TESI DOCTORAL - La Salle

3.2. Random hierarchical consensus architectures
s : number of stages
10 1
Number of RHCA stages as a function of b
10 0
10 0
10 1
10 2
b
10 3
100
200
500
1000
2000
5000
10000
10 4
(a) Evolution of the number of
RHCA stages s as a function
of b
10 4
10 3
10 2
10 1
10 0
10 0
10 1
10 2
b
10 3
100
200
500
1000
2000
5000
10000
10 4
sum(K i ) : total number of consensus Number of consensus in a RHCA as a function of b
(b) Evolution of the total
number of RHCA consensus
s
Ki as a function of b
i=1
10 4
10 3
10 2
10 1
b ij : mean mini−ensemble size Mini−ensembles size of a RHCA as a function of b
10 0
10 0
10 1
10 2
b
10 3
100
200
500
1000
2000
5000
10000
10 4
(c) Evolution of the mean
mini-ensembles size as a function
of b
Figure 3.3: Evolution of RHCA parameters as a function of the mini-ensembles size b for
cluster ensembles sizes ranging from 100 to 10000.
3.2.3 Running time minimization
PTCRHCA O (log b (l)(2b) w )=O (b w log b (l)) (3.11)
In light of the expressions of the upper bounds of the serial and parallel time complexities
of RHCA, a naturally arising question is which particular RHCA configuration yields, for
a given cluster ensemble, the minimal running time —notice that the user’s election of the
mini-ensembles size b determines both the number of stages and of consensus computed per
stage, see equations (3.1) and (3.2), which ultimately determines the running time of the
RHCA.
In fact, there exists a trade-off between the value of b and the execution time of the
whole RHCA, as selecting a small value for b simultaneously reduces the time complexity of
each consensus while increasing the total number of stages (s) and of consensus processes
of the RHCA ( s
Ki), and vice versa.
i=1
With the purpose of visualizing the dependence between b and these factors, figure 3.3
depicts their value for different cluster ensembles sizes l ∈ [100, 10000] as a function of the
mini-ensembles size b ∈ 2, ⌊ l
2 ⌋ .
Firstly, figure 3.3(a) shows the exponential decay of the number of RHCA stages s as a
function of b, caused by the fact that s is computed as the b-base logarithm of the cluster
ensemble size l. Secondly, the evolution of the total number of consensus processes follows
a fast exponential decay (hard to appreciate in this doubly logarithmic chart), as depicted
in figure 3.3(b). And finally, figure 3.3(c) presents the mean value of the effective miniensembles
size bij across the whole RHCA (which obviously scales linearly with b)asarough
indicator of the complexity of each halfway consensus process, which will approximately be
(linearly or quadratically) proportional to this value.
Allowing for the evident dependence between the user defined mini-ensembles size b and
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