Submit your .ss
file via OWL-Space. You will need to use the "Intermediate Student" language to do Problem 18.1.15. If you want to use explicit lambda
notation (anywhere the right hand side of a define
statement), you will need to use the "Intermediate Student with lambda" language. You may use either intermediate level language for the entire assignment if you choose.
Required problems:
du-dir
. The final sentence should read "storing a file or a directory in a dir structure costs 1 storage unit." In other words, given a dir structure, each directory entry (a file or a directory) contained therein costs 1 unit of storage for the bookkeeping data. For a file, this bookkeeping overhead is in addition to the size of its data.merge
, should be recursive (call itself directly or indirectly) and it may need to deviate slightly from the structural recursion template. The top level merge
function is not recursive.; expression ; An expression is one of: ; - a number ; - a symbol ; - (make-mul e1 e2) where e1 and e2 are expressions ; - (make-add e1 e2) where e1 and e2 are expressions |
(* (f (+ 15 x)) (g x)) |
Optional problem for extra credit: [50 pts]
The fibonacci function fib is defined by the following rules (in Scheme notation):
(fib 0) = 0 ;; formerly (fib 0) = 1; revised to match the prevailing mathematical conventions for defining fib (fib 1) = 1 (fib (+ n 1)) = (+ (fib n) (fib (- n 1))) |
A naive program for computing fib (lifted directly from the definition) runs in exponential time, i.e. the running time for (fib n)
is proportional to K*b**n
for some constants K
and b
). It is easy to write a program that computes (fib n)
in time proportional to n
. Your challenge is to write a program that computes (fib n)
in log time assuming that all multiplications and additions take constant time, which is unrealistic for large n
. More precisely, your program should compute (fib n)
using only O(log n)
addition and multiplication operations (less than K * log n
operations for some constant K
).
Hints: assume n = 2**m
. Derive a recurrence for fib 2**(m + 1)
in terms of fib 2**m
and fib 2**m - 1
. Initially write a program that works when n
is a power of 2. Then refine it to a program that works for all n
.