16890 unit 2 heuristic search techniques

Artificial Intelligence
Course Code: ECE434

Heuristic Search Techniques

Suryender Kumar Sharma
Suryender.16890@lpu.co.in
Asst. Prof. D: EEE - SEE- R & A
Lovely Professional University

Review
 General Purpose Problem Solving
 Production System
 State-Space Search
 Control Strategies
 Characteristics of Problems
 Exhaustive Search Techniques
(BFS,DFS,DFID,BS)
 Analysis of Search Methods

Travelling Salesman Problem (TSP)

 Statement:
To find the shortest route of visiting all the
cities once and returning back to the starting
point.
Assume that there are n cities and distance
between each pair of cities is given.
Factorial n-1 paths for n cities

Travelling Salesman Problem (TSP)

Start Generating complete paths, keep track of
the shortest path found so far

Stop exploring any path as soon as its partial
length becomes greater than the shortest path
found so far

TSP
n=5
No of paths  (n-1)! = 4! = 24 C1

7 11

12
15
(C2) 20
C3
10 13

C4
12
17
5

C5

Using problem specific knowledge
to aid searching
• Without incorporating
knowledge into searching, one
can have no bias (i.e. a Search
everywhere!!
preference) on the search space.

• Without a bias, one is forced to
look everywhere to find the
answer. Hence, the complexity
of uninformed search is
intractable.

Using problem specific knowledge
to aid searching
• With knowledge, one can search the state space as if he was
given “hints” when exploring a maze.
– Heuristic information in search = Hints
• Leads to dramatic speed up in efficiency.
A

B C D E

F G H I J
Search only in
this subtree!! K L M N

O

More formally, why heuristic
functions work?

• In any search problem where there are at most b choices at
each node and a depth of d at the goal node, a naive search
algorithm would have to, in the worst case, search around
O(bd) nodes before finding a solution (Exponential Time
Complexity).

• Heuristics improve the efficiency of search algorithms by
reducing the effective branching factor from b to (ideally)
a low constant b* such that
– 1 =< b* << b


 General Purpose Heuristics:
– Are useful in various problem domains
 Special purpose Heuristics:
– Are domain specific

 General-purpose heuristics.
 Best-first search.
 Branch and bound search (uniform cost search).
 A* algorithm
 Hill climbing.
 Beam search.

General Purpose Heuristics

1. For combinatorial is nearest neighbor algorithms that
work by selecting the locally superior alternative
2. Mathematical analysis is not possible to perform
 It is fun to see a program do something intelligent
than to prove it
 AI Problem domain are complex, not possible to
produce analytical proof that will work
 not possible to make statistical analysis of
program behavior

Heuristic Functions
• A heuristic function is a function f(n) that gives an estimation on the “cost” of
getting from node n to the goal state – so that the node with the least cost
among all possible choices can be selected for expansion first.

• Three approaches to defining f:

– f measures the value of the current state (its “goodness”)

– f measures the estimated cost of getting to the goal from the current state:
• f(n) = h(n) where h(n) = an estimate of the cost to get from n to a goal

– f measures the estimated cost of getting to the goal state from the current state and
the cost of the existing path to it. Often, in this case, we decompose f:
• f(n) = g(n) + h(n) where g(n) = the cost to get to n (from initial state)

Approach 1: f Measures the
Value of the Current State
• Usually the case when solving optimization problems
– Finding a state such that the value of the metric f is optimized

• Often, in these cases, f could be a weighted sum of a set of
component values:

– N-Queens
• Example: the number of queens under attack …

Approach 2: f Measures the Cost to the
Goal

A state X would be better than a state Y if the estimated
cost of getting from X to the goal is lower than that of Y
– because X would be closer to the goal than Y

• 8–Puzzle
h1: The number of misplaced tiles
(squares with number).
h2: The sum of the distances of the tiles
from their goal positions.

Approach 3: f measures the total cost of the
solution path (Admissible Heuristic Functions)

• A heuristic function f(n) = g(n) + h(n) is admissible if h(n) never
overestimates the cost to reach the goal.
– Admissible heuristics are “optimistic”: “the cost is not that much …”
• However, g(n) is the exact cost to reach node n from the initial state.
• Therefore, f(n) never over-estimate the true cost to reach the goal state
through node n.
• Theorem: A search is optimal if h(n) is admissible.
– I.e. The search using h(n) returns an optimal solution.
• Given h2(n) > h1(n) for all n, it’s always more efficient to use h2(n).
– h2 is more realistic than h1 (more informed), though both are optimistic.

Traditional informed search
strategies
• Greedy Best first search
– “Always chooses the successor node with the best f value”
where f(n) = h(n)
– We choose the one that is nearest to the final state among all
possible choices

• A* search
– Best first search using an “admissible” heuristic function f
that takes into account the current cost g
– Always returns the optimal solution path

Informed Search Strategies

Best First Search


Greedy Search
eval-fn: f(n) = h(n)

Greedy Search
Start State Heuristic: h(n)
A 75
118 A 366
140 B B 374
C
111 C 329
E
D 80 99 D 244
E 253
G F
F 178
97
G 193
H 211 H 98
101 I 0
I
Goal f(n) = h (n) = straight-line distance heuristic

Greedy Search: Tree Search
Start
A

Start
A 75
118

[329] 140 [374] B
C

[253] E

Start
A 75
118

[329] 140 [374] B
C

[253] E
80 99

[193] [178]
G F
[366] A

Start
A 75
118

[329] 140 [374] B
C

[253] E
80 99

[193] [178]
G F
[366] A
211

[253] I [0]
E

Goal

Start
A 75
118

[329] 140 [374] B
C

[253] E
80 99

[193] [178]
G F
[366] A
211

[253] I [0]
E

Goal
Path cost(A-E-F-I) = 253 + 178 + 0 = 431
dist(A-E-F-I) = 140 + 99 + 211 = 450

Greedy Search: Optimal ?
A 75
118 A 366
140 B B 374
C
111 C 329
E
D 80 99 D 244
E 253
G F
F 178
97
G 193
H 211 H 98
101 I 0
I
dist(A-E-G-H-I) =140+80+97+101=418

Greedy Search: Complete ?
A 75
118 A 366
140 B B 374
C
111 ** C 250
E
D 80 99 D 244
E 253
G F
F 178
97
G 193
H 211 H 98
101 I 0
I

Start
A 75
118

[250] 140 [374] B
C

[253] E

Start
A 75
118

[250] 140 [374] B
C

[253] E
111
[244] D

Start
A 75
118

[250] 140 [374] B
C

[253] E
111
[244] D

Infinite Branch !
[250] C

Start
A 75
118

[250] 140 [374] B
C

[253] E
111
[244] D

Infinite Branch !
[250] C

[244] D

Greedy Search: Time and Space
Complexity ?
Start
A
118 75 • Greedy search is not optimal.
140 B
C • Greedy search is incomplete
111
E without systematic checking of
D 80 99
repeated states.
G F
• In the worst case, the Time
97 and Space Complexity of
H 211 Greedy Search are both O(bm)
101 Where b is the branching factor and m the
I maximum path length
Goal


Branch and bound search
Uniform Cost Search
f(n)=g(n)

Uniform Cost Search (UCS)

5 2
[5] [2]
1 4 1 7
[6] [3] [9]
[9]
Goal state
4 5

[x] = g(n) [7] [8]
path cost of node n


5 2
[5] [2]


5 2
[5] [2]

1 7
[3] [9]


5 2
[5] [2]

1 7
[3] [9]
4 5

[7] [8]


5 2
[5] [2]
1 4 1 7
[6] [3] [9]
[9]
4 5

[7] [8]


5 2
[5] [2]
Goal state
path cost 1 4 1 7
g(n)=[6]
[3] [9]
[9]
4 5

[7] [8]


 In case of equal step costs, Breadth First search finds
the optimal solution.

 For any step-cost function, Uniform Cost search
expands the node n with the lowest path cost.

 UCS takes into account the total cost: g(n).

 UCS is guided by path costs rather than depths. Nodes
are ordered according to their path cost.


 Main idea: Expand the cheapest node. Where the cost is the path
cost g(n).

 Implementation:
Enqueue nodes in order of cost g(n).
QUEUING-FN:- insert in order of increasing path cost.
Enqueue new node at the appropriate position in the queue so that we
dequeue the cheapest node.

 Complete? Yes.
 Optimal? Yes, if path cost is non decreasing function of depth
 Time Complexity: O(bd)
 Space Complexity: O(bd), note that every node in the fringe keep in
the queue.

Branch and bound search
(uniform cost search).
• Cost function g(X) is designed that gives
cumulative expense to the path from start
node to the current node X.
• Least cost path obtained to far is expanded at
each iteration till we reach to the Goal State
• There can be incomplete paths as the shortest
one is always extended one level further.
• Can create as many new incomplete paths

Algorithm: Branch and Bound
Input: START and GOAL states
Local variables: OPEN, CLOSED, NODE, SUCCs, FOUND;
OUTPUT: Yes Or No
Method:
• Initially store the root node with g(root)=0 in a open list and CLOSED= and
FOUND= false;
• While(OPEN≠ and FOUND= false) do
{
• Remove the top element from OPEN and call it NODE;
• If NODE= GOAL Node then FOUND= true else
• {
• Put NODE in closed list:
• Find SUCC’s of NODE. If any and compute their ‘g’ values and store them in
OPEN list
• Sort all the nodes in the open list based on their cost function values:
}
} /* end of while*/
If FOUND= true then return Yes else return No
• Stop


A* Search

eval-fn: f(n)=g(n)+h(n)

A* (A Star)

• Greedy Search minimizes a heuristic h(n) which is an
estimated cost from a node n to the goal state. However,
although greedy search can considerably cut the search time
(efficient), it is neither optimal nor complete.

• Uniform Cost Search minimizes the cost g(n) from the
initial state to n. UCS is optimal and complete but not
efficient.

• New Strategy: Combine Greedy Search and UCS to get an
efficient algorithm which is complete and optimal.

A* (A Star)

• A* uses a heuristic function which combines
g(n) and h(n): f(n) = g(n) + h(n)

• g(n) is the exact cost to reach node n from
the initial state. Cost so far up to node n.

• h(n) is an estimation of the remaining cost to
reach the goal.

A* (A Star)

g(n)

f(n) = g(n)+h(n) n

h(n)

A* Search
A 75
118 A 366
140 B B 374
C
111 C 329
E
D 80 99 D 244
E 253
G F
F 178
97
G 193
H 211 H 98
101 I 0
I
Goal f(n) = g(n) + h (n)
g(n): is the exact cost to reach node n from the initial state.

A* Search: Tree Search

A Start


A Start

118 75
140

[447] C E [393] B [449]


A Start

118 75
140

[447] C E [393] B [449]
80 99

[413] G F [417]


A Start

118 75
140

[447] C E [393] B [449]
80 99

[413] G F [417]

97
[415] H


A Start

118 75
140

[447] C E [393] B [449]
80 99

[413] G F [417]

97
[415] H

101
Goal I [418]


A Start

118 75
140

[447] C E [393] B [449]
80 99

[413] G F [417]

97
[415] H I [450]

101
Goal I [418]

A* with h() not Admissible

h() overestimates the cost to reach
the goal state

A* Search: h not admissible !
A 75
118 A 366
140 B B 374
C
111 C 329
E
D 80 99 D 244
E 253
G F
F 178
97
G 193
H 211 H 138
101 I 0
I
Goal f(n) = g(n) + h (n) – (H-I) Overestimated
g(n): is the exact cost to reach node n from the initial state.


A Start

118 75
140

[447] C E [393] B [449]
80 99

[413] G F [417]

97
[455] H


A Start

118 75
140

[447] C E [393] B [449]
80 99

[413] G F [417]

97
[455] H Goal I [450]


A Start

118 75
140

[447] C E [393] B [449]
80 99

[473] [413] G F [417]
D

97
[455] H Goal I [450]


A Start

118 75
140

[447] C E [393] B [449]
80 99

[473] [413] G F [417]
D

97
[455] H Goal I [450]

A* not optimal !!!

A* Search: Analysis
•A* is complete except if there is an infinity of
Start nodes with f < f(G).
A 75
118
•A* is optimal if heuristic h is admissible.
140 B
C •Time complexity depends on the quality of
111 heuristic but is still exponential.
E
•For space complexity, A* keeps all nodes in
D 80 99
memory. A* has worst case O(bd) space
complexity, but an iterative deepening version is
G F possible (IDA*).
97

H 211
101
I
Goal

A* Algorithm
1. Search queue Q is empty.
2. Place the start state s in Q with f value h(s).
3. If Q is empty, return failure.
4. Take node n from Q with lowest f value.
(Keep Q sorted by f values and pick the first element).
5. If n is a goal node, stop and return solution.
6. Generate successors of node n.
7. For each successor n’ of n do:
a) Compute f(n’) = g(n) + cost(n,n’) + h(n’).
b) If n’ is new (never generated before), add n’ to Q.
c) If node n’ is already in Q with a higher f value, replace it with
current f(n’) and place it in sorted order in Q.
End for


Iterative Deepening A*

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

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’) N

c(N,N’)

(triangular inequality) N’ h(N)

• A consistent heuristic is admissible. h(N’)

IDA* Algorithm

• In the first iteration, we determine a “f-cost limit” – cut-off value
f(n0) = g(n0) + h(n0) = h(n0), where n0 is the start node.

• We expand nodes using the depth-first algorithm and backtrack whenever
f(n) for an expanded node n exceeds the cut-off value.

• If this search does not succeed, determine the lowest f-value among the
nodes that were visited but not expanded.

• Use this f-value as the new limit value – cut-off value and do another depth-
first search.

• Repeat this procedure until a goal node is found.

Hill Climbing
Local variables: OPEN, CLOSED, NODE, SUCCs, FOUND;
OUTPUT: Yes Or No
Method:
• Initially store the root node in a open list (maintained as stack)
FOUND= false;
• While(OPEN≠ and FOUND= false) do
{
• Remove the top element from OPEN and call it NODE;
• If NODE= GOAL Node then FOUND= true else
{
• Find SUCC’s of NODE If any:
• Sort SUCC’s by estimated cost from NODE to goal state and add them to the
front of OPEN list:
} /* end of while*/
If FOUND= true then return Yes else return No
• Stop

Hill Climbing: Disadvantages
• Fail to find a solution
• Either Algo may terminate not by finding a
goal state but by getting to a state from
which no better state can be generated.

• This happen if program reached
– Local maximum,
– Plateau,
– Ridge.
86

Local maximum
A state that is better than all of its
neighbours, but not better than some
other states far away.

88

Plateau
A flat area of the search space in which all
neighbouring states have the same value.
• requiring random walk

89

Ridge
Special kind of local maximum.
The orientation of the high region, compared
to the set of available moves, makes it
impossible to climb up.
Many moves executed serially may increase
the height.

90

Ways Out
• Backtrack to some earlier node and try going in
a different direction. (good way in dealing with
local maxima)
• Make a big jump to try to get in a new section.
(good way in dealing with plateaus)
• Moving in several directions at once. (good
strategy for dealing with ridges)
91

• Hill climbing is a local method:
Decides what to do next by looking only at the
“immediate” consequences of its choices rather
than by exhaustively exploring all the
consequences.
• Global information might be encoded in
heuristic functions.

Beam search
Local variables: OPEN, NODE, SUCC’s, W_OPEN, FOUND;
OUTPUT: Yes Or No
Method:
• NODE= Root_node: FOUND= false:
If NODE= GOAL Node then FOUND= true else Find SUCC’s of NODE If any: with
estimated cost and store in OPEN list.
While(FOUND=False and not able to proceed further) do
{
• Sort OPEN list:
• Select top W elements from OPEN list and put it in W_OPEN list and empty
open list:
• For each node from W_OPEN list
{
• If NODE=Goal State then found = true else find SUCC’s of NODE if any with
its estimated cost and store in open list
}
} /* end of while*/
• If FOUND= true then return Yes else return No
• Stop

Beam search

Continue till goal state is found
or not able to proceed further

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

Constraint Satisfaction
• Constraint Satisfaction problems in AI have
goal of discovering some problem state
that satisfies a given set of constraints.

• Design tasks can be viewed as constraint
satisfaction problems in which a design
must be created within fixed limits on
time, cost, and materials.

Constraint satisfaction
• Constraint satisfaction is a search procedure that operates in a space of
constraint sets. The initial state contains the constraints that are
originally given in the problem description. A goal state is any state
that has been constrained “enough” where “enough”must be defined
for each problem. For example, in cryptarithmetic, enough means that
each letter has been assigned a unique numeric value.

• Constraint Satisfaction is a two step process:
– First constraints are discovered and propagated as far as possible
throughout the system.
– Then if there still not a solution, search begins. A guess about
something is made and added as a new constraint.

97

Constraint Satisfaction: Example
• Cryptarithmetic Problem:
SEND
+MORE
-----------
MONEY
Initial State:
• No two letters have the same value
• The sums of the digits must be as shown in the
problem
Goal State:
• All letters have been assigned a digit in such a way that
all the initial constraints are satisfied.

Cryptasithmetic Problem: Constraint
Satisfaction
• The solution process proceeds in cycles. At each cycle, two
significant things are done:
1. Constraints are propagated by using rules that correspond to the
properties of arithmetic.
2. A value is guessed for some letter whose value is not yet
determined.

A few Heuristics can help to select the best guess to try first:

• If there is a letter that has only two possible values and other with
six possible values, there is a better chance of guessing right on
the first than on the second.
• Another useful Heuristic is that if there is a letter that participates
in many constraints then it is a good idea to prefer it to a letter
that participates in a few.

Solving a Cryptarithmetic Problem
Initial state
M=1
SEND
S= 8 or 9
O = 0 or 1 => O =0
+MORE
N= E or E+1 -> N= E+1
C2 = 1
-------------
N+R >8
E≠ 9
MONEY

E=2
N=3
R= 8 or 9
2+D = Y or 2+D = 10 +Y
C1= 0 C1= 1
2+D =Y 2+D = 10 +Y
N+R = 10+E D = 8+Y
R =9 D = 8 or 9
S =8
D=8 D=9
Y= 0 ; Conflict Y =1 ; Conflict

Two-step process:
1. Constraints are discovered and
propagated as far as possible.
2. If there is still not a solution, then search
begins, adding new constraints.

Two kinds of rules:
1. Rules that define valid constraint propagation.
2. Rules that suggest guesses when necessary.

When to Use Search Techniques

• The search space is small, and
– There are 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

Conclusions

• Frustration with uninformed search led to the idea
of using domain specific knowledge in a search so
that one can intelligently explore only the relevant
part of the search space that has a good chance of
containing the goal state. These new techniques are
called informed (heuristic) search strategies.

• Even though heuristics improve the performance of
informed search algorithms, they are still time
consuming especially for large size instances.

16890 unit 2 heuristic search techniques

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