1. Self-Improving Algorithms for Delaunay Triangulations

    Nir Alon, Bernard Chazelle, Ken Clarkson*, Ding Liu, C. Seshadhri, Wolfgang Mulzer

    *IBM Almaden
    (all others: Princeton Univ., or both)

  2. Outline

  3. Sequences of Computational Problems

  4. Self-Improvement

  5. Random Data and Comparisons

  6. Identifying the Output using Comparisons

  7. Entropy Lower Bounds

  8. Meeting the Entropy Lower Bounds

  9. These Results Are Not About

  10. An Additional Condition: Independence

  11. Sorting : The Typical Set `V`

  12. Sorting : Search Trees `T_i`

  13. Sorting : The Algorithm

  14. Analysis

  15. Analysis via Encoding

  16. Delaunay Triangulations

  17. Sorting vs. Triangulation

  18. Triangulation: the Typical Set `V`

  19. More About `V`

  20. Triangulation: Search Trees `T_i`

  21. Triangulation: the Algorithm

  22. Analysis: Encoding

  23. All the Analogies

    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)`
  24. Concluding Remarks

    Thank you for your attention