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Focusing on the 2D plane:

Lawson's Flip Algorithm works in worst-case $O(n^2)$ flips. I have seen it mentioned that (other?) flip-based algorithms work in expected $O(nlogn)$ time for two dimensions. What is a simple example of an algorithm like this (perhaps with proof of complexity)? How would this be found?

Edit: If there are none in expected $O(nlogn)$, are there average-case $O(nlogn)$ flip-based algorithms?

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The proof that edge flipping is expected to perform well goes back to this paper below as well as a couple related one around the same time-frame:

Guibas, Leonidas J., Donald E. Knuth, and Micha Sharir. "Randomized incremental construction of Delaunay and Voronoi diagrams." Algorithmica 7.1-6 (1992): 381-413.

A relatively concise algorithm and proof is given in the paper below by Edelsbrunner and Shah which describes a flip-based algorithm that runs in expected O(n log n) time.

Edelsbrunner, Herbert, and Nimish R. Shah. "Incremental topological flipping works for regular triangulations." Algorithmica 15.3 (1996): 223-241.

The algorithm involves adding points to the triangulation one at a time and using some history of the construction process to locate the new points. The paper is slightly more general, dealing with regular triangulations for which the Delaunay triangulation is a special case.

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