In the following, we state, in Subsection 3.1, Theorems 1 to 8 to derive the lower bounds of Roperations in PSCM and, in Subsection 3.2, Theorems 9 and 10 for PMCM, along with their corresponding proofs. The pipelining operation, which has not been alluded in the previous works [3] and [4], is explicitly included in the proposed lower bounds through the Roperations.
3.1 PSCM case
Whenever a constant c is mentioned in the theorems of this subsection (Theorems 1 to 8), we consider that the MNSD of that constant is S and its number of prime factors is Ω.
Theorem 1 provides the upper limit of nonzero digits that can be generated by any graph with a given number of depth levels, regardless of its number of Roperations. From this, we can know the minimum number of depth levels that a graph must have to implement a constant with a given S.
Theorems 2 and 3 prove the properties of the completely multiplicative graphs, namely, generating the upper limit of nonzero digits mentioned in Theorem 1 with the minimum possible number of Roperations. From them, we have that the completely multiplicative graph is a solution with the lower bound for the number of Roperations. However, as it is known, this graph has articulation points, and every articulation point represents the union between two cascaded subgraphs, i.e., the product of two smaller constants. Therefore, Theorem 4 uses Ω to identify what constants can be implemented with the completely multiplicative graph (for example, prime constants cannot be factorized into smaller constants; thus, they cannot be implemented by a completely multiplicative graph).
Theorem 5 identifies the minimum number of Roperations needed in any nonmultiplicative graph with a given number of depth levels, and Theorem 6 proves that nonmultiplicative graphs can generate the upper limit of nonzero digits mentioned in Theorem 1 with its minimum number of Roperations. Then, Theorem 7 establish the lower bound for the number of Roperations needed to implement a prime constant (Ω = 1).
Finally, Theorem 8 completes the information of Theorems 4 and 7, namely, the lower bound of Roperations needed to implement nonprime constants that have fewer number of factors than the number of subgraphs used in a completely multiplicative graph.
Theorem 1
A graph with p depth levels can provide at most n
^{p}
nonzero digits for a constant.
Proof
The proof is given by induction (see proof of Theorem 6.9 in [39] for the case of 2input Aoperations):

1)
The base case corresponds to the first depth level, where a ninput Aoperation can form a constant with at most n nonzero digits. This is true since the input of any graph has one nonzero digit [3, 4, 39].

2)
As inductive step, we assume that, in the pth level, there are n
^{p} nonzero digits at most. In the (p + 1)th level, an Aoperation can form a constant whose number of nonzero digits is the sum of the numbers of nonzero digits at every input of that Aoperation. This is at most n times the maximum number of nonzero digits available in the previous level, i.e., n × n
^{p} = n
^{p + 1} nonzero digits.
Since assuming that the theorem is true for p implies that the theorem is also true for p + 1, and since the base case is also true, the proof is complete. An adder, regardless of its number of inputs, cannot generate more nonzero digits than the sum of the numbers of nonzero digits in every one of its inputs. Thus, the MNSD can be, at most, nplicate if the inputs of the ninput adder placed in any depth level come from the immediately previous depth level. ■
Theorem 2
A completely multiplicative graph with p Aoperations can generate n
^{p}
nonzero digits.
Proof
This proof is an straightforward extension of the proof of Theorem 6.8 in [39], which corresponds to completely multiplicative graphs with 2input Aoperations. As stated earlier, the input of a graph has one nonzero digit. In the completely multiplicative graph, there are at most n nonzero digits after the Aoperation placed at the 1st depth level. Cascading an Aoperation to that output yields at most n × n nonzero digits, and so on. The number of nonzero digits at the depth level p is at most the ntuple of the number of nonzero digits of a fundamental at the (p − 1)th depth level. Consequently, the maximum number of nonzero digits at the pth depth level is n
^{p}. ■
Theorem 3
A completely multiplicative graph with p depth levels needs only p Roperations.
Proof
The completely multiplicative graph with p depth levels has p Aoperations, and every Aoperation forms a subgraph. Pipelining between two subgraphs needs only one register, according to [38], because the pipelining occurs on the articulation point. This results in every Aoperation being followed by a register. Since an Aoperation followed by a register is considered an Roperation, there are only p Roperations in total. This is illustrated in Fig. 7. ■
Theorem 4
A constant with (n
^{p − 1}
+ 1) ≤ S ≤ n
^{p}
and Ω ≥ p needs at least p Roperations.
Proof
From Theorem 2, we have that a constant with (n
^{p − 1}
+ 1) ≤ S ≤ n
^{p} nonzero digits can be implemented with at least p depth levels, which implies at least p Aoperations. From Theorem 3, we have that a completely multiplicative graph can generate those values for S with only p Roperations. The completely multiplicative graph with p Roperations consists of p cascaded subgraphs; thus, a constant implemented with that graph must have at least p prime factors. Since Ω ≥ p holds, the completely multiplicative graph can be employed to implement that constant using p Roperations. ■
Theorem 5
A nonmultiplicative graph with p depth levels needs at least (2p − 1) Roperations.
Proof
According to Theorem 3, if a graph with p depth levels has only p Roperations in total, it must be a pipelined completely multiplicative graph. According to Theorem 2, that graph can generate the maximum possible number of nonzero digits, namely, n
^{p}. To make nonmultiplicative that optimal graph, the (p − 1) articulation points must be eliminated. From [38], it is known that at least one additional Roperation must be added for every eliminated articulation point. Therefore, at least (2p − 1) Roperations are required, i.e., the original p minimum number of Roperations in the form of additiondelay pairs plus the additional (p − 1) Roperations in the form of pure delays. Figure 8 shows an example with p = 3. ■
Theorem 6
A nonmultiplicative graph with p depth levels and (2p − 1) Roperations can generate n
^{p}
nonzero digits.
Proof
Consider a graph with p depth levels formed by two completely multiplicative graphs of (p − 1) levels each, connected in parallel from the input of the graph, and one Aoperation placed in the pth level summing up the outputs of the aforementioned graphs. The output of one of these graphs is connected to the n − 1 inputs of the last Aoperation, and the output of the other graph is connected to the remaining input of the last Aoperation. This is a nonmultiplicative graph because it is not formed by cascading subgraphs, and it is composed by (2p − 1) Aoperations. According to Theorem 2, we can obtain n
^{p − 1} nonzero digits from the completely multiplicative graphs, and according to Theorem 3, these graphs can be pipelined without requiring extra registers. Since the last Aoperation can add n times the n
^{p − 1} nonzero digits in each one of its inputs and can be pipelined without extra cost, the resulting graph generates n
^{p} nonzero digits using (2p − 1) Roperations. An example of this is shown in Fig. 9. ■
Theorem 7
A constant with (n
^{p − 1}
+ 1) ≤ S ≤ n
^{p}
and Ω = 1 needs at least 2p − 1 Roperations.
Proof
Since Ω = 1 holds, the nonmultiplicative graph must be employed to implement that constant. From Theorem 6, we have that a constant with (n
^{p − 1}
+ 1) ≤ S ≤ n
^{p} nonzero digits can be implemented with at least p depth levels and at least 2p − 1 Roperations. This is a lower bound for the number of Roperations, since from Theorem 5, we have that a nonmultiplicative graph with plevels needs at least 2p − 1 Roperations. ■
Theorem 8
A constant with (n
^{p−1}
+ 1) ≤ S ≤ n
^{p}
and 1 < Ω < p needs at least (2p − Ω) Roperations.
Proof
From Theorem 1, we have that p depth levels are necessary to achieve the values of S in the specified range. Since Ω < p holds, we can take advantage of a completely multiplicative graph with Ω−1 Roperations at most, which, according to Theorem 2, generates n
^{Ω−1} nonzero digits at most, and represents the product of Ω − 1 factors. The last factor can be formed with a nonmultiplicative subgraph with [p − (Ω − 1)] depth levels. According to Theorem 5, this subgraph needs at least 2[p − (Ω − 1)] − 1 Roperations, and according to Theorem 6, it can generate n
^{[p − (Ω − 1)]} nonzero digits. The total graph, illustrated in Fig. 10, can generate at most n
^{Ω − 1} × n
^{[p − (Ω − 1)]} = n
^{p} nonzero digits and uses at least (Ω − 1) + 2[p − (Ω − 1)] − 1 = 2p − 2(Ω − 1) + (Ω − 1 − 1 = 2p − (Ω − 1) − 1 = (2p − Ω) Roperations. ■
Finally, from Theorem 1, we have that the number of depth levels necessary to achieve S is p = ⌈ log_{
n
}(S)⌉. Substituting this value for p and using Theorems 4, 7, and 8, we obtain the lower bound for the number of Roperations needed to form a PSCM block as follows:
$$ {L}_{SCM}=\left\{\begin{array}{l}2\left\lceil { \log}_n(S)\right\rceil \varOmega; \kern3.5em \varOmega <\left\lceil { \log}_n(S)\right\rceil, \\ {}\left\lceil { \log}_n(S)\right\rceil; \kern2.25em \varOmega \ge \left\lceil { \log}_n(S)\right\rceil .\kern3.25em \end{array}\right. $$
(6)
3.2 PMCM case
The theorems in this section are stated for N constants c
_{1}, c
_{2}, …, c
_{
N
}, whose respective MNSDs are S
_{1}, S
_{2}, …, S
_{
N
}, and their respective numbers of prime factors are Ω_{1}, Ω_{2}, …, Ω_{
N
}, such that S
_{1} ≤ S
_{2} ≤ … ≤ S
_{
N
}.
Theorem 9 indicates the lower bound for the number of ninput Aoperations needed to form an MCM block. If pipelining is added, more Roperations than the aforementioned lower bound may be needed because the constants with fewer prime factors may use nonmultiplicative graphs, which require extra Roperations (see Theorems 5 to 8). Besides, all the outputs of the PMCM block must have equal number of depth levels to balance the input–output delay, which also may require extra Roperations. Based on these observations, Theorem 10 extends the lower bound provided in Theorem 9 by identifying at least how many extra Roperations would be needed. From these theorems, we obtain the lower bound for the number of Roperations needed to form a PMCM block.
Theorem 9
At least K ninput Aoperations are needed to build an MCM block, where K is given by
$$ K=\left\lceil { \log}_n\left({S}_1\right)\right\rceil +{\displaystyle \sum_{i=1}^{N1} E\left({S}_i,{S}_{i+1}\right)}, $$
(7)
with
\( E\left({S}_i,{S}_{i+1}\right)=\left\{\begin{array}{c}\hfill 1;\kern5em {S}_i={S}_{i+1},\hfill \\ {}\hfill \left\lceil { \log}_n\frac{S_{i+1}}{S_i}\right\rceil; \kern0.75em {S}_i<{S}_{i+1}.\hfill \end{array}\right. \)
Proof
Recall that every Aoperation has only one possible configuration and therefore can generate only one fundamental. Simply shifted (i.e., scaled by a power of two) versions of that fundamental can be obtained from that Aoperation. Since the target constants are integer and odd by definition, it is not possible to obtain two target constants from the same Aoperation. Therefore, there must be at least N ninput Aoperations for the N constants. Note that, since the terms S
_{
i
} are sorted in ascendant order, S
_{1} corresponds to the simplest constant, i.e., the one with the smallest number of nonzero digits. From Theorem 1, we have that with p depth levels we can obtain n
^{p} nonzero digits at most. By using the relation n
^{p} ≥ S
_{1}, we have that the minimum number of levels necessary to generate S
_{1} nonzero digits is ⌈ log_{
n
}(S
_{1})⌉, which implies the existence of at least ⌈ log_{
n
}(S
_{1})⌉ Aoperations for that constant. Finally, if S
_{
i+1} > n × S
_{
i
} holds, we have that a single Aoperation is not able to generate the constant c
_{
i+1} if there are only coefficients with at most S
_{
i
} digits available because the number of nonzero digits at the output of an Aoperation is at most the sum of the number of nonzero digits at its inputs. Therefore, at least ⌈ log_{
n
}(S
_{
i + 1}/S
_{
i
})⌉ Aoperations will be required. This proof is a straightforward extension of the proof given in [3] for the lower bound of 2input Aoperations that form an MCM block. ■
Theorem 10
At least L Roperations are needed to build a PMCM block, where L = K + F + G, with
$$ F=\left\{\begin{array}{c}\hfill {\displaystyle \underset{i}{ \max }}\left\{\left\lceil { \log}_n\left({S}_i\right)\right\rceil {\varOmega}_i\right\};\kern0.5em \forall\ i\kern0.5em \mathrm{such}\ \mathrm{that}\kern0.75em {\varOmega}_i<\left\lceil { \log}_n\left({S}_i\right)\right\rceil, \hfill \\ {}\hfill 0;\kern8.25em \mathrm{otherwise}.\hfill \end{array}\right. $$
$$ G={\displaystyle \sum_{i=1}^{N1}\left\lceil { \log}_n\left({S}_N\right)\right\rceil \left\lceil { \log}_n\left({S}_i\right)\right\rceil } $$
and K given in (7).
Proof
Consider that there is a constant c
_{
m
} that satisfies Ω_{
m
} < ⌈ log_{
n
}(S
_{
m
})⌉ and, if there are more constants that satisfy such condition, c
_{
m
} has the greatest difference [⌈ log_{
n
}(S
_{
m
})⌉ − Ω_{
m
}]. From Theorem 8, we have that the constant can be formed by cascading a nonmultiplicative graph with a completely multiplicative graph, where the nonmultiplicative graph needs 2[⌈ log_{
n
}(S
_{
m
})⌉ − (Ω_{
m
} − 1)] − 1 Roperations. Since Theorem 9 has not taken into consideration the number of prime factors, only [⌈ log_{
n
}(S
_{
m
})⌉ − (Ω_{
m
} − 1)] Aoperations have been accounted in that theorem, under the assumption that the constant c
_{
m
} can be constructed with the optimal completely multiplicative graph. Therefore, at least [⌈ log_{
n
}(S
_{
m
})⌉ − (Ω_{
m
} − 1)] − 1 extra Roperations must be included when pipelining is applied, which explains the term F. The term G is explained by the fact that extra Roperations may be needed to achieve the same number of pipelined stages from input to output in every constant. Since the minimum depth level of a constant is given by ⌈ log_{
n
}(S)⌉, the differences between the minimum depth level of the constant c
_{
N
} (which has the greatest depth level among other constants) and the minimum depth levels of the other constants are accumulated in the term G. ■
From Theorem 10, we can express the lower bound for the number of Roperations in the PMCM case as
$$ {L}_{PMCM}=\left\lceil { \log}_n\left({S}_1\right)\right\rceil +{\displaystyle \sum_{i=1}^{N1}\left(\left\lceil { \log}_n\left({S}_N\right)\right\rceil \left\lceil { \log}_n\left({S}_i\right)\right\rceil \right)}+{\displaystyle \sum_{i=1}^{N1} E\left({S}_i,{S}_{i+1}\right)}+ F, $$
(8)
with \( E\left({S}_i,{S}_{i+1}\right)=\left\{\begin{array}{c}\hfill 1;\kern5em {S}_i={S}_{i+1},\hfill \\ {}\hfill \left\lceil { \log}_n\frac{S_{i+1}}{S_i}\right\rceil; \kern0.75em {S}_i<{S}_{i+1},\hfill \end{array}\right. \)
and \( F=\left\{\begin{array}{c}\hfill {\displaystyle \underset{i}{ \max }}\left\{\left\lceil { \log}_n\left({S}_i\right)\right\rceil {\varOmega}_i\right\};\kern0.5em \forall\ i\kern0.5em \mathrm{such}\ \mathrm{that}\kern0.75em {\varOmega}_i<\left\lceil { \log}_n\left({S}_i\right)\right\rceil, \hfill \\ {}\hfill 0;\kern8.25em \mathrm{otherwise}.\hfill \end{array}\right. \)