Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Algorithm: Addition of integers
procedure add(a, b: integers)
(a n-1 a n-2 …a 1 a 0 ) 2 := base_2_expansion(a)
(b n-1 b n-2 …b 1 b 0 ) 2 := base_2_expansion(b)
c := 0
for j := 0 to n-1
d := 7(a j +b j +c)/28
s j := a j +b j +c – 2d
c := d
s n := c
{the binary expansion of the sum is (s k-1 s k-2 …s 1 s 0 ) 2 }
Questions:
• What is the complexity of this algorithm?
• Is this the fastest way to compute the sum?
57
Algorithm: Mutiplication of integers
procedure multiply(a, b: integers)
(a n-1 a n-2 …a 1 a 0 ) 2 := base_2_expansion(a)
(b n-1 b n-2 …b 1 b 0 ) 2 := base_2_expansion(b)
for j := 0 to n-1
if b j = 1 then c j := a shifted j places else c j := 0
{c 0 ,c 1 ,…,c n-1 are the partial products}
p := 0
for j := 0 to n-1
p := p + c j
{p is the value of ab}
Examples:
• (10) 2 $(11) 2 = (110) 2 . Note that there are more bits than the original integers.
• (11) 2 $(11) 2 = (1001) 2 . Twice as many binary digits!
58