Finding Approximate Partitions And Splitters In External Memory

This paper studies two fundamental problems both of which are defined on a set S of elements drawn from an ordered domain. In the first problem-called approximate K-partitioning-we want to divide S into K disjoint partitions P-1,...,P-K such that (i) every element in P-i is smaller than all the elements in P-j for any i, j satisfying 1 <= i < j <= K, and (ii) the size of each P-i (1 <= i <= K) falls in a given range [a, b]. In the second problem-called approximate K-splitters-we want to find K-1 elements s1,..., s (K-1) from S, such that the size of S boolean AND (si, s(i-1)] falls in a given range [a, b] (define dummy s 0 =-infinity and s(K) =infinity) .We present I/O-efficient comparison-based algorithms for solving these problems, and establish their optimality by proving matching lower bounds. Our results reveal that the two problems are separated in terms of I/O complexity when K is small, but have the same hardness when K is large.