Question

I have to implement Fortunes algorithm for constructing Voronoi diagrams.

Important part of the algorithm is a data structure called “Beach Line Data Structure”.

It is a binary balanced tree, similar to AVL, but different in a way that data is stored only on the leafs (there are other differences, but are unimportant for the question).

I am not sure how to implement it. Obviously using AVL “as is” will not work because when balancing AVL tree leaf node can become inner node and vice versa.

I also tried to look at some other known data structures at wikipedia, but none suits the needs.
I have seen some implementations that do this with a linked list, but this is not good because searching linked list is O(n), and it needs to be O(log n) for the algorithm to be efficient.

Answer 1

The leaves indeed store (single) points and the inner nodes of the event structure (the “beach line tree”) stores ordered tuples of points whose parabolas/arcs lie next to each other. If the parabola that point P_a forms lies to the left of the parabola formed by P_b (and these two parabola’s intersect), the inner node stores the ordered tuple (P_a, P_b).

Obviously using AVL “as is” will not work because when balancing AVL tree leaf node can become inner node and vice versa.

If you’re worried about storing different types of objects in the AVL tree, a simple scheme would be to store the leaves as tuples too. So don’t store point P_j as a leaf, but store the tuple (P_j, P_j) instead. If P_j as a leaf disappears from the event tree (beach line), and its parent is (P_i, P_j), simply change the parent into (P_j, P_j), and of course its parent will also needs to be changed from (P_j, P_?) to (P_i, P_?) etc. Just as with a regular AVL tree: you walk up the tree and modify the inner nodes that need to be changed and/or re-balanced.

Note that a good implementation of the algorithm can’t be easily written down in a SO answer (at least, not by me!). For a proper explanation of the entire algorithm, including a description of the data structures used by it, see Computational geometry: algorithms and applications by Mark de Berg et al.. Chapter 7 is devoted solely to Voronoi diagrams.