TESI DOCTORAL - La Salle

3.2. Random hierarchical consensus architectures
1. Given the cluster ensemble size l, create a set of mini-ensembles sizes b with values
sweeping from 2 to ⌊ l
2 ⌋.
2. For each b value –which corresponds to a RHCA variant– compute the number of
stages of the RHCA s according to equation (3.1). With these results in hand, limit
the sweep of values of b according to two criteria:
i) as there exist multiple values of b that yield RHCA variants with the same number
of stages, consider only the largest and smallest values of b that yield the same
number of RHCA stages s.
ii) keep those values of b that uniquely give rise to RHCA variants with a specific
number of stages.
3. For the reduced set of b values, compute the total number of consensus processes
s
Ki, and the real mini-ensembles sizes bij of the corresponding RHCA variants,
i=1
according to equations (3.2) and (3.3), respectively.
4. Measure the time required for executing the consensus function F on c randomly
picked mini-cluster ensembles of the sizes bij corresponding to each value of b.
5. Employ the computed parameters of each RHCA variant (i.e. number of stages s,
s
total number of consensus processes Ki and the running times of the consensus
i=1
function F) to estimate the running times of the whole hierarchical architecture,
using equations (3.4) or (3.9) depending on whether its fully serial or parallel version
is to be implemented in practice.
Table 3.2: Methodology for estimating the running time of multiple RHCA variants.
Experimental design
– What do we want to measure?
i) The time complexity of random hierarchical consensus architectures.
ii) The ability of the proposed methodology for predicting the computationally optimal
RHCA variant, in both the fully serial and parallel implementations.
– How do we measure it?
i) The time complexity of the implemented serial and parallel RHCA variants is
measured in terms of the CPU time required for their execution —serial running
time (SRTRHCA) and parallel running time (PRTRHCA).
ii) The estimated running times of the same RHCA variants –serial estimated running
time (SERTRHCA) and parallel estimated running time (PERTRHCA)– are
computed by means of the proposed running time estimation methodology, which
is based on the measured running time of c = 1 consensus clustering process. Predictions
regarding the computationally optimal RHCA variant will be successful
56