Heuristic Search
Russell and Norvig: Chapter 4
Slides adapted from:
robotics.stanford.edu/~latombe/cs121/2003/home.htm
Heuristic Search
Blind search totally ignores where the
goals are.
A heuristic function that gives us an
estimate of how far we are from a goal.

Example:
How do we use heuristics?
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) = 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
Greedy Best-First Search
Path search(start, operators, is_goal) {
fringe = makeList(start);
while (state=fringe.popFirst()) {
if (is_goal(state))
return pathTo(state);
S = successors(state, operators);
fringe =
insert(S,
fringe);
}
return NULL;
}
Order the nodes in the fringe in increasing values of h(N)
Robot Navigation
Robot Navigation
f(N) = h(N), with h(N) = Manhattan distance to the goal
8
7
7
6
5
4
5
4
3
3
2
6
7
6
8
7
3
2
3
4
5
6
5
1
0
1
2
4
5
6
5
4
3
2
3
4
5
6
Robot Navigation
f(N) = h(N), with h(N) = Manhattan distance to the goal
8
7
7
6
5
4
5
4
3
3
2
6
77
6
8
7
3
2
3
4
5
6
5
1
00
1
2
4
5
6
5
4
3
2
3
4
5
6
Properties of the Greedy
Best-First Search
If the state space is finite and we avoid
repeated states, the best-first 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 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
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 = 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
A* Search
A* search combines Uniform-cost and Greedy
Best-first 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
f(N) is the estimated cost of cheapest solution
THROUGH N
0 <   c(N,N’) (no negative cost steps).
Order the nodes in the fringe in increasing
values of f(N)
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
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
goal
0+4
3+2
1+3
2+3
5+0
3+4
1+5
2+4
4+1
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
4 3+8
3 2+9
2 3+10
3 4
7+2
7
6+1
6
5
5+6
5 4+7
4 3+8
3
3 2+9
2 1+10
1 0+11
0 1
6
5
2
4
7+0
7 6+1
6
8+1
8 7+2
7 6+3
6 5+4
5 4+5
4 3+6
3 2+7
2 3+8
3 4
5
5
6
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:
N
c(N,N’)
h(N)  c(N,N’) + h(N’)
N’
h(N’)
(triangular inequality)
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
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
Complexity of Consistent A*
Let s be the size of the state space
Let r be the maximal number of states
that can be attained in one step from
any state
Assume that the time needed to test if
a state is in CLOSED is O(1)
The time complexity of A* is O(s r log s)
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

h2 dominates h1
Which heuristic for 8-puzzle is better?
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
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=4
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=4
4
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=4
4
5
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
5
4
Cutoff=4
4
5
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
6
5
4
5
6
6
4
Cutoff=4
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=5
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=5
4
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=5
4
5
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=5
4
5
7
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=5
5
4
5
7
6
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
4
Cutoff=5
5
4
5
7
6
6
5
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
8-Puzzle
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
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
Greedy Best-first search
Admissible heuristic and A*
A* is complete and optimal
Consistent heuristic and repeated states
Heuristic accuracy
IDA*