Algorithms and Data Structures for External Memory

5.1 Sorting by Distribution 31
the Aggarwal–Vitter model discussed in Section 2.3 by use of PDM
with the same number of I/Os, up to a constant factor.
In Section 5.5, we consider the situation in which the items in the
input file do not have unique keys. In Sections 5.6 and 5.7, we consider
problems related to sorting, such as permuting, permutation networks,
transposition, and fast Fourier transform. In Chapter 6, we give lower
bounds for sorting and related problems.
5.1 Sorting by Distribution
Distribution sort [220] is a recursive process in which we use a set of
S − 1 partitioning elements e1, e2, ..., eS−1 to partition the current
set of items into S disjoint subfiles (or buckets), as shown in Figure 5.1
for the case D = 1. The ith bucket, for 1 ≤ i ≤ S, consists of all items
with key value in the interval [ei−1,ei), where by convention we let
e0 = −∞, eS =+∞. The important property of the partitioning is that
all the items in one bucket precede all the items in the next bucket.
Therefore, we can complete the sort by recursively sorting the individual
buckets and concatenating them together to form a single fully
sorted list.
File
on disk
Input buffer
Output buffer 1
Internal Memory
Output buffer S
S buckets
on disk
Fig. 5.1 Schematic illustration of a level of recursion of distribution sort for a single disk
(D = 1). (For simplicity, the input and output operations use separate disks.) The file on
the left represents the original unsorted file (in the case of the top level of recursion) or
one of the buckets formed during the previous level of recursion. The algorithm streams the
items from the file through internal memory and partitions them in an online fashion into
S buckets based upon the key values of the S − 1 partitioning elements. Each bucket has
double buffers of total size at least 2B to allow the input from the disk on the left to be
overlapped with the output of the buckets to the disk on the right.