SICP Exercise 3.57 fibs using add-streams

Exercise 3.57.  How many additions are performed when we compute the nth Fibonacci number using the definition of fibs based on the add-streams procedure? Show that the number of additions would be exponentially greater if we had implemented (delay <exp>) simply as (lambda () <exp>), without using the optimization provided by the memo-proc procedure described in section 3.5.1.64

 SOLUTION

 The Racket code and explanation is here.

 The Scheme code and tests are here.

 EXPLANATION

 (define fibs
(stream-cons
0
(stream-cons
1
(add-streams (stream-rest fibs) fibs)
)
)
)

 Number of additions performed:

 0th Fibonacci number (0): 0 additions
1st Fibonacci number (1): 0 additions
2nd Fibonacci number (1): 1 addition
3rd Fibonacci number (2): 2 additions
4th Fibonacci number (3): 3 additions

 As we can see above, for every higher Fibonacci number, one additional addition is performed. So when we compute the nth Fibonacci number, n - 1 additions are performed in total.

Now, if we had implemented (delay <exp>) simply as (lambda () <exp>) without memoization, then, upon accessing the nth Fibonacci number using (stream-ref fibs n) where n > 1, add-streams is executed which accesses the fibs stream twice. Since accessing fibs results in a call to add-streams, the result is that every add-streams call results in two calls to add-streams. This pattern results in 2^n additions. 

 Therefore the number of additions will be exponentially greater compared to when we use memoization in the 'delay' implementation.

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