The.Algorithm.Design.Manual.Springer-Verlag.1998

Fibonacci numbers
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Fibonacci numbers
The tradeoff between space and time exploited in dynamic programming is best illustrated in evaluating
recurrence relations, such as the Fibonacci numbers. The Fibonacci numbers were originally defined by
the Italian mathematician Fibonacci in the thirteenth century to model the growth of rabbit populations.
Rabbits breed, well, like rabbits. Fibonacci surmised that the number of pairs of rabbits born in a given
year is equal to the number of pairs of rabbits born in each of the two previous years, if you start with
one pair of rabbits in the first year. To count the number of rabbits born in the nth year, he defined the
following recurrence relation:
with basis cases and . Thus , , and the series continues
. As it turns out, Fibonacci's formula didn't do a very good job of counting
rabbits, but it does have a host of other applications and interesting properties.
Since they are defined by a recursive formula, it is easy to write a recursive program to compute the nth
Fibonacci number. Most students have to do this in one of their first programming courses. Indeed, I
have particularly fond memories of pulling my hair out writing such a program in 8080 assembly
language. In pseudocode, the recursive algorithm looks like this:
Fibonacci[n]
if (n=0) then return(0)
else if (n=1) then return(1)
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else return(Fibonacci[n-1]+Fibonacci[n-2])