Next: References Up: Triangle: Engineering a 2D Mesh Generator Previous: Correct Adaptive Tests

The sweepline and incremental Delaunay triangulation implementations compared by Su and Drysdale  each use some variant of uniform bucketing to locate points. Bucketing yields fast implementations on uniform point sets, but is easily defeated; a small, dense cluster of points in a large, sparsely populated region may all fall into a single bucket. I have not used bucketing in Triangle, preferring algorithms that exhibit good performance with any distribution of input points. As a result, Triangle may be slower than necessary when triangulating uniformly distributed point sets, but will not exhibit asymptotically slower running times on difficult inputs.

Fortune's sweepline algorithm uses two nontrivial data structures in addition to the triangulation: a priority queue to store events, and a balanced tree data structure to store the sequence of edges on the boundary of the mesh. Fortune's own implementation, available from Netlib, uses bucketing to perform both these functions; hence, an O(n log n) running time is not guaranteed, and Su and Drysdale  found that the original implementation exhibits O(n3/2) performance on uniform random point sets. By modifying Fortune's code to use a heap to store events, they obtained O(n log n) running time and better performance on large point sets (having more than 50,000 points). However, bucketing outperforms a heap on small point sets.

Triangle's implementation uses a heap as well, and also uses a splay tree  to store mesh boundary edges, so that an O(n log n) running time is attained, regardless of the distribution of points. Not all boundary edges are stored in the splay tree; when a new edge is created, it is inserted into the tree with probability 0.1. (The value 0.1 was chosen empirically to minimize the triangulation time for uniform random point sets.) At any time, the splay tree contains a random sample of roughly one tenth of the boundary edges. When the sweepline sweeps past an input point, the point must be located relative to the boundary edges; this point location involves a search in the splay tree, followed by a search on the boundary of the triangulation itself.

Splay trees adjust themselves so that frequently accessed items are near the top of the tree. Hence, a point set organized so that many new vertices appear at roughly the same location on the boundary of the mesh is likely to be triangulated quickly. This effect partly explains why Triangle's sweepline implementation triangulates points on the boundary of a circle more quickly than the other point sets, even though there are many more boundary edges in the cocircular point set and the splay tree grows to be much larger (containing O(n) boundary edges instead of O(n1/2)).

Triangle's incremental insertion algorithm for Delaunay triangulation uses the point location method proposed by Mücke, Saias, and Zhu . Their jump-and-walk method chooses a random sample of O(n1/3) vertices from the mesh (where n is the number of nodes currently in the mesh), determines which of these vertices is closest to the query point, and walks through the mesh from the chosen vertex toward the query point until the triangle containing that point is found. Mücke et al. show that the resulting incremental algorithm takes expected O(n4/3) time on uniform random point sets. Table 1 appears to confirm this analysis. Triangle uses a sample size of 0.45n1/3; the coefficient was chosen empirically to minimize the triangulation time for uniform random point sets. Triangle also checks the previously inserted point, because in many practical point sets, any two consecutive points have a high likelihood of being near each other.

A more elaborate point location scheme such as that suggested by Guibas, Knuth, and Sharir  could be used (along with randomization of the insertion order) to obtain an expected O(n log n) triangulation algorithm, but the data structure used for location is likely to take up as much memory as the triangulation itself, and unlikely to surpass the performance of the divide-and-conquer algorithm; hence, I do not intend to pursue it.

Note that all discussion in this paper applies to Triangle version 1.2; earlier versions lack the sweepline algorithm and many optimizations to the other algorithms.

Next: References Up: Triangle: Engineering a 2D Mesh Generator Previous: Correct Adaptive Tests

Jonathan Richard Shewchuk
Mon Aug 12 10:28:49 EDT 1996