A* algorithm
- 1. A* Algorithm Submitted By: Komal Samdariya
- 2. • A* Algorithm is one of the best and popular techniques used for path finding and graph traversals. • A lot of games and web-based maps use this algorithm for finding the shortest path efficiently. • It is essentially a best first search algorithm.
- 3. Working- • It maintains a tree of paths originating at the start node. • It extends those paths one edge at a time. • It continues until its termination criterion is satisfied. • A* Algorithm extends the path that minimizes the following function- f(n) = g(n) + h(n)
- 4. • ‘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
- 5. 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.
- 6. • 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.
- 7. • 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.
- 8. Problem-1 • Given an initial state of a 8puzzle problem and final state to be reached- Initial State Final State 2 8 3 1 6 4 7 5 1 2 3 8 4 7 6 5
- 9. • Find the most cost-effective path to reach the final state from initial state using A* Algorithm. • Consider g(n) = Depth of node and h(n) = Number of misplaced tiles. g g=0 h=4 f=0+4=4 g=1 g=1 g=1 h=5 h=3 h=5 f=1+5=6 f=1+3=4 f=1+5=6 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 4 7 6 5 2 8 3 1 6 4 7 5
- 10. 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 4 7 6 5 2 8 3 1 6 4 7 5 2 3 1 8 4 7 6 5 2 8 3 1 4 7 6 5 2 8 3 1 4 7 6 5 g=2 g=2 g=2 h=3 h=3 h=5
- 11. g=3 g=3 g=3 g=3 h=3 h=4 h=2 h=4 8 3 2 1 4 7 6 5 2 8 3 7 1 4 6 5 2 8 3 1 4 7 6 5 2 3 1 8 4 7 6 5 2 3 1 8 4 7 6 5 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 g=4 h=1 g=5 h=0 g=5 h=2 1 2 3 7 8 4 6 5 1 2 3 8 4 7 6 5
- 12. Problem-2 • 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.
- 13. • 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). • f(B) = 6 + 8 = 14 • f(F) = 3 + 6 = 9 • Since f(F) < f(B), so it decides to go to node F. • Path- A → F
- 14. • 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. • Path- A → F → G
- 15. • 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. • Path- A → F → G → I
- 16. • 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. • Path- A → F → G → I → J
- 17. • This is the required shortest path from node A to node J. •