Theoretical and Experimental DNA Computation (Natural ...

98 4 Complexity Issues
1)), ADDR[]) is given by
S(n)−1
∨
FETCH[0...r − 1] = k=0 Mk[] ∧ (bin(k) ≡ ADDR[])
The total number of two-input gates required to realize this operation is
Size(FETCH) ≤ S(n) − 1+S(n)(r + m) =O(S(n)logS(n))
Depth(FETCH) ≤ m + log(m) =O(log S(n)))
We now describe in detail the combinational circuit associated with each operation
in Table 4.4.
Load constant x
This merely involves transcribing the r-bit binary representation of the
constant x onto the r-wires representing the accumulator contents.
Size(LDC) =0;Depth(LDC) =0
Load direct x
This is equivalent to FETCH(M[],x) and transcribing the output to the
accumulator wires.
Size(LDA) =O(S(n)logS(n)); Depth(LDA) =O(log S(n))
Load indirect x
Indirect addressing involves two successive invocations of the FETCH
module, i.e., FETCH(M[],FETCH[M[],x)) and transcribing to the accumulator.
Size(LDI)=O(S(n)logS(n)); Depth(LDI) =O(log S(n))
ADD x
ADD(ACC[],FETCH(M[],x)). An addition module, ADD(x, y) adds
two r-bit values and returns an r-bit value (overflow is assumed to cause
an error). A number of combinational circuits exist which achieve this in O(r)
gates and O(log r) depth, e.g., [91]. Hence, the dominant complexity contribution
arises from fetching the operand. In total we have
Size(ADD) =O(S(n)logS(n)); Depth(ADD) =O(log S(n))
SUB x
Similar to ADD using, for example, complement arithmetic.