A* algorithm

Hristian Hristov

Telerik Corporationhttp:/telerikacademy.

com

Table of Contents

1. Dijkstra algorithm

2. Greedy search

3. A*

Heuristics

Admissible heuristics

Consistent heuristic

4. Additional information

2

Dijkstra algorithm Short Revision

Dijkstra algorithm Uninformed search- We do not know where the goal is!

- Guaranteed to find the goal!

Dijkstra algorithm Short explanation

1. Start from a certain state and add onto fringe

2. Pop the state with highest priority from the fringe, mark as used and add its unmarked successors to the fringe

3. If fringe is empty - end the algorithm, else go to step 2

Complexity – O(nlogn)

5

Dijsktra algorithm - Example

6

Greedy search

Greedy search We are informed for the goal

Use some greedy function to reach the goal

Does NOT always work!

Greedy search Short explanation

For the labyrinth example:1. Start from certain state

2. Move towards the goal minimizing the Manhattan distance.

3. If cannot move – end algorithm, else go to step 2

Greedy search - Example

A* algorithm

A* algorithm

Why not mixing greedy and Dijkstra?

12

A* algorithm

How to mix them? - Hello heuristic functions!

- admissible heuristics

- consistant heuristics

13

Heuristics

What is a heuristic?

14

Heuristics Some interesting values for graph searchh(n) – heuristic of state nC(n) – actual minimum cost from ng(n) – cost to current state from startf(n) – evaluation functionc(n , p) – edge between states n and p

f(n) = g(n) + h(n)15

Heuristics Admissible heuristics

for every state n: C(n) >= f(n)

16

HeuristicsConsistent heuristic

for every n and its successor a:

h(n) <= c(n , a) + h(a) and h(goal)=O

17

A* algorithm

Let’s use the heuristics!- Basically same idea as Dijkstra- Instead of the distance travelled we use the heuristics

- Heuristics should be admissible and consistent

18

A* algorithm

Pseudocode

1. Push start state with its heuristics onto the fringe

2. While fringe is not empty or goal is not reached:

- Pop highest-priority state from the fringe

- Push its non-visited successors onto the fringe calculating their heuristics and mark current state as parent

3. Print the path if needed 19

AlgorithmLive demo

Additional information Dijkstra ALgorithms: “Programming = ++Algorithms” by P. Nakov;http://en.wikipedia.org/wiki/Dijkstra's_algorithm

Heuristics:http://en.wikipedia.org/wiki/Heuristic;http://en.wikipedia.org/wiki/Admissible_heuristic

http://en.wikipedia.org/wiki/Consistent_heuristic

21

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