On Sorting an Intransitive Total Ordered Set Using SemiHeap
Jie Wu
Department of Computer Science and Engineering
Florida Atlantic University
Boca Raton, FL 33431
[email protected]
Abstract
1
The problem of sorting an intransitive total ordered set,
a generalization of regular sorting, is considered. This gen
eralized sorting is based on the fact that there exists a spe
cial linear ordering for any intransitive total ordered set.
A new data structure called semiheap is proposed to con
struct an optimal
sorting algorithm. Finally, we
propose a costoptimal parallel algorithm using semiheap.
The run time of this algorithm is
with
pro
cessors under the EREW PRAM model.
1. Introduction
Sorting is one of the fundamental problems in computer
science and many different solutions for sorting have been
proposed [5, 6]. Basically, given a sequence of
numbers
as an input, a sorting algorithm generates
a permutation (reordering)
of the input se
quence such that
.
We consider a generalization of the sorting problem by
replacing
with
, where
is a total order without the
transitive property, i.e., it is intransitive. That is, if
and
, it is not necessary that
. The total
order requires that for any two elements
and
, either
or
, but not both (antisymmetric).
The set
of
elements exhibiting intransitive total or
der can be represented by a directed graph, where
represents a directed edge from vertex
to vertex
. The
underlying graph is a complete graph. This graph is also
called a
tournament
[2], representing a tournament of
players where every possible pair of players plays one game
to decide the winner (and the loser) between them. Sorting
on
corresponds to finding a Hamiltonian path in the tour
nament.
1
This work was support in part by NSF grant CCR 9900646.
Hell and Rosenfeld [4] proved that the bound of find
ing a Hamiltonian path is
, the same complex
ity as the regular sorting. They also considered bounds on
finding some generalized Hamiltonian paths. It is easy to
prove that many regular sorting algorithms can be used to
find a Hamiltonian path in a tournament, such as bubble
sort, insertion sort, binary insertion sort, and merge sort.
Among parallel sorting algorithms, evenodd merge sort can
still be applied. However, heapsort and quicksort cannot be
used. BarNoy and Naor [1] studied different parallel solu
tions based on different models and the number of proces
sors. They showed that under the CRCW PRAM model, the
generalized sorting problem can be solved in
us
ing
processors. Other fast parallel algorithms can be
found in [7].
In this paper, we propose a new data structure called
semiheap
, which is an extension of a regular heap struc
ture. We introduce an optimal
algorithm to de
termine a Hamiltonian path in a tournament based on the
semiheap structure. Then, we propose a costoptimal par
allel algorithm based on the semiheap structure that takes
in run time using
processors in the EREW
PRAM model. An implementation of the costoptimal par
allel algorithm in the network model with a linear array of
processors is also shown.
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 Spring '09
 STAFF
 Algorithms, Sort, complete binary tree, 021Cn log n1D, 021Clog n1D processors

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