Algorithms and Data Structures for External Memory

6.2 Lower Bounds for Sorting and Other Problems 63
by N!/ � (N/K)! � K , which is the maximum number of permutations
of N numbers having K distinct values. Solving for I + O gives the
lower bound ٠� nmax � 1,log m(K/B) �� , which is equal to the desired
lower bound for BundleSort(N,K) when K = B 1+Ω(1) or M = B 1+Ω(1) .
Matias et al. [249] derive the remaining case of the lower bound for
BundleSort(N,K) by a potential argument based upon the derivation
of the transposition lower bound (Theorem 7.2). Dividing by D gives
the lower bound for D disks.
Chiang et al. [105], Arge [30], Arge and Miltersen [45], Munagala
and Ranade [264], and Erickson [150] give models and lower bound
reductions for several computational geometry and graph problems.
The geometry problems discussed in Chapter 8 are equivalent to sorting
in both the internal memory and PDM models. Problems such
as list ranking and expression tree evaluation have the same nonlinear
I/O lower bound as permuting. Other problems such as connected
components, biconnected components, and minimum spanning forest
of sparse graphs with E edges and V vertices require as many I/Os as
E/V instances of permuting V items. This situation is in contrast with
the (internal memory) RAM model, in which the same problems can
all be done in linear CPU time. In some cases there is a gap between
the best known upper and lower bounds, which we examine further in
Chapter 9.
The lower bounds mentioned above assume that the data items are
in some sense “indivisible,” in that they are not split up and reassembled
in some magic way to get the desired output. It is conjectured
that the sorting lower bound (5.1) remains valid even if the indivisibility
assumption is lifted. However, for an artificial problem related to
transposition, Adler [5] showed that removing the indivisibility assumption
can lead to faster algorithms. A similar result is shown by Arge
and Miltersen [45] for the decision problem of determining if N data
item values are distinct. Whether the conjecture is true is a challenging
theoretical open problem.