TESI DOCTORAL - La Salle

Chapter 3. Hierarchical consensus architectures
i # inner loop iterations
1 K1
2 K2
... ...
s 1
Table 3.1: Number of inner loop iterations as a function of the outer’s loop index i.
s
Ki ≤
i=1
s
i=1
l
= l ·
bi s
i=1
1 1
1 b − b = l ·
bi s+1
1 − 1
bs − 1
= l ·
b
b
s
(3.6)
(b − 1)
. Therefore, the
which equals the partial sum of a geometric series whose common ratio is 1
b
upper bound of the time complexity STCRHCA can be rewritten as:
bs − 1
STCRHCA
Ki, ∀i ∈ [2,s]). In this case of maximum parallelism, the parallel time complexity of a
RHCA (PTCRHCA) ofs stages is formulated according to equation (3.9).
PTCRHCA =
s
i=1
max
j∈[1,Ki] (O (bij w )) (3.9)
That is, the parallel time complexity of a RHCA is equal to the sum of the time complexities
of the most time-consuming consensus process of each RHCA stage. Notice that
it is not difficult to find an upper bound to PTCRHCA, as finding the maximum of O (bij w )
just requires taking into account that bij < 2b, ∀i, j. Thus:
PTCRHCA <
s
i=1
O ((2b) w )=s · O ((2b) w )=O (s · (2b) w ) (3.10)
If the number of RHCA stages is approximated as s ≈ log b (l), and constants are
dropped, equation (3.10) can be rewritten as a function of l and b:
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