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    <p>There is space for improvement because in unweighted graphs, you gain an additional attribute which does not hold for weighted graphs, namely:</p> <blockquote> <p>For any edge directly connecting A to C, you know for sure that there is no shorter path via a third node B.</p> </blockquote> <p>With this in mind, you should be able to simplify Dijkstra's Algorithm: As you may know, it works with three sets of nodes: </p> <ol> <li>Those of which the definitive shortest path is known, </li> <li>those of which a preliminary distance has been calculated and </li> <li>those which have not yet been explored. </li> </ol> <p>When following an edge <code>e</code> from <code>A</code> (1.) to <code>C</code> (3.), original Dijkstra would move node <code>C</code> from (3.) to (2.). Since the above attribute holds in all your graphs, you can however add it directly to the set (1.), which is more efficient.</p> <p>Here's the essential observation: The procedure outlined above is basically a BFS (breadth first search), i.e. you can find the distance from some fixed node <code>v</code> to any other node in <code>O(|V| + |E|)</code>.</p> <p>You did not mention in your original question that the graph was basically a grid with some holes in it. This is an even stronger requirement, and I am sure you can exploit it. Doing a quick search for "grid graph shortest path" yields <a href="http://onlinelibrary.wiley.com/doi/10.1002/net.3230070404/abstract" rel="nofollow">this paper</a> which promises <code>O(sqrt(n))</code> in the best case. As the problem you specify is fairly well-structured, I'm almost sure there are several more papers which you might want to look into.</p>
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    1. COOk , That's something . I mean that the shortest path to all my neighbors is 1 . That might help to dynamic programming algorithm . But I think when can get something better out of it .
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    2. CO@RanZilber: I added an idea for a specific algorithm to my answer, maybe that is more useful.
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