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I'm new to stack overflow, but I'm here because I've searched everywhere and can't seem to find much info on the time complexity of A*, besides off the wiki. I would also like to compare it to Dijkstra's algorithm and see how adding a heuristic in A* improves it's performance.

I know it's a very advanced topic, but I just can't fully understand it from the info given on wiki (Even the analysis of Dijkstra's algorithm on wiki seems quite advanced).

https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm https://en.wikipedia.org/wiki/A*_search_algorithm

I would greatly appreciate it if anyone could explain the time complexity in more detail, or suggest any reading / learning material on the topic. I do have a good understanding of the A* algorithm, but I've just started learning the analysis thereof now.

melpomene
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Brandon Kynoch
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    you might ask here instead: https://cs.stackexchange.com/ – Erix Jun 30 '17 at 15:05
  • Being E the number of edges and V the number of vertices, Dijkstra complexity is `O(E + V log V)` ( `O(V^2)` for the naive implementation), while A* is `O(E)` ... what's not clear from the wiki ? – digEmAll Jun 30 '17 at 15:12
  • If an user answered your question please also accept his answer ([Accepting Answers: How does it work?](https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work)). If not than please specify what remains unanswered, this is a really crucial part of StackOverflow, thank you very much. – Zabuzard Sep 24 '17 at 15:37

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The answer is simply it depends. A star by itself is no complete algorithm. A star is Dijkstra with a heuristic that fulfills some properties (like triangle inequality). You can select different heuristic functions that lead to different time complexities. The simplest heuristic is straight line distance. However there is also more advanced stuff like landmarks heuristic for example.

In the worst case you always need to explore the whole neighborhood so you won't get better than Dijkstra from a general point of analysis. However in most practical applications you can achieve much better bounds. This is only when you know some properties of your graph and of your heuristic function. You then can make some assumptions which lead to a better complexity, but only for those instances.

For example if you know that the straight line distance is always the correct distance in your graph and you use a straight line distance heuristic, then your A star will have the best possible complexity with Theta(1). However this is a much to strong assumption for most applications. But you can think of where this goes.

The bottom line is: It extremely depends on the structure of your graph and your heuristic function.

Here's a lecture on A star as you ask for learning material: Efficient Route Planning (A*, Landmarks, Set Dijkstra) - University of Freiburg

There is also much on the internet, the algorithm is pretty popular as it is very easy to implement and for most cases already fast enough (non-complex games for example).

Zabuzard
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