TESI DOCTORAL - La Salle

Chapter 3. Hierarchical consensus architectures
Notice that, as opposed to what was observed in random HCA, the distinct DHCA
variants do not differ in their number of stages (which is, in all cases, equal to the number
of diversity factors, i.e. s = f), but in the time complexity of each stage of the architecture.
Thus, in order to determine which is the computationally optimal DHCA variant, it is
necessary to analyze the dependence between the ordering of the diversity factors and the
total number of consensus processes executed and their complexity. In this section, we tackle
this issue for both the fully serial and parallel implementation of deterministic hierarchical
consensus architectures.
Without loss of generality, let us assume that consensus clustering is to be conducted
on a cluster ensemble of size l generated upon f = 3 mutually crossed diversity factors. By
means of an ordered list O, these three factors are associated to one of the stages of the
DHCA, i.e. O = {df1,df2,df3} —recall that, according to the definition of DHCA, i) the
numerical subindex of each diversity factor identifies the stage it is associated to, and ii)
Ki consensus processes of complexity O (|dfi| w )(wherew = {1, 2}) are conducted in the ith
stage, with i = {1, 2, 3} in this case.
As aforementioned, the total number of consensus processes depends on the cardinality
of the diversity factors, which in this particular case amounts to the expression presented
in equation (3.18):
f
Ki =
i=1
3
Ki =
i=1
3
|dfk| +
k=2
3
|dfk| +1=|df2||df3| + |df3| + 1 (3.18)
k=3
where the number of consensus per stage Ki is computed according to equation (3.13).
Firstly, let us analyze the running time of the fully parallel DHCA implementation. As
in section 3.2, we assume that sufficient computing resources allow the concurrent execution
of all the consensus processes of any of the DHCA stages —notice that this amounts to
having as many as |df2||df3| parallel computation units capable of running simultaneously
all the consensus processes of the first stage, which is the one with the largest number of
consensus.
If this condition is met, the running time of any parallel DHCA variant becomes independent
of the ordering of the diversity factors. This is due to the fact that, assuming
the fully simultaneous execution of all the consensus processes corresponding to the same
DHCA stage, the running time of the whole consensus architecture will be proportional to
O (|df1| w )+O (|df2| w )+O (|df3| w ) –as the running time of each DHCA stage is equivalent to
the execution of a single consensus process–, which is independent of which diversity factor
is assigned to each stage.
However, despite the diversity factors ordering does not affect the running time of parallel
DHCA variants, this factor has a significant impact on the dimensioning of the necessary
resources for the entirely parallel execution of all the consensus processes involved, as it is
directly related to the total number of consensus that must be executed in the DHCA.
f
From equation (3.18), it is straightforward to see that Ki is independent of the
cardinality of the diversity factor associated to the first DHCA stage (df1). Thus, notice
that the total number of consensus of a DHCA is minimized if the diversity factors are
arranged in the ordered list O according to their cardinality and in decreasing order, i.e.
|df1| ≥|df2| ≥|df3|. By doing so, the number of consensus processes conducted in the
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i=1