Shortest path using A Algorithm Introduction History Components - PowerPoint PPT Presentation

Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Shortest path using A ∗ Algorithm Introduction History Components of A ∗ Algorithm Phaneendhar Reddy Vanam Steps Involved in A ∗ Algorithm Example showing A ∗ December 7, 2011 Search Applications Time Complexity

Shortest path using A ∗ Algorithm Introduction 1 Phaneendhar Reddy Vanam History 2 Introduction Components of A ∗ Algorithm History 3 Components of A ∗ Steps Involved in A ∗ Algorithm Algorithm 4 Steps Involved in A ∗ Example showing A ∗ Search Algorithm 5 Example showing A ∗ Search Applications 6 Applications Time Time Complexity 7 Complexity

Introduction Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam The A ∗ Algorithm is a best-first search algorithm that Introduction finds the least cost path from an initial configuration to a History final configuration. Components of A ∗ The most essential part of the A ∗ Algorithm is a good Algorithm Steps Involved heuristic estimate. in A ∗ Algorithm This Heuristic estimate function can improve the efficiency Example and performance of the algorithm. showing A ∗ Search Applications Time Complexity

History of A ∗ Algorithm Shortest path using A ∗ Algorithm Phaneendhar In 1964 Nils Nilson invented a heuristic based approach to Reddy Vanam increase the speed of Dijkstra’s algorithm.This algorithm is Introduction called A1. History In 1967 Bertram Rapheal made dramatic improvements Components of A ∗ upon this algorithm, but failed to show optimatility.He Algorithm called this algorithm as A2. Steps Involved in A ∗ Then in 1968 Peter E.Hart introduced an argument that Algorithm proved A2 was optimal when using a consistent heuristic. Example showing A ∗ Search Finally, He named new algorithm in Kleene star syntax to Applications be the algorithm that starts with A called A ∗ . Time Complexity

Components of A ∗ Algorithm Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction A ∗ Algorithm mainly uses the function f(n)=g(n)+h(n). History Components g(n) is path-cost function, which is the cost from the of A ∗ Algorithm starting node to the current node. Steps Involved h(n) is the heuristic estimate of the distance to the goal. in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity

Contd Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History A search algorithm is said to be complete, if it is Components guaranteed to find a goal state if one exits. of A ∗ Algorithm A search algorithm is said to be optimal, if the first goal Steps Involved state it finds is the least cost. in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity

Steps Involved in A ∗ Algorithm Shortest path using A ∗ Algorithm Let’s characterize a class of admissible heuristic search Phaneendhar strategies, using the evaluation function: Reddy Vanam f ′ ( n ) = g ( n ) + h ( n ). Introduction A ∗ can be implemented more efficiently roughly speaking, History no node needs to be processed more than once. Components of A ∗ As A ∗ traverses the graph, it follows a path of the lowest Algorithm Steps Involved known cost, keeping a sorted priority queue of alternate in A ∗ Algorithm path segments along the way. Example If, at any point, a segment of the path being traversed has showing A ∗ Search a higher cost than another encountered path segment, it Applications abandons the higher-cost path segment and traverses the Time lower-cost path segment. Complexity

Steps Contd. Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Starting with the initial node, it maintains a priority queue of nodes to be traversed, known as the open set. Introduction History The lower f(x) for a given node x, the higher its priority. At each step of the A ∗ algorithm, the node with the Components of A ∗ Algorithm lowest f(x) value is removed from the queue, the f and h Steps Involved values of its neighbors are updated accordingly, and these in A ∗ Algorithm neighbors are added to the queue. Example The A ∗ algorithm continues until a goal node has a lower showing A ∗ Search f value than any node in the queue. Applications Time Complexity

Steps Contd. Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam The f value of the goal is then the length of the shortest Introduction path, since h at the goal is zero in an admissible History heuristic.If the actual shortest path is desired, the Components algorithm may also update each neighbor with its of A ∗ Algorithm immediate predecessor in the best path found. Steps Involved in A ∗ A closed set of nodes already traversed may be used to Algorithm make the search more efficient.This process continues until Example showing A ∗ the goal is reached. Search Applications Time Complexity

Example Shortest path using A ∗ Values of Heuristic i.e straight line distance to Bucharest Algorithm Phaneendhar City Hueristic value City Heuristic Reddy Vanam value Introduction Arad 366 Mehadia 241 History Bucharest 0 Neamt 234 Components of A ∗ Craiova 160 Oradea 380 Algorithm Eforie 161 Pitesti 100 Steps Involved in A ∗ Fagaras 176 Rimnicu Vilcea 193 Algorithm Example Dobreta 242 Timisoara 329 showing A ∗ Search Hirsova 151 Urziceni 80 Applications Iasi 226 Vaslui 199 Time Lugoj 244 Zerind 374 Complexity

Example Contd. Steps in Map of Romania with Arad as start state Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example1

Example Contd. After expanding Arad Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example2

Example Contd. After expanding Sibiu Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example3

Example Contd. After expanding Rimnicu Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example4

Example Contd. After expanding Fagaras Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example5

Example Contd. After expanding Pitesti Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example6

Example Contd. Finally, the shortest path between Arad and Bucharest Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction History Components of A ∗ Algorithm Steps Involved in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity Figure: Example7

Applications Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam Introduction The A ∗ algorithm for heuristic search is applied to History Components construct a Neural Network structure (NS). of A ∗ Algorithm This algorithm mostly used in Computer Gaming, Robotics Steps Involved and Google Maps. in A ∗ Algorithm Example showing A ∗ Search Applications Time Complexity

Time Complexity Shortest path using A ∗ Algorithm Phaneendhar Reddy Vanam The time complexity of A ∗ depends on the heuristic.In the Introduction worst case, the number of nodes expanded is exponential in the History length of the solution. Components of A ∗ Algorithm | h ( x ) − h ∗ ( x ) | = O ( log ( h ) ∗ ( x )) Steps Involved in A ∗ where h ∗ is the optimal heuristic and it is also defined as true Algorithm Example cost of getting from n to the goal. showing A ∗ Search Applications Time Complexity

References Shortest path using A ∗ Algorithm Phaneendhar Stuart Russell,Peter Norvig Artificial intelligence a modern Reddy Vanam approach Introduction David L.Poole and Alan K.Mackworth Foundations of History computational agents Cambridge University Press, 2010 Components of A ∗ Algorithm Ben Coppin Artificial Intelligence Illuminated Steps Involved in A ∗ wikipedia A ∗ search algorithm Algorithm http://en.wikipedia.org/wiki/A* algorithm Example showing A ∗ Search A ∗ algorithm for beginners Applications http://www.policyalmanac.org/games/aStarTutorial.htm Time Complexity