TESI DOCTORAL - La Salle

Chapter 3. Hierarchical consensus architectures
the mini-ensembles size, notice that the size of the third mini-ensemble of the first RHCA
stage is increased (b13 =3) so that all the l = 7 components of the cluster ensemble are
involved in one of the K1 = 3 consensus processes of the first RHCA stage. This also
happens in the second stage, where b21 =3andK2 =1,which,asjustmentioned,yieldsa
single consensus at its output.
The interested reader will find a more detailed description of these and other RHCA
configuration examples in appendix C.1.
3.2.2 Computational complexity
In the following paragraphs, we present a study of the asymptotic computational complexity
of RHCA, considering both its fully serial and parallel implementations, which, as
aforementioned, constitute the upper and lower bounds of the RHCA execution time.
Serial RHCA
For starters, the time complexity of the fully serialized implementation is considered. This
means that the intermediate consensus tasks of each RHCA stage must be sequentially executed
on a single computation unit. Recall that the time complexity of consensus functions
typically grows linearly or quadratically with the cluster ensemble size, that is, it can be
expressed as O (l w ), where w ∈{1, 2}. Therefore, the serial time complexity of a RHCA
(STCRHCA) withs stages boils down to systematically adding the time complexities of all
the consensus processes executed across the whole RHCA, as defined in equation (3.4).
STCRHCA =
s Ki
O (bij w ) (3.4)
i=1 j=1
where Ki refers to the number of consensus processes executed in the ith RHCA stage, bij is
the mini-ensemble size corresponding to the jth consensus process executed at the ith stage
of the hierarchy —the exact value of these parameters is computed according to equations
(3.2) and (3.3), respectively—, and O (bij w ) reflects the complexity of each intermediate
consensus process.
Equation (3.4) can be reformulated so as to obtain a compact expression of an upper
bound of STCRHCA as a function of the user defined mini-ensembles size b. This requires
recalling that, in the current RHCA implementation, the effective mini-ensembles size is
bounded, that is, bij < 2b, ∀i ∈ [1,s]and∀j ∈ [1,Ki]. Therefore, we can write:
STCRHCA <
s Ki
O ((2b) w ) (3.5)
i=1 j=1
Notice that, from an algorithmic viewpoint, equation (3.5) can be regarded as two nested
loops where the number of iterations of the inner loop (Ki) depends on the value of the
outer loop’s index (i). The number of iterations of the inner loop as a function of the outer
loop’s index is presented in table 3.1.
Thus, it can be observed that the total number of times the mini-ensemble consensus of
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