The document discusses various informed search algorithms including A*, greedy search, and uniform cost search. It provides instructional objectives for learning about heuristic functions, designing heuristics for problems, and comparing heuristic functions. Key aspects of A* search are summarized, including that it uses an admissible heuristic function to find optimal solutions, and conditions like admissibility and consistency that guarantee its optimality.

1. Informed Search
1AI, Subash Chandra Pakhrin

2. Instructional Objectives
• The students will learn the following strategies
for informed search:
– A*
– Greedy search
– Uniform Cost Search
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3. Instructional Objectives
• At the end of this lesson the student should be
able to do the following:
– Understand what is a heuristic function.
– They should be able to design heuristic functions
for a given problem.
– They should be able to determine if A* uses an
admissible heuristic function, it will produce an
optimum solution.
– They should learn how to compare two heuristic
functions.
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4. Informed Search
• We have seen un-informed search methods
that systematically explore the state space and
find the goal.
• Inefficient in most cases
• Informed search methods use problem
specific knowledge, are more efficient
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5. Heuristics
• “Heuristics are criteria, methods or principles
for deciding which among several alternatives
course of action promises to be the most
effective in order to achieve some goal”
Judea Pearl
Can we use heuristics to identify the most
promising search path
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6. Heuristic Distance
• The distance as the crow fly between two
places even though there is no road that goes
between those two places.
• Being close is actually not a good thing
because it may result in dead end.
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3
S
G
D
E
B
A
C
5
4
4
6
3
5
7+
6
7+

7. Heuristic Distance
• Objective of enlisted search algorithm is to be
close to the goal.
– Hill Climbing
– Beam Search
• When Beam Search value (b) = 2
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C B A D
Reject
Keep Keep
Reject

8. Remember
• FRINGE: Nodes not yet expanded
• CLOSED: Nodes that are expanded
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9. Branch and Bound
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S
A B
B D
C G
A C
D E
3 5
7
1111 12 15
996
C, D, E need not be
extended any
further because
their accumulated
length so far is less
than or equal to the
length of the goal

10. Branch and Bound + Extended List
• Don’t extend the node
which has been
extended
• Saved some work
because no extension
• Vast area of tree has
been pruned away
and don’t have to be
examined at all
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S
A B
B D
G
A C
E
3 5
7
11 15
996

11. Example of Heuristic Function
• A heuristic function at a node n is an estimate of
the optimum cost from the current node to a
goal. Denoted by h(n)
h(n) = estimated cost of the cheapest path from
node n to a goal node
• Example
– Want path from Kathmandu to Delhi
– Heuristic from Kathmandu may be straight – line
distance between Kathmandu to Delhi
– H(Kathmandu) = Euclidean Distance (Kathmandu,
Delhi)
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12. Heuristics - example
8 – puzzle: Number of tiles out of place
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h1(n) = 5
h1(n) = 5; is an underestimate of the actual
number of states to move into goal state

13. Heuristics - example
8-puzzle: Manhattan Distance (distance tile is out of
place)
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h2(n) = 6; is an underestimate of the actual number of
move required to move from initial state to goal state
h n(n) = max{h1(n), h2(n)}= 6

14. Best – First Search
Priority queue of nodes to be explored
Cost function f(n) applied to each node
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5
A
B
C
D
3 1
E F
4 6HG
5
7
I J
STEP 1:
CLOSED: A
OPEN: BCD
STEP 2: D expanded
CLOSED: AD
OPEN: BCEF

15. Best – First Search
Let Fringe be a priority queue containing the initial state
LOOP
if Fringe is empty return failure
Node <- remove-first (Fringe)
if Node is a goal
then return the path from initial state to Node
else generate all successors of Node, and
put the newly generated nodes into fringe
according to their f values
End LOOP
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16. Greedy Search
• Idea: Expand node with the smallest
estimated cost to reach the goal
• Use heuristic function f(n) = h(n)
– Not optimal
– Incomplete
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17. Greedy Best First Search
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18. Solution using Greedy Best First Search
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Arad
366

19. Solution using Greedy Best First Search
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20. A* Search
• Hart, Nilsson & Rafael 1968
– Best first search with f(n) = g(n) + h(n)
– Where g(n) = sum of edge cost from start to n
– And h(n) = estimate of lowest cost path from node n
to goal
– If h(n) is admissible then search will find optimal
solution.
• A* = Branch and Bound + Extended List +
Admissible Heuristic
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21. Consistency ( Monotonicity )
• A heuristic is said to be consistent if for any
node N and any successor N’ of N , estimated
cost to reach to the goal from node N is less
than the sum of step cost from N to N’ and
estimated cost from node N’ to goal node.
h(n) ≤ c(n, n’) + h(n’)
Where;
h(n) = Estimated cost to reach to the goal node from
node n
• c(n, n’) = actual cost from n to n’
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22. Admissible
• A heuristic h(n) is admissible 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
the cost to reach the goal, i.e., it is optimistic.
• Theorem: If h(n) is admissible, A* using TREE-
SEARCH is optimal.
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23. Optimality of A* (Proof)
• Suppose some suboptimal goal G2 has been
generated and is in the fringe. Let n be an
unexpanded node in the fringe such that n is on a
shortest path to an optimal goal G.
We want to prove:
f(n) < f(G2)
(then A* will prefer n over G2)
f (G2) = g(G2) since h(G2) = 0
g(G2) > g(G) since G2 is suboptimal
f(G) = g(G) since h(G) = 0
f(G2) > f (G) from aboveAI, Subash Chandra Pakhrin 23

24. Optimality of A* (Proof)
f(G2) > f(G) (from above)
h(n) ≤ h*(n), (since h is admissible(under –
estimate))
g(n) + h(n) ≤ g(n) + h*(n)
Since g(n) + h(n) = f(n) &
g(n) + h*(n) = f(G)
f(n) ≤ f(G) < f (G2)
Hence f (G2) > f(n), and A* will never select G2 for
expansion
• Therefore A* does find the optimal solutionAI, Subash Chandra Pakhrin 24

25. Theorem: If h(n) is consistent , then the values of f(n)
along the path are non-decreasing.
• A heuristic is consistent (or monotonic) if for
every node n, every successor n' of n
generated by any action a:
h(n) ≤ c(n, a, n’) + h(n’)
• If h is consistent, we have:
f(n’) = g(n’) + h(n’)
= g(n) + c(n, a, n’) + h(n’)
≥ g(n) + h(n) = f(n)
i.e., f(n) is non-decreasing along any path.
Theorem: If h(n) is consistent, A* using GRAPH-
SEARCH is optimal
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Triangle Inequality

26. A* Search
Lower Bound on the path (crow fly) =
Accumulated Distance + Airline Distance
*** Forget Everything Use the concept of Airline
Distance ***
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S
G
D
E
B
A
C
5
4
4
6
3
5
7+
6
7+
3

27. A* Search
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S
A B
B D
G
A C
3 + 7+ = 10+
7 + 6 = 13
6 + 5 = 11
5 + 6 = 11
9 + 7+ = 16+
11 + 0 = 11
9 + 7+ = 16+
Tie: Lexically Least

28. A* Search
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29. A* Search
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30. A* Search
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31. Algorithm A* (graphs)
OPEN = nodes on frontier. CLOSED = expanded nodes.
OPEN = {<s, nil>}
While OPEN is not empty
remove from OPEN the node <n, p> with minimum f(n)
place <n, p> on CLOSED
if n is a goal node, return success (path p)
for each edge connecting n & m with cost c
if <m, q> is on CLOSED and { p| e} is cheaper than q
then remove n from CLOSED, put <m, { p | e} > on OPEN
else if <m, q> is on OPEN and {p| e} is cheaper than q
then replace q with {p| e}
else if m is not on OPEN put <m, {p| e} > on OPEN
Return failure
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32. A* Search
• Optimal – optimally efficient
• Complete
• Number of nodes searched still exponential in
the worst case
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33. Admissible Heuristics
• An admissible heuristic is one that never
overestimates the cost to reach the goal.
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S
C
A
B
G
1
1
10
1
100
100
0
0
The graph satisfy admissibility but does not satisfy consistency
Estimated Distance {H( x , G)} ≤ Actual Distance {D(x , G)}
| H( x , G) - H( y , G)| ≤ D ( x , y)

34. A*
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S
G
C
BB
C
1+0 = 1
11+0 = 11
111+0 = 111
1+100 = 101
2+0 = 2
Do not extend C because it has been extended

35. Conditions
• Admissibility
Estimated Distance {H( x , G)} ≤ Actual Distance {D(x
, G)}
• Consistency
| H( x , G) - H( y , G)| ≤ D ( x , y)
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36. Admissible + Consistence
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S
C
A
B
G
1
1
10
2
100
2
0
0

37. A* Search
• Admissibility: Provided a solution exists, the
first solution found is an optimal solution
• Condition for admissibility
– State space graph
• Every node has a finite number of successors
• Every arc in the graph has a cost greater than some є > 0
– Heuristic function
• For every node n, h(n) < h*(n)
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38. Admissibility of A*
• A* is optimally efficient for a given heuristic of the
optimal search algorithms that expand search
paths from the root node, it can be shown that no
other optimal algorithm will expand fewer nodes
and find a solution
• Monotone heuristic: along any path the f - cost
never decreases.
– If this property does not hold we can use the following
trick ( m is a child of n)
f(m) = max ( f(n), g(m)+h(m))
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39. Completeness of A*
• Let G be an optimum goal state.
• A* cannot reach a goal state only if there are
infinitely many nodes where f(n) ≤ f*
• Can only happen if either happens:
– A node with infinite branching factor
– A path with finite cost but infinitely many nodes
• Thus A* is complete.
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40. Optimality
• Lemma: A* expands nodes in order of increasing f
value.
• Gradually adds “f-contours” of nodes.
• Contour I has all nodes with f= f i, where f i< fi+1.
• That is to say, nodes inside a given contour have
f-costs less than or equal to contour value.
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41. Maze Traversal (for A* Search)
Problem: To get from square A3 to
square E2, one step at a time,
avoiding obstacles (black squares).
Operators: (in order)
Go left(n), Go down(n), Go right(n)
Each operator costs 1.
Heuristic: Manhattan distance
• Start Position: A3
• Goal: E2
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42. Maze Traversal (for A* Search)
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43. Using multiple heuristics
• Suppose you have identified a number of non-
overestimating heuristics for a problem:
h1(n), h2(n), … , h k(n)
Then
max (h1(n), h2(n), … , h k(n))
Is a more powerful non-overestimating
heuristic.
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