On the average complexity of some bucketing algorithms

Abstract

Consider n independent uniform (0, 1) random variables, and let N 1, …, N n be the cardinalities of the intervals [ (i - 1) n) , (i n) ], 1 ≤ i ≤ n . Then E( max N 1)∼ ( log n log log 4 ) as n → ∞ . This result (proved in the paper) and related results about the asymptotical behavior of E(g( max i N i)) for increasing functions g allow us to draw some conclusions about the average complexity of some bucketing algorithms in computational geometry. We illustrate this point by showing that Shamos' unpublished bucketing algorithm for finding the convex hull of n independent identically distributed random vectors X 1 … X n in R 2 has an average complexity 0( n) whenever the X i 's have a bounded density with compact support.