A* Search Algorithm

May 6, 20213 likes18,102 views

Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search, Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A * algorithm, Iterative Deepening A* , Recursive Best First Search, Pruning the CLOSED and OPEN Lists

Topic To Be Covered:
A* Search Algorithm
Jagdamba Education Society's
SND College of Engineering & Research Centre
Department of Computer Engineering
SUBJECT: Artificial Intelligence & Robotics
Lecture No-9(ii)
Prof.Dhakane Vikas N

A* Search Algorithm
 This is informed search technique also called as HEURISTIC search.
 This algo. Works using heuristic value.
 A* uses h(n)->Heuristic function & g(n)->Cost to reach the node ‘n’ from
start state.
 Find shortest path though search spaces.
 Estimated Cost f(n)=g(n)+h(n)
 A* gives Fast & Optimal result as compared with previous algorithms.
 Space & Time Complexity of BFS is also O(V+E) where V is vertices and E
is edges.
 Also Written as:-O(b) ^d Where, b->Branching factor d->depth

A* Search Algorithm
 A* Algorithm extends the path that minimizes the following function-
f(n) = g(n) + h(n)
Here,
 ‘n’ is the last node on the path
 g(n) is the cost of the path from start node to node ‘n’
 h(n) is a heuristic function that estimates cost of the cheapest path from
node ‘n’ to the goal node
Algorithm-
 The implementation of A* Algorithm involves maintaining two lists-
OPEN and CLOSED.
 OPEN contains those nodes that have been evaluated by the heuristic
function but have not been expanded into successors yet.
 CLOSED contains those nodes that have already been visited.
The algorithm is as follows-

A* Search Algorithm
The algorithm is as follows-
Step-01:
 Define a list OPEN.
 Initially, OPEN consists solely of a single node, the start node S.
Step-02:
If the list is empty, return failure and exit.
Step-03:
 Remove node n with the smallest value of f(n) from OPEN and move it
to list CLOSED.
 If node n is a goal state, return success and exit.
Step-04:
Expand node n.

A* Search Algorithm
Step-05:
 If any successor to n is the goal node, return success and the solution by
tracing the path from goal node to S.
 Otherwise, go to Step-06.
Step-06:
For each successor node,
 Apply the evaluation function f to the node.
 If the node has not been in either list, add it to OPEN.
Step-07:
Go back to Step-02.

A* Search Algorithm
Example with Solution:
Consider the following graph-
 The numbers written on edges
represent the distance between
the nodes.
 The numbers written on nodes
represent the heuristic value.
 Find the most cost-effective path
to reach from start state A to
final state J using A* Algorithm.

A* Search Algorithm
Example with Solution:
Solution-
Step-01:
We start with node A.
Node B and Node F can be reached from
node A.
A* Algorithm calculates f(B) and f(F).
Estimated Cost f(n)=g(n)+h(n)
f(B) = 6 + 8 = 14
f(F) = 3 + 6 = 9
Since f(F) < f(B), so it decides to go to
node F.
->Closed list(F)
Path- A → F

A* Search Algorithm
Example with Solution:
Solution-
Step-02:
Node G and Node H can be reached from
node F.
A* Algorithm calculates f(G) and f(H).
f(G) = (3+1) + 5 = 9
f(H) = (3+7) + 3 = 13
Since f(G) < f(H), so it decides to go to
node G.
->Closed list(G)
Path- A → F → G

A* Search Algorithm
Example with Solution:
Solution-
Step-03:
Node I can be reached from node G.
A* Algorithm calculates f(I).
f(I) = (3+1+3) + 1 = 8
It decides to go to node I.
->Closed list(I)
Path- A → F → G → I

A* Search Algorithm
Example with Solution:
Solution-
Step-04:
Node E, Node H and Node J can be reached
from node I.
A* Algorithm calculates f(E), f(H) and f(J).
f(E) = (3+1+3+5) + 3 = 15
f(H) = (3+1+3+2) + 3 = 12
f(J) = (3+1+3+3) + 0 = 10
Since f(J) is least, so it decides to go to
node J.
->Closed list(J)
Shortest Path- A → F → G → I → J

A* Search Algorithm
Advantages of BFS:
A* Algorithm is one of the best path finding algorithms.
It is Complete & Optimal
Used to solve complex problems.
Disadvantages of BFS:
Requires more memory

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The use of huge quantity of natural fine aggregate (NFA) and cement in civil construction work which have given rise to various ecological problems. The industrial waste like Blast furnace slag (GGBFS), fly ash, metakaolin, silica fume can be used as partly replacement for cement and manufactured sand obtained from crusher, was partly used as fine aggregate. In this work, MATLAB software model is developed using neural network toolbox to predict the flexural strength of concrete made by using pozzolanic materials and partly replacing natural fine aggregate (NFA) by Manufactured sand (MS). Flexural strength was experimentally calculated by casting beams specimens and results obtained from experiment were used to develop the artificial neural network (ANN) model. Total 131 results values were used to modeling formation and from that 30% data record was used for testing purpose and 70% data record was used for training purpose. 25 input materials properties were used to find the 28 days flexural strength of concrete obtained from partly replacing cement with pozzolans and partly replacing natural fine aggregate (NFA) by manufactured sand (MS). The results obtained from ANN model provides very strong accuracy to predict flexural strength of concrete obtained from partly replacing cement with pozzolans and natural fine aggregate (NFA) by manufactured sand.

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A* Search Algorithm

1. Topic To Be Covered: A* Search Algorithm Jagdamba Education Society's SND College of Engineering & Research Centre Department of Computer Engineering SUBJECT: Artificial Intelligence & Robotics Lecture No-9(ii) Prof.Dhakane Vikas N

2. A* Search Algorithm  This is informed search technique also called as HEURISTIC search.  This algo. Works using heuristic value.  A* uses h(n)->Heuristic function & g(n)->Cost to reach the node ‘n’ from start state.  Find shortest path though search spaces.  Estimated Cost f(n)=g(n)+h(n)  A* gives Fast & Optimal result as compared with previous algorithms.  Space & Time Complexity of BFS is also O(V+E) where V is vertices and E is edges.  Also Written as:-O(b) ^d Where, b->Branching factor d->depth

3. A* Search Algorithm  A* Algorithm extends the path that minimizes the following function- f(n) = g(n) + h(n) Here,  ‘n’ is the last node on the path  g(n) is the cost of the path from start node to node ‘n’  h(n) is a heuristic function that estimates cost of the cheapest path from node ‘n’ to the goal node Algorithm-  The implementation of A* Algorithm involves maintaining two lists- OPEN and CLOSED.  OPEN contains those nodes that have been evaluated by the heuristic function but have not been expanded into successors yet.  CLOSED contains those nodes that have already been visited. The algorithm is as follows-

4. A* Search Algorithm The algorithm is as follows- Step-01:  Define a list OPEN.  Initially, OPEN consists solely of a single node, the start node S. Step-02: If the list is empty, return failure and exit. Step-03:  Remove node n with the smallest value of f(n) from OPEN and move it to list CLOSED.  If node n is a goal state, return success and exit. Step-04: Expand node n.

5. A* Search Algorithm Step-05:  If any successor to n is the goal node, return success and the solution by tracing the path from goal node to S.  Otherwise, go to Step-06. Step-06: For each successor node,  Apply the evaluation function f to the node.  If the node has not been in either list, add it to OPEN. Step-07: Go back to Step-02.

6. A* Search Algorithm Example with Solution: Consider the following graph-  The numbers written on edges represent the distance between the nodes.  The numbers written on nodes represent the heuristic value.  Find the most cost-effective path to reach from start state A to final state J using A* Algorithm.

7. A* Search Algorithm Example with Solution: Solution- Step-01: We start with node A. Node B and Node F can be reached from node A. A* Algorithm calculates f(B) and f(F). Estimated Cost f(n)=g(n)+h(n) f(B) = 6 + 8 = 14 f(F) = 3 + 6 = 9 Since f(F) < f(B), so it decides to go to node F. ->Closed list(F) Path- A → F

8. A* Search Algorithm Example with Solution: Solution- Step-02: Node G and Node H can be reached from node F. A* Algorithm calculates f(G) and f(H). f(G) = (3+1) + 5 = 9 f(H) = (3+7) + 3 = 13 Since f(G) < f(H), so it decides to go to node G. ->Closed list(G) Path- A → F → G

9. A* Search Algorithm Example with Solution: Solution- Step-03: Node I can be reached from node G. A* Algorithm calculates f(I). f(I) = (3+1+3) + 1 = 8 It decides to go to node I. ->Closed list(I) Path- A → F → G → I

10. A* Search Algorithm Example with Solution: Solution- Step-04: Node E, Node H and Node J can be reached from node I. A* Algorithm calculates f(E), f(H) and f(J). f(E) = (3+1+3+5) + 3 = 15 f(H) = (3+1+3+2) + 3 = 12 f(J) = (3+1+3+3) + 0 = 10 Since f(J) is least, so it decides to go to node J. ->Closed list(J) Shortest Path- A → F → G → I → J

11. A* Search Algorithm EXAMPLE-02

12. A* Search Algorithm Advantages of BFS: A* Algorithm is one of the best path finding algorithms. It is Complete & Optimal Used to solve complex problems. Disadvantages of BFS: Requires more memory

A* Search Algorithm

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