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

Fibonacci numbers
Figure: The computation tree for computing Fibonacci numbers recursively
How much time does this algorithm take to compute Fibonacci[n]? Since
, this means that . Since our recursion tree, illustrated in
Figure , has only 0 and 1 as leaves, summing up to such a large number means we must have at least
leaves or procedure calls! This humble little program takes exponential time to run!
In fact, we can do much better. We can calculate in linear time by storing all values. We trade space
for time in the algorithm below:
Fibonacci[n]
For i=1 to n,
Because we evaluate the Fibonacci numbers from smallest to biggest and store all the results, we know
that we have and ready whenever we need to compute . Thus each of the n values is computed
as the simple sum of two integers in total O(n) time, which is quite an improvement over exponential
time.
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