### Table 2: Rectangular initial triangulation At this point some remarks concerning the practical use of both kinds of algorithms are in order. Even when in theory (as discussed throughout this paper), these are competitive linear algorithms for dealing with the triangulation re nement problem, it should be stressed here that, the computational work involved with the use of the mixed Delaunay algorithm is higher, since it involves the repetitive use of the incremental Delaunay procedure (based on the Delaunay circle test) locally required to insert each new point. In the case of the longest-side re nement algorithms, in exchange, the insertion of each new point only means the partition of the individual triangle (and the update of the data structure), an essentially clean and robust task. This is also a clear advantage with respect to the Delaunay algorithm, which is known as an algorithm su ering of non-robustness issues (due to the oat point computations involved with the circle test).

### Table 2. Speed comparison of calculating Voronoi diagram

### Table 6: Voronoi-diagram | simulation results

### Table 3. The proposed Voronoi diagram algorithm

### Table 2 Delaunay Triangulation: running time in secs; 400000 random points, 32{128 bit 32 40 52 80 100 128

"... In PAGE 17: ... As input data we used rat points with homogeneous integer coordinates of difierent bit-lengths. As to be expected, for random inputs ( Table2 ), the dt search version gains about 10-15 % in the overall running time against the dt exact version, due to not having to compute the error bounds for most predicates. The dt ip version, though, performs much worse since the additional check over all edges of the triangulation is rather expensive in that case, even if no ips take place.... ..."

### Table 5. Triangulation of improvement issues.

"... In PAGE 10: ...ocumentation, i.e. Issue-2 and Issue-4 to Issue-8. Table 4 offers an overview of this. 4.3 Triangulation of Results Table5 offers an overview of all issues pre- sented, and information about substantiation from the different data sources, i.... ..."

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### Table 5. Triangulation of improvement issues.

"... In PAGE 9: ...ocumentation, i.e. Issue-2 and Issue-4 to Issue-8. Table 4 offers an overview of this. 4.3 Triangulation of Results Table5 offers an overview of all issues presented, and information about substantiation from the different data sources, i.... ..."

### Table 1: The running times of the two algorithms as a function of the size n of the input set and the number of non-trivial sites h. T1 indicates the time for the algorithm with one level of the AW-Voronoi diagram and T2 indicates the running time for an hierarchy of AW-Voronoi diagrams. Unless otherwise indicated, both T1 and T2 are given in seconds. The experiments were performed on a Pentium-III 1GHz running Linux.

"... In PAGE 21: ... T1 denotes the running time of the algorithm that uses only one level of the AW-Delaunay graph, and T2 denotes the running time of the algorithm that uses the AW-Delaunay hierarchy. The last two columns of Table1 have been added for convenience. They show the ratios of the running times of the two algorithms over the quantity n log h.... ..."

### Table 1: The running times of the two algorithms as a function of the size n of the input set and the number of non-trivial sites h. T1 indicates the time for the algorithm with one level of the AW-Voronoi diagram and T2 indicates the running time for an hierarchy of AW-Voronoi diagrams. Unless otherwise indicated, both T1 and T2 are given in seconds. The experiments were performed on a Pentium-III 1GHz running Linux.

"... In PAGE 18: ...1.00, 0.95, 0.80, 0.50}. T1 denotes the running time of the algorithm that uses only one level of the AW- Delaunay graph, and T2 denotes the running time of the algorithm that uses the AW-Delaunay hierarchy. The last two columns of Table1 have been added for convenience. They show the ratios of the running times of the two algorithms over the quantity n log h.... ..."