Fast Node Overlap Removal in Graph Layout Adjustment

Most graph layout algorithms in the field of graph drawing treat nodes as points. The problem of node overlap removal is to adjust the layout generated by such methods so that nodes of non-zero width and height do not overlap, yet are as close as possible to their original positions. We give an O(nlogn) algorithm for achieving this assuming that the number of nodes overlapping any single node is bounded by some constant. This method has two parts, a constraint generation algo- rithm which generates a linear number of "separation" constraints and an algorithm for finding a solution to these constraints "close" to the original node placement values. We also extend our constraint solving algorithm to give an active set based algorithm which is guaranteed to find the optimal solution but which has considerably worse theoretical complexity. We compare our method with convex quadratic optimization and force scan approaches and find that it is faster than either, gives re- sults of better quality than force scan methods and similar quality to the quadratic optimisation approach.