Discussion Algorithm for Z

12.4 Topological Simplification

In this section, we first present a case study of the five reordering algorithms described in Chapter 8 and illustrated in Figure 8.7. We then provide experi- mental evidence of the utility of the algorithms, as well as the rarity of basic and recursive conflicts. We end this chapter with visualizations of persistent complexes.

12.4.1 A Case Study

In this brief picturesque study, we show the effect of the reordering algorithms in the presence of conflicts. Figure 12.8a displays the k-triangles of the data set SOD. This zeolite does not contain any basic conflicts, but it does have 26 recursive conflicts. We are interested in the tip of the region of large overlap- ping 1-triangles, shown in Figure 12.8b. The rest of the figures in c–l show how this area changes with the different reordering algorithms in Figure 8.7. Note that the differences for the pseudo-triangle algorithm cancel, as each cy- cle is given its due influence, given its persistence. Consequently, we will use this algorithm as the default method for simplification.

12.4.2 Timings and Statistics

We have implemented all of the reordering algorithms for experimentation. The algorithms have the basic structure and therefore take about the same time. So, we only give the time taken for the Pseudo-triangle algorithm in Table 12.13. All timings were done on a Sun Ultra-10 with a 440 MHz Ultra- SPARC IIi processor and 256 megabyte RAM, running Solaris 8. Here, each complex is reordered with p equal to the size of the filtration. Generally, the reordering algorithms encounter the same number of conflicts, so we only list the number of basic and recursive conflicts for the pseudo-triangle algorithm in Table 12.13. The time taken for reordering correlates very well with the size of the filtration, as all algorithms make a single pass through the filter. A simplex may move multiple times during reordering, however, because of the recursive nature of the algorithms. The number of recursive conflicts is one indication of the complexity of the reordering. The table shows that the data sets with a large number of recursive conflicts, namely BOG, bearing, TAO, and bone, all have large reordering times. a SOD b Zoomed c Naive d Difference e Shift f Difference g Wormhole h Difference i Pseudo- triangle j Difference k Sudden Death l Difference Fig. 12.8. Reordering algorithms on SOD. a displays the k-triangles of SOD with the region of interest boxed and zoomed in b. c–l show the results of each reordering algorithm and the image difference between these results and b. The difference be- tween images is shown in shades of gray. I have increased the saturation by 25 for better viewing.