TESI DOCTORAL - La Salle

3.3. Deterministic hierarchical consensus architectures
Serial DHCA
Firstly, let us consider the fully serialized version of the DHCA. In this case, the time
complexity amounts to the sum of all the consensus processes, as defined by equation (3.14).
Notice that STCDHCA can be expressed ultimately in terms of the number and cardinalities
of the diversity factors employed in the generation of the cluster ensemble.
STCDHCA =
s Ki
i=1 j=1
O (bij w )=
s
i=1
Ki · O (bij w )=
f
f
i=1
k=i+1
Keeping the higher order terms, serial DHCA time complexity is:
Parallel DHCA
f
STCDHCA = O |dfk| |df1| w
k=2
|dfk| · O (|dfi| w ) (3.14)
(3.15)
And secondly, the time complexity of the parallelized execution of the DHCA is presented
in equation (3.16). As all the consensus processes in a given DHCA stage are equally costly,
the value of PTCDHCA amounts to the addition of the complexities of one of the consensus
processes run on each of the s stages of the hierarchy.
PTCDHCA =
s
i=1
O (bij w )=
f
O (|dfi| w ) (3.16)
Notice that the parallel execution of a DHCA can be regarded as a sequence of f
instructions of complexity O (|dfi| w ) , ∀i ∈ [1,f]. Therefore, applying the sum rule of
assymptotic notation, PTCDHCA can be rewritten as:
PTCDHCA = O
3.3.3 Running time minimization
i=1
max (|dfi|)
i
w
(3.17)
As in section 3.2, a naturally arising question regarding the practical implementation of
deterministic hierarchical consensus architectures is the following: given a cluster ensemble
of size l created upon a set of diversity factors dfi (for i = {1,...,f}), which is the least
time consuming DHCA variant that can be built?
Indeed, as the topology of a deterministic hierarchical consensus architecture is ultimately
determined by an ordered list O of the f diversity factors indicating upon which
diversity factor consensus is conducted at each DHCA stage, there exist f! distinctDHCA
variants for a given consensus clustering problem —one for each of the possible ways of ordering
the f diversity factors. Then, the question transforms into: how should the diversity
factors be ordered so as to minimize the total running time of the DHCA?
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