Sorting | Delaunay Triangulation |
---|---|
Intervals `(x_i, x_{i'})` containing no values of `I` | Delaunay disks |
Typical set `V` | Range space `epsilon`-net `V` [MRW90, CV07], Ranges are disks, `epsilon = 1//n` |
`log n` training instance points in each bucket | `log n` training instance points in each disk |
Expect `O(1)` values of `I` in each bucket | Expect `O(1)` points in each D. disk of `V` |
Optimal weighted binary trees `T_i` | Entropy-optimal planar point location data structures `T_i` [AMMW07] |
Sorting within buckets `->` sorted list of `V cup I` | Triangulation within small regions `-> T(V cup I)` |
Removal of `V` from sorted `V cup I` (trivial) | Construction of `T(I)` from `T(V cup I)` [ChDHMST02] |
In analysis: merge of sorted `V` and `I` | In analysis: merge of `T(V)` and `T(I)` [Ch92] |
In analysis: recovery of buckets `b_i` from sorted `V cup I` (trivial) | In analysis: recovery of triangles `b_i` in `T(I)` from `T(V cup I)` |