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

Shortest path using A∗ Algorithm

Phaneendhar Reddy Vanam

December 7, 2011

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

1 Introduction

2 History

3 Components of A∗ Algorithm

4 Steps Involved in A∗ Algorithm

5 Example showing A∗ Search

6 Applications

7 Time Complexity

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

Introduction

The A∗ Algorithm is a best-first search algorithm thatfinds the least cost path from an initial configuration to afinal configuration.

The most essential part of the A∗ Algorithm is a goodheuristic estimate.

This Heuristic estimate function can improve the efficiencyand performance of the algorithm.

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

History of A∗ Algorithm

In 1964 Nils Nilson invented a heuristic based approach toincrease the speed of Dijkstra’s algorithm.This algorithm iscalled A1.

In 1967 Bertram Rapheal made dramatic improvementsupon this algorithm, but failed to show optimatility.Hecalled this algorithm as A2.

Then in 1968 Peter E.Hart introduced an argument thatproved A2 was optimal when using a consistent heuristic.

Finally, He named new algorithm in Kleene star syntax tobe the algorithm that starts with A called A∗.

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

Components of A∗ Algorithm

A∗ Algorithm mainly uses the function f(n)=g(n)+h(n).

g(n) is path-cost function, which is the cost from thestarting node to the current node.

h(n) is the heuristic estimate of the distance to the goal.

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

Contd

A search algorithm is said to be complete, if it isguaranteed to find a goal state if one exits.

A search algorithm is said to be optimal, if the first goalstate it finds is the least cost.

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

Steps Involved in A∗ Algorithm

Let’s characterize a class of admissible heuristic searchstrategies, using the evaluation function:f ′(n) = g(n) + h(n).

A∗ can be implemented more efficiently roughly speaking,no node needs to be processed more than once.

As A∗ traverses the graph, it follows a path of the lowestknown cost, keeping a sorted priority queue of alternatepath segments along the way.

If, at any point, a segment of the path being traversed hasa higher cost than another encountered path segment, itabandons the higher-cost path segment and traverses thelower-cost path segment.

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

Steps Contd.

Starting with the initial node, it maintains a priority queueof nodes to be traversed, known as the open set.

The lower f(x) for a given node x, the higher its priority.

At each step of the A∗ algorithm, the node with thelowest f(x) value is removed from the queue, the f and hvalues of its neighbors are updated accordingly, and theseneighbors are added to the queue.

The A∗ algorithm continues until a goal node has a lowerf value than any node in the queue.

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

Steps Contd.

The f value of the goal is then the length of the shortestpath, since h at the goal is zero in an admissibleheuristic.If the actual shortest path is desired, thealgorithm may also update each neighbor with itsimmediate predecessor in the best path found.

A closed set of nodes already traversed may be used tomake the search more efficient.This process continues untilthe goal is reached.

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

Example

Values of Heuristic i.e straight line distance to Bucharest

City Hueristic value City Heuristicvalue

Arad 366 Mehadia 241

Bucharest 0 Neamt 234

Craiova 160 Oradea 380

Eforie 161 Pitesti 100

Fagaras 176 Rimnicu Vilcea 193

Dobreta 242 Timisoara 329

Hirsova 151 Urziceni 80

Iasi 226 Vaslui 199

Lugoj 244 Zerind 374

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

Example Contd.

Steps in Map of Romania with Arad as start state

Figure: Example1

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

Example Contd.

After expanding Arad

Figure: Example2

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

Example Contd.

After expanding Sibiu

Figure: Example3

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

Example Contd.

After expanding Rimnicu

Figure: Example4

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

Example Contd.

After expanding Fagaras

Figure: Example5

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

Example Contd.

After expanding Pitesti

Figure: Example6

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

Example Contd.

Finally, the shortest path between Arad and Bucharest

Figure: Example7

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

Applications

The A∗ algorithm for heuristic search is applied toconstruct a Neural Network structure (NS).

This algorithm mostly used in Computer Gaming, Roboticsand Google Maps.

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

Time Complexity

The time complexity of A∗ depends on the heuristic.In theworst case, the number of nodes expanded is exponential in thelength of the solution.

|h(x) − h∗(x)| = O(log(h)∗(x))

where h∗ is the optimal heuristic and it is also defined as truecost of getting from n to the goal.

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

References

Stuart Russell,Peter Norvig Artificial intelligence a modern

approach

David L.Poole and Alan K.Mackworth Foundations of

computational agents Cambridge University Press, 2010

Ben Coppin Artificial Intelligence Illuminated

wikipedia A∗ search algorithmhttp://en.wikipedia.org/wiki/A* algorithm

A∗ algorithm for beginnershttp://www.policyalmanac.org/games/aStarTutorial.htm