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• Do the following programming problems:
  1. [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 the Binary Search Trees contain both keys and values in each node and hence represents a finite mapping from keys to nodes.   The stub file contains describes 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 to symbol to match the explication of Binary Search Trees in the book. Your task is to 
     • Give some examples of the symbol-BSTM type.
     • Devise a set of test cases (input output pairs expressed using check-expect) for the getBSTM function.
       
     • Write a Template Instantiation for getBSTM (based on the general template for functions that process BSTMs symbol-BSTMs)
     • 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 symbol-BSTM that is well balanced (maximum depth is proportional to the log N where N is the number of keys in the symbol-BSTM) and function searches an ordered list of (key value) pairs pairs represented as two element lists (as in Problem 2 below).
       
       
     Each of these five subtasks, except for devising the collection of test cases, takes only a few lines.  A good set of test cases might take as many as 10 lines.
     
     
  2. [30 pts] The stub file HW02.rkt provides a detailed description of how to develop the function cross (and supporting function cross-help) that consumes a number-list and a symbol-list and produces a number-symbol-pair-list where a  number-symbol-pair is represented by a two element list containing a number and a symbol.
     
     
  3. [30 pts] The stub file HW02.rkt provides a detailed description of how to develop the function merge (and supporting function merge-help) that consumes two ascending [ascending (technically non-descending} number-lists lists and merges them to form an ascending number-list.
     
     

  4. [10pts]  The ubiquitous Fibonacci function defined by the trivial fib program given in the stub tile HW02.rkt is interminably slow (exponential running time) for large inputs.  Develop a Racket function fastFib that consumes a natural number input n, produces the same answer as the fib 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 large n).  Hint: write a help function fastFibHelp that accumulates the result in an accumulator argument performing essentially the same computation as an imperative program relying on a loop that maintains fib(k-1) and fib(k-2) in mutable variables as k increases from 2 to n.  The poor efficiency the trivial functional program for fib is due to the fact that it repeatedly computes the Fibonacci function for small k exponentially many times.

     • Show Type Contracts, Purpose Statements, Examples, and Template Instantiations for fastFibHelp and fastFib.  (The answers for the Template Instantiations can vary; only the salient features (primarily recursive calls) are 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 and fastFibHelp) is shaded in the DrRacket definitions panel.  Such shading indicates a failure to evaluate the shaded expressions in any test cases.
       
       

  5. Optional problem for extra credit: [50 pts]
     The fibonacci 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 for (fib n) is proportional to C*2^n for some constant C. It is straightforward 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(loglog n) addition and multiplication operations (less than C *loglog n operations for some constant C).

     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 this prototype to a program that works for all n.  

    • 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".  Of course you can define pairs using (define-struct pair (left right)).  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 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 effect.