TESI DOCTORAL - La Salle

Chapter 3. Hierarchical consensus architectures
the running of RHCA, it seems necessary to accurately choose the value of this parameter
—regardless of whether the serial or parallel version of RHCA is implemented (in this latter
case, notice that the RHCA running time still depends on b (via s) although it becomes
s
insensitive to the total number of consensus processes, Ki)—,asRHCAvariantswith
different values of b may have dramatically different running times.
As mentioned earlier, we aim to design an automatic mechanism that allows selecting
thespecificvalueofb that gives rise to the RHCA variant of minimal running time, making
it also possible to decide aprioriwhether it is more computationally efficient than flat
consensus.
To do so, we propose a simple yet effective methodology for comparing distinct RHCA
variants (i.e. with different b values) on computational grounds, a detailed description of
which is presented in table 3.2. In a nutshell, the proposed strategy is based on estimating
the RHCA variants running time using equations (3.4) or (3.9) (depending on whether the
serial or parallel version is to be implemented), replacing the theoretical time complexity of
intermediate consensus processes (O (bij w )) by the average real execution time of c consensus
processes using a specific consensus function F on a mini-ensemble of size bij —recall that
the values of bij can be computed by means of equation (3.3).
It is to note that, for a specific RHCA variant (that is, for a given mini-ensemble size
b), the parameter bij may take a few different values —for instance, in the three toy RHCA
variants depicted in figure 3.2, bij = {2, 3} although b = 2 in all of them. That is, the
number of consensus processes to be executed for estimating the running time of the RHCA
is usually small, which makes the proposed procedure little computationally demanding.
Futhermore, notice that a more robust estimation can be obtained by averaging the running
times of c>1 executions of the consensus function F on mini-ensembles of size bij, although
this would increase the time required for completing the estimation procedure.
By means of the proposed methodology, the user is provided with an estimation of
which is the most computationally efficient RHCA configuration and which is its running
time for the consensus clustering problem at hand. However, it is necessary to decide
whether this allegedly optimal RHCA variant is faster than flat consensus. The running
time of flat consensus can be estimated by extrapolating from the running times of the
consensus processes executed upon mini-ensembles of size bij. A simpler although less
efficient alternative consists of launching the execution of flat consensus, halting it as soon
as its running time exceeds the estimated execution time of the allegedly optimal RHCA
variant.
The next section is devoted to illustrate the performance of the proposed running time
estimation methodology by means of several experiments, as well as for evaluating the
computational efficiency of RHCA in front of flat consensus.
3.2.4 Experiments
In the following paragraphs, we present a set of experiments oriented to i) evaluate the
performance of the computationally optimal consensus architecture prediction methodology,
and ii) analyze the computational efficiency of random hierarchical consensus architectures.
To do so, we have designed several experiments which are outlined next.
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i=1