Lecture Notes: Distribution Sweeping

The skyline problem can be easily solved in linear I/Os in 2D space, after sorting the points of P by x-coordinate (think: how). We will show that the 3D version of the problem can also be settled in O(NB logM/B N B ) I/Os. Our description below follows that of [3]. Algorithm. Our algorithm accepts as input a dataset P whose points are sorted by x-coordinate. If N ≤ M (i.e., the memory capacity), we simply find the skyline of P using a memory-resident algorithm. The I/O cost incurred is O(N/B). Now consider N > M . We divide P into s = Θ(M/B) partitions P (1), ..., P (s) with the same size ⌈N/s⌉ (except perhaps one partition), such that each point in P (i) has a smaller x-coordinate than all points in P (j) for any 1 ≤ i < j ≤ s. As P is already sorted on the x-dimension, the partitioning can be done in linear cost, while leaving the points of each P (i) sorted in the same way. The next step is to obtain the skyline Σ(i) of each P (i), i.e., Σ(i) = SKY (P (i)). Since this is identical to solving the original problem (only on a smaller dataset), we recursively invoke our algorithm on P (i). Now consider the moment when all Σ(i) have been returned from recursion. Our algorithm proceeds by performing a skyline merge, which finds the skyline of the union of all Σ(i), that is, SKY (Σ(1) ∪ ... ∪ Σ(s)), which is exactly SKY (P ). We enforce an invariant that, SKY (P ) be returned in an array where the points are sorted in ascending order of y-coordinate (to be used by the upper level of recursion, if any). Due to recursion, the invariant implies that, the same ordering has been applied to all the Σ(i) at hand.