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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 or-dered domain. In the first problem—called approximate K-partitioning—we want to divide S into K disjoint partitions P1,..., PK such that (i) every element in Pi is smaller than all the eleme ..."
Abstract
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This paper studies two fundamental problems both of which are defined on a set S of elements drawn from an or-dered domain. In the first problem—called approximate K-partitioning—we want to divide S into K disjoint partitions P1,..., PK such that (i) every element in Pi is smaller than all the elements in Pj for any i, j satisfying 1 ≤ i < j ≤ K, and (ii) the size of each Pi (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,..., sK−1 from S, such that the size of S ∩ (si, si−1] falls in a given range [a, b] (define dummy s0 = − ∞ and sK =∞). 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.
