Intrepid Blog » directed acyclic graph http://blog.affien.com A few thoughts Mon, 23 Jan 2012 08:47:37 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 Online Cycle Detection in Directed Graphs http://blog.affien.com/archives/2007/07/02/online-cycle-detection-in-directed-graphs/ http://blog.affien.com/archives/2007/07/02/online-cycle-detection-in-directed-graphs/#comments Mon, 02 Jul 2007 21:01:45 +0000 Bas Westerbaan http://blog.w-nz.com/archives/2007/07/02/online-cycle-detection-in-directed-graphs/ A short while ago I came across a quite interesting problem. Design a datastructure (and algorithms) to maintain a Directed Acyclic Graph.

There has to be only one operation: adding a link between two given nodes. This operation must be able to detect and deny any new link that would cause a cycle. For simplicity, nodes are identified by sequential ids starting with 0.

An example:

addLink 0, 1 -> True
addLink 1, 2 -> True
addLink 2, 0 -> False # Fails because it would create a cycle

addLink 0, 1 -> True
addLink 1, 2 -> True
addLink 0, 2 -> True # This isn't a real cycle, so it's perfectly fine

It’s rather trivial to create a \mathcal{O}\left(n+\ell\right) (where n is the number of nodes and \ell the number of links). I conjecture there exists a \mathcal{O}\left(\log n\right) algorithm.

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