Computing Voronoi Treemaps: Faster, Simpler, and Resolution‐independent

Abstract

Voronoi treemaps represent hierarchies as nested polygons. We here show that, contrary to the apparent popular belief, utilization of an algorithm for weighted Voronoi diagrams is not only feasible, but also more efficient than previous low-resolution approximations, even when the latter are implemented on graphics hardware. More precisely, we propose an instantiation of Lloyd's method for centroidal Voronoi diagrams with Aurenhammer's algorithm for power diagrams that yields an algorithm running in O(n log n) rather than Ω(n 2 ) time per iteration, with n the number of sites. We describe its implementation and present evidence that it is faster also in practice.

Organizing Search Results with a Reference Map

“…As demonstrated in [39], running time could be reduced significantly by using heuristic techniques, which lower the number of iterations.…”

“…An easy solution for this would be to collapse parts of the hierarchy, which would also limit the number of shown objects on the screen. With a fast implementation [39], the computation time of a Voronoi treemap consisting of a hierarchy with 1300 nodes, would be below one second with the CPU mentioned above.…”

“…For c ∈ C, n c = |children(c)| it holds that ∑ c∈C n c = n. It is clear that each node in C is in the queue exactly once. Thus, V is computed for each c ∈ C. Further, the computation of the single layer Voronoi treemap V can be done in O(k · |S| log |S|) [39], where k is the constant fixed number of iterations one allows. It follows that the Algorithm 2: Voronoi treemap computation for a hierarchically clustered graph Input: At this point the search results have not been positioned yet.…”

“…We adapt the approach from Nocaj and Brandes [39] by using Fortune's algorithm [19] for weighted Voronoi diagrams.…”

The four-level nested model revisited

“…For example, a comparison guideline at the visual encoding level is to choose node-link diagrams when visualizing small networks and matrix diagrams when visualizing large ones [14], for reasons of avoiding visual clutter. An example of a comparison guideline at the algorithm level is to choose the newer Voronoi treemap algorithm of Nocaj and Brandes [26] over the original algorithm of Balzer and Deussen [5] because it is independent of display resolution and faster to compute.…”