TESI DOCTORAL - La Salle

3.3. Deterministic hierarchical consensus architectures
1. Given the cluster ensemble size l generated upon a set of f mutually crossed diversity
factors, create f! ordered lists corresponding to all the possible permutations of the
diversity factors, each giving rise to a DHCA variant.
2. For each one of the ordered lists, compute the total number of consensus processes
per stage Ki, according to equation (3.13).
4. Measure the time required for executing the consensus function F on c randomly
picked mini-cluster ensembles of sizes |dfi| for i ∈{1,...,f}.
5. Employ the computed parameters of each DHCA variant (i.e. number of stages s =
f, total number of consensus processes s
Ki and the running times of the consensus
i=1
function F) to estimate the running times of the whole hierarchical architecture,
using equations (3.14) or (3.16) depending on whether its fully serial or parallel
version is to be implemented in practice.
Table 3.6: Methodology for estimating the running time of multiple DHCA variants.
first stage of the DHCA is minimized, which is equivalent to minimizing the necessary
computation units for the fully parallel implementation of the DHCA.
If the running time of the serial implementation of the DHCA is now considered, the
total number of consensus to be executed is not the only factor to take into account. In
fact, arranging the diversity factors in decreasing order, while minimizing the total number
of consensus to be executed across the DHCA, brings about a contradictory collateral effect
if the complexity and number of the consensus processes run at each stage are considered.
Indeed, notice that it is in the first DHCA stage where the largest number of consensus
processes is executed (K1 = |df2||df3|), and they have the highest complexity (O (|df1| w ),
where w = {1, 2}), as |df1| ≥|df2| ≥|df3|. Moreover, a single minimum complexity (i.e.
O (|df3| w )) consensus process is conducted at the third stage, as K3 =1.
Thus, as far as the running time of the serial implementation of DHCA is concerned,
there exists an apparent trade-off involving the order of diversity factors, and the number
and complexity of the associated consensus processes. It is important to note that the computationally
optimal solution ultimately depends on the growth rates of the total number
of consensus and of their time complexities with respect to the cardinality of the diversity
factors (|dfi|, fori = {1,...,f}). However, while the latter grow according to a linear or
quadratic law, the former follows a usually steeper multiplicative growth rate.
Similarly to what has been discussed in section 3.2, table 3.6 presents a methodology
for estimating the running times of the f! DHCA variants, which allows comparing them
and, as a consequence, predicting which is the most computationally efficient.
74