Repeated States in A*
If the heuristic h is consistent, then:
Let CLOSED be the list of states
associated with expanded nodes
When a new node N is generated:


If its state is in CLOSED, then discard N
If it has the same state as another node in
the fringe, then discard the node with the
largest f
Heuristic Accuracy
h(N) = 0 for all nodes is admissible and
consistent. Hence, breadth-first and uniformcost are particular A* !!!
Let h1 and h2 be two admissible and consistent
heuristics such that for all nodes N: h1(N) 
h2(N).
Then, every node expanded by A* using h2 is
also expanded by A* using h1.
h2 is more informed than h1
Iterative Deepening A* (IDA*)
Use f(N) = g(N) + h(N) with admissible
and consistent h
Each iteration is depth-first with cutoff
on the value of f of expanded nodes
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=4
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=4
4
6
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=4
4
5
6
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
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4
Cutoff=4
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8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
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5
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6
4
Cutoff=4
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
4
6
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
4
5
6
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
4
5
7
6
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
5
4
5
7
6
6
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
5
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5
7
6
6
5
8-Puzzle
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
4
Cutoff=5
5
4
5
7
6
6
5
About Heuristics
Heuristics are intended to orient the search along
promising paths
The time spent computing heuristics must be
recovered by a better search
After all, a heuristic function could consist of solving
the problem; then it would perfectly guide the
search
Deciding which node to expand is sometimes called
meta-reasoning
Heuristics may not always look like numbers and
may involve large amount of knowledge
What’s the Issue?
Search is an iterative local procedure
Good heuristics should provide some
global look-ahead (at low computational
cost)
Another approach…
for optimization problems

rather than constructing an optimal
solution from scratch, start with a
suboptimal solution and iteratively improve
it
Local Search Algorithms




Hill-climbing or Gradient descent
Potential Fields
Simulated Annealing
Genetic Algorithms, others…
Hill-climbing search
If there exists a successor s for the current state n
such that


h(s) < h(n)
h(s) <= h(t) for all the successors t of n,
then move from n to s. Otherwise, halt at n.
Looks one step ahead to determine if any successor
is better than the current state; if there is, move to
the best successor.
Similar to Greedy search in that it uses h, but does
not allow backtracking or jumping to an alternative
path since it doesn’t “remember” where it has been.
Not complete since the search will terminate at "local
minima," "plateaus," and "ridges."
Hill climbing on a surface of
states
Height Defined
by Evaluation
Function
Hill climbing
Steepest descent (~ greedy best-first
with no search)  may get stuck into
local minimum
Robot Navigation
Local-minimum problem
f(N) = h(N) = straight distance to the goal
Examples of problems with HC
applet
Drawbacks of hill climbing
Problems:
 Local Maxima: peaks that aren’t the
highest point in the space
 Plateaus: the space has a broad flat region
that gives the search algorithm no direction
(random walk)
 Ridges: flat like a plateau, but with dropoffs
to the sides; steps to the North, East, South
and West may go down, but a step to the
NW may go up.
Remedy:

Introduce randomness
 Random restart.
Some problem spaces are great for hill climbing
and others are terrible.
What’s the Issue?
Search is an iterative local procedure
Good heuristics should provide some
global look-ahead (at low computational
cost)
Hill climbing example
start
2 8 3
1 6 4
7
5
-5
h = -4
-5
2 8 3
1
4 h = -3
7 6 5
-3
h = -3
goal
1 2 3
8
4 h=0
7 6 5
-2
1 2 3
8 4 h = -1
7 6 5
-4
2
3
1 8 4
7 6 5
2 3
1 8 4 h = -2
7 6 5
-4
f(n) = -(number of tiles out of place)
Example of a local maximum
start
1 2 5
7 4
8 6 3
1 2 5
7 4
8 6 3
1 2 5
7 4 -4
8 6 3
-3
-4
1 2 5
7 4 -4
8 6 3
goal
1 2 5
7 4 0
8 6 3
Potential Fields
Idea: modify the heuristic function
Goal is gravity well, drawing the robot toward
it
Obstacles have repelling fields, pushing the
robot away from them
This causes robot to “slide” around obstacles
Potential field defined as sum of attractor
field which get higher as you get closer to the
goal and the indivual obstacle repelling field
(often fixed radius that increases
exponentially closer to the obstacle)
Does it always work?
No.
But, it often works very well in practice
Advantage #1: can search a very large
search space without maintaining fringe
of possiblities
Scales well to high dimensions, where
no other methods work
Example: motion planning
Advantage #2: local method. Can be
done online
Example: RoboSoccer
All robots have same field: attracted to the ball
Repulsive potential to other players
Kicking field: attractive potential to the ball
and local repulsive potential if clase to the
ball, but not facing the direction of the
opponent’s goal. Result is tangent, player
goes around the ball.
Single team: kicking field + repulsive field to
avoid hitting other players + player position
fields (paraboilic if outside your area of the
field, 0 inside). Player nearest to the ball has
the largest attractive coefficient, avoids all
players crowding the ball.
Simulated annealing
Simulated annealing (SA) exploits an analogy between
the way in which a metal cools and freezes into a
minimum-energy crystalline structure (the annealing
process) and the search for a minimum [or maximum] in
a more general system.
SA can avoid becoming trapped at local minima.
SA uses a random search that accepts changes that
increase objective function f, as well as some that
decrease it.
SA uses a control parameter T, which by analogy with the
original application is known as the system
“temperature.”
T starts out high and gradually decreases toward 0.
Simulated annealing (cont.)
A “bad” move from A to B is accepted with
a probability
e
(f(B)-f(A)/T)
The higher the temperature, the more
likely it is that a bad move can be made.
As T tends to zero, this probability tends to
zero, and SA becomes more like hill
climbing
If T is lowered slowly enough, SA is
complete and admissible.
The simulated annealing
algorithm
Simulated Annealing
applet
Successful application: circuit routing,
traveling sales person (TSP)
Summary: Local Search Algorithms
Steepest descent (~ greedy best-first
with no search)  may get stuck into
local minimum
Better Heuristics: Potential Fields
Simulated annealing
Genetic algorithms
When to Use Search Techniques?
The search space is small, and


There is no other available techniques, or
It is not worth the effort to develop a more
efficient technique
The search space is large, and


There is no other available techniques, and
There exist “good” heuristics
Summary
Heuristic function
Best-first search
Admissible heuristic and A*
A* is complete and optimal
Consistent heuristic and repeated
states
Heuristic accuracy
IDA*