Since the A-depth must be preserved equal to p, only p of these operations can be sequentially connected.
Since the A-depth is equal to p, only p of these operations can be sequentially connected.
Similarly, we will denote the new lower bounds as LA,new for the A-cost and Ld,new for the A-depth, which can be obtained using (4) and (5 - 6), respectively.
Let us denote the current lower bounds from [12] as LA,c for the A-cost and Ld,c for the A-depth, which can be obtained using (2).
However, according to Theorem 3, the A-depth can be kept equal to ⌈ log2S c)⌉ by using only one additional A-operation if the condition S(c) ≤ 3 × 2 log 2 S c − 2 + 1 holds.
This paper has presented an extension of the current theoretical lower bounds for the A-cost and the A-depth in single constant multiplication (SCM) blocks constructed with shifts and A-operations (additions and subtractions).
The theoretical lower bound for the A-cost required when the minimum A-depth is preserved, as well as the lower bound for the A-depth necessary to obtain the minimum A-cost, can be revealed.
If the required multiplicative characteristics for the corresponding graph of the constant c, highlighted in the previous theorems, cannot be accomplished due to the value Ω(c), the lower bounds for the A-cost and the A-depth are affected.
The lower bounds L A and L d for the A-cost and the A-depth of the graph for c, respectively, are given in [12] as L A = L d = log 2 S c (2).
Relatively high concentrations of certain elements (e.g. S, P, Pb, Hg, Cd), particularly in the A-depth, are attributed to anthropogenic sources such as fertilisers, paints, vehicle emissions or industrial emissions.
It is worth highlighting that, whereas the lower bound L A only depends on the relation between Ω(c) and S(c), the lower bound L d depends on the A-cost N A. As pointed out in [12], for many problem instances, there is a tradeoff between the A-cost and the A-depth.
