16.7: Memorized Fibonacci
- Page ID
- 46426
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As a small application of the techniques we have seen, consider the Fibonacci function (\( \mathit{fib}(n) = \mathit{fib}(n - 1) + \mathit{fib}(n - 2) \text{ with } \mathit{fib}(0) = 0, \mathit{fib}(1) = 1 \)). We will study two versions of it: a recursive version and a memorized version. Memoizing consists in introducing a cache to avoid redundant computation.
Consider the following definition, close to the mathematical definition:
Integer»fibSlow
self assert: self >= 0.
(self <= 1) ifTrue: [ ^ self].
^ (self - 1) fibSlow + (self - 2) fibSlow
The method fibSlow is relatively inefficient. Each recursion implies a duplication of the computation. The same result is computed twice, by each branch of the recursion.
A more efficient (but also slightly more complicated) version is obtained by using a cache that keeps intermediary computed values. The advantage is to not duplicate computations since each value is computed once. This classical way of optimizing program is called memoizing.
Integer»fib
^ self fibWithCache: (Array new: self)
Integer»fibLookup: cache
| res |
res := cache at: (self + 1).
^ res ifNil: [ cache at: (self + 1) put: (self fibWithCache: cache ) ]
Integer»fibWithCache: cache
(self <= 1) ifTrue: [ ^ self].
^ ((self - 1) fibLookup: cache) + ((self - 2) fibLookup: cache)
As an exercise, profile 35 fibSlow and 35 fib to be convinced of the gain of memoizing.