I recently read Lev Nachmanson, Arlind Nocaj, Sergey Bereg, Leishi Zhang, and Alexander Holroyd’s article on “Node Overlap Removal by Growing a Tree,” which presents a really interesting method.

Using a minimum spanning tree to deal with overlapping nodes seems like a really innovative technique. It made me wonder how the authors came up with this approach!

As outlined in the paper, the algorithm begins with a Delaunay triangulation on the node centers – more information on Delaunay triangulations here – but its essentially a maximal planar subdivision of the graph: eg, you draw triangles connecting the centers of all the nodes.

From here, the algorithm finds the minimal spanning tree, where the cost of an edge is defined so that greater node overlap the lower the cost. The minimal spanning tree, then, find the maximal overlaps in the graph. The algorithm then “grows” the tree: increasing the cost of the tree by lengthening edges. Starting at the root, the lengthening propagates outwards. The algorithm repeats no overlaps exist on the edge of the triangulation.

Impressively, this algorithm runs in O(|V|) time per iteration, making it a fast as well as an effective algorithm.