...
- [30 pts] Section 14.2 in the HTDP First Edition book describes what it calls Binary Search Trees. The terminology in this section of the book is non-standard because a Binary Search Tree contains both keys and values in each node and hence represents a finite mapping from keys to values. The stub file contains a simple Racket programming problem (with a solution consisting of only a few lines of executable Racket code) based on essentially the same inductive data definition as Binary Search Trees but the type of the
value
field is parametric (alpha
) which must be instantiated tosymbol
to match the explication of Binary Search Trees in the book. Using the stub code for Problem 1 inHW02.rkt
as a starter template, your task is to- Give some examples of the (
BSTM-of symbol)
type. - Devise a set of test cases (input-output pairs expressed using
check-expect
) for thegetBSTM
function. - Write a Template Instantiation for
getBSTM
(based on the general template for functions that process (BSTM-of symbol).
- Develop the code for the function
getBSTM
that satisfies the contract given in the stub file. - Briefly compare the asymptotic worst-case running time of searching a (
BSTM-of symbol)
that is well balanced (maximum depth is proportional to the log N where N is the number of keys in the (BSTM-of symbol)
) and function searches an ordered(list-of
(pair-of key value))
, where each pair is represented as a two-element lists (as in Problem 2 below).
- Give some examples of the (
- [30 pts] The stub file
HW02.rkt
provides a detailed description of how to develop the functioncross
(and supporting functioncross-help
) that consumes a(list-of number)
and a(list-of
symbol)
and produces a(pair-of number symbol)
where a(pair-of number symbol)
is represented by a two-elementlist
containing anumber
and asymbol
. - [30 pts] The stub file
HW02.rkt
provides a detailed description of how to develop the functionmerge
(and supporting functionmerge-help
) that consumes two ascending (technically non-descending}(list-of number)
inputs merges them to form an ascending(list-of number)
. [10pts] The ubiquitous Fibonacci function defined by the trivial
fib
program given in the stub tileHW02.rkt
is interminably slow (exponential running time) for large inputs. Develop a Racket functionfastFib
that consumes a natural numbern
, produces the same answer as thefib
function defined in the stub file, and runs in linear time (assuming that the primitive addition operation runs in constant time, which fails for very largen
). Hint: write a help functionfastFibHelp
that accumulates the result in an accumulator argument performing essentially the same computation as an imperative program relying on a loop that maintainsfib(k-1)
andfib(k-2)
in mutable variables ask
increases from2
to n. The poor efficiency of the trivial functional program forfib
is due to the fact that it repeatedly computes the Fibonacci function for smallk
exponentially many times.- Show Type Contracts, Purpose Statements, Examples, and Template Instantiations for
fastFibHelp
andfastFib
. (The answers for the Template Instantiations can vary; only the salient features (primarily recursive calls) matter.) - As usual, testing comes for free given that you provided input-output examples. Make sure that after you run your program that no source code text (definitions of
fastFib
andfastFibHelp)
is shaded in the DrRacket definitions panel. Such shading indicates a failure to evaluate the shaded expressions in any test case.
- Show Type Contracts, Purpose Statements, Examples, and Template Instantiations for
Optional problem for extra credit: [50 pts (a challenging problem)]
The Fibonacci function fib
is defined in the stub for Problem 4 in HW02.rkt. The naive program for computing fib
coded in the file HW02.rkt runs in exponential time, i.e. the running time ofh (fib n)
is proportional to C*bn
for some proportionality constant C and base b > 1. It is easy to write a program that computes (fib n)
in time proportional to n
as assigned in Problem 4. Your challenge is to write a program that computes (fib n)
in log n
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 C*
log
n
operations for some constant C).
...
Derive a recurrence for
fib(2*m)
in terms offib(m)
andfib(m-1)
. Derive a similar recurrence for fib(2*m+1). To produce an algorithm that runs in log operations you need to reduce computing the pair(fib(2*m),fib(2*m-1))
to computing(fib(m),fib(m-1))
using a bounded number of arithmetic operations and tests.Initially write a program that works when n is a power of 2. Then refine this prototype to a program that works for all n based on determining whether n is even or odd.
This is a challenging problem. Make sure that you have thoroughly completed the regular homework problems before attempting it.
- In my solution, I used "dotted pairs" to reduce overhead. The "dotted pair" representation of a pair (a,b) is (cons a b) which is illegal in all of the HTDP dialects when b is not a list, It is supported in the "other language" called "Pretty Big". This design choice is a bit of a stunt to minimize obvious overhead. Of course, you can define pairs using
(define-struct pair (left right))
without compromising the goal of a solution that only requiresO(
log
n)
operations. My intuition was that such pairs have more overhead than dotted pairs but I did not perform any benchmark comparisons. If you decide to use a language other than Intermediate student with lambda, please Please put your solution to the challenge problem in a separate file called Chal02.txt and put a comment in your regular solution file HW02.rkt for problem 5 to that effectrkt and submit as instructed in the Canvas announcement (11:59 pm on Monday, Sept 23).