| Summary: | 博士 === 國立清華大學 === 資訊工程學系 === 87 === In recent years, many interconnection networks, such as the hypercubes,
the group graphs and de bruijn networks, have been proposed for parallel
processing. These networks have many attractive properties, such as
regularity, symmetry, potential for parallel processing. However, their
allowable system sizes are restricted. Thus, a practical system can not
always be implemented by these interconnection networks. It motivates us
to study the properties of the networks that can be interconnected as
arbitrary system size. In this dissertation, we focus our attention on the
incomplete hypercubes and the degree four chordal rings. The former is a
generalization of a hypercube and the latter is a variation of a ring.
Both of these two networks can be used to interconnect an arbitrary system
size. In the first part of this dissertation, we propose a new decomposition
methodology, the vertical decomposition, for the incomplete hypercubes.
Moreover, we investigate the properties of the vertical decomposition and
horizontal decomposition and show that they are useful for solving the
allocation problems on incomplete hypercubes. In the second part of this
dissertation, by the aid of these two decomposition methodologies, we study the
Hamiltonian properties of the incomplete hypercubes. Furthermore, we
propose a new code called generalized Gray code(GGC, for short), a
generalization of Gray code, which is a permutation from 0 to N-1 for
any integer N such that any two consecutive codewords differ by exactly
one bit. In fact, we show that the problem of finding a Hamiltonian path
on an incomplete hypercubes with system size N is equivalent to the problem
of how to generate a GGC with N codewords. Since the Gray code has been
applied in many regions such as coding theory, analogue-to-digital conversions,
neural networks, software engineering, image processing, pattern recognition,
communications, signal processing, nuclear science and electronic circuits.
It''s our opinion that the generalized Gray code can be also applied to the
related regions.
As a variety of interconnection networks are being proposed, portability of
algorithms across various interconnection networks is receiving a great deal
of attention. Therefore, one of the important properties of an interconnection
network H is that it should be able to embed other interconnection networks G
with low overheads. Thus the parallel algorithms designed for G can be
implemented on H systematically and efficiently. On the other hand, a multiprocessing
system may only allocate a part of the whole system for a task. Hence, we are
motivated to study the problem of how to embed these interconnection networks of
an arbitrary size into the incomplete hypercubes. In the third part of this
dissertation, we discuss how to embed a ring and a mesh of an arbitrary size
to the incomplete hypercube, since cycles and meshes demonstrate popular
importance. We have proposed an algorithm to enumerate rings on the incomplete
hypercubes optimally and definitely by the GGC. Moreover, We have improved the
expansion for the embedding a mesh to an incomplete hypercube.
In the forth part of this dissertation, we discuss the degree four chordal rings. First, we show that ILLIAC networks are isomorphic to a subclass of the degree four chordal rings. Therefore, ILLIAC networks can be regarded as a special class of the degree four chordal rings. We investigate the topological properties, such as diameter and average distance of ILLIAC networks and the optimal degree four chordal rings, another special class of degree four chordal rings. In addition, we study the abilities of degree four chordal rings and ILLIAC networks to execute parallel program using graph-embedding techniques. We discuss the mapping functions, simple mapping and snake-like mapping, of embedding meshes and torus networks onto the degree four chordal rings in detail. Comparisons of ILLIAC networks and optimal chordal rings in these embedding issues are also given.
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