Homework 04 Solution

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1    Introduction

 

This homework will focus on lists, trees, sorting, and work-span analysis.

 

 

1.1     Getting The Homework Assignment

 

The starter files for the homework assignment have been distributed through  our git repos- itory, as usual.

 

 

1.2     Submitting The Homework Assignment

 

Submissions will be handled  through  Autolab,  at

 

https://autolab.cs.cmu.edu

 

In preparation for submission, your hw/04 directory  should contain  a file named exactly

hw04.pdf containing  your written  solutions to the homework.

To submit your solutions, run make from the hw/04 directory (that contains a code folder and a file hw04.pdf). This should produce a file hw04.tar, containing  the files that  should be handed  in for this homework assignment.   Open the Autolab  web site, find the page for this assignment,  and submit  your hw04.tar file via the “Handin  your work” link.

 

The Autolab handin script does some basic checks on your submission:  making sure that the file names are correct; making sure that  no files are missing; making sure that  your code compiles cleanly.  Note that  the handin  script  is not  a grading script—a  timely submission that  passes the handin script will be graded, but will not necessarily receive full credit.  You can view the results of the handin script by clicking the number corresponding to the “check” section of your latest  handin  on the “Handin  History”  page.  If this number is 0.0, your submission failed the check script; if it is 1.0, it passed.

Remember  that  your written  solutions  must  be submitted in PDF  format—we  do not accept MS Word files or other formats.

 

Your  hw04.sml file must  contain  all the  code that  you want  to  have  graded  for this assignment,  and must compile cleanly.  If you have a function that  happens to be named the same as one of the  required  functions  but  does not  have the  required  type, it will not  be graded.

 

 

 

 

1.4     Methodology

 

You must  use the five step methodology  discussed in class for writing functions,  for every

function you write in this assignment.  Recall the five step methodology:

 

  1. In the first line of comments, write the name and type of the function.

 

  1. In the second line of comments, specify via a REQUIRES clause any assumptions about the arguments  passed to the function.

 

  1. In the third line of comments, specify via an ENSURES clause what  the function com- putes (what  it returns).

 

  1. Implement the function.

 

  1. Provide testcases, generally in the format

val <return value> = <function> <argument value>. For example, for the factorial function presented  in lecture:

(* fact : int -> int

* REQUIRES: n >= 0

* ENSURES: fact(n) ==> n!

*)

 

fun fact (0 : int) : int = 1

| fact (n : int) : int = n * fact(n-1) (* Tests: *)

val 1 = fact 0

val 720 = fact 6

 

2    Fastest Fib

 

The Fibonacci function fib : int -> int, as defined below, has exponential  runtime.

 

fun fib n = if n<=2 then 1 else fib(n-1) + fib(n-2)

 

In lab last  week you wrote a function  fibber : int -> int that  does the  same job but  in linear time.

 

Task 2.1 (10%). Write  a function

 

fastfib : int -> int

 

that  does the same job but  in logarithmic time!

Your function should be based on the facts that,  for all integers k ≥ 0,

 

 

 

fib(2k) = fib(k)(2fib(k + 1) − fib(k))

fib(2k + 1) = fib(k + 1)2 + fib(k)2

 

For example:

 

– fib 10;

val it = 55 : int

– fib(5) * (2*fib(6)-fib(5));

val it = 55 : int

– fib 11;

val it = 89 : int

– fib(5)*fib(5) + fib(6)*fib(6);

val it = 89 : int

 

The apparent runtime  (in the ML interpreter) for fib(42) is noticeably slow.

 

– fib 42;

val it = 267914296 : int (* after a few seconds! *)

 

Your function should evaluate  fastfib(42) much faster!

By the way, design your function so that  it reproduces exactly these results.  Sometimes the Fibonacci series is taken to start  at n=0 rather  than n=1, and that  would lead to a slightly different  function  that’s  always “off by 1”.  The  function  defined above has fib(1)=1 and fib(2)=1, and computes the Fibonacci numbers  for n ≥ 1.

 

3    Series Trees

 

Consider the below datatype

 

datatype rtree = rEmpty

| rNode of rtree * real * rtree

 

2
2

Recall  the  definition  of a  “partial sum”  from Homework  02.   In  that   assignment we considered the partial  sum of the geometric series for 1 . Now we are interested  in expanding this definition to any real r, not just  1 .  We denote the  partial  sum of the  first n terms  of the series to be Sn. Also, we define the behavior of Sn(r) as follows:

 

Sn(r) = 1.0 if n = 0

 

n

Sn(r) = X rk in general (for k ≥  0)

k=0

 

We also define the following:

 

r0 = 1.0

 

rm+1 = r· rm

 

 

3

for all real numbers  r. For example, if r = 1 ,

 

n

Sn = X(

i=0

1 )i =

3

1     1      1

+   +

3     9     27

+ . . .

 

 

 

Task 3.1 (5%).  Write  a recursive ML function

 

geometricTree : int * real -> rtree

 

such that  for all non-negative  integers  n, and all reals r, geometricTree(n, r) returns  the “complete”  (or “full”)  binary  tree of size 2n − 1 and depth  n, with the  read value at  each node being Si(r), where i is the depth  of that  node in the tree.

Your function  design should exploit  symmetry:   when n > 0, the  complete  binary  tree

with depth  n has identical  left- and right children,  each of which is also a complete binary tree.  It is possible to build geometricTree(n, r) with a linear number of recursive calls; you will lose credit if your function makes an exponential  number  of recursive calls.

Remember to use Real.== to test  for the equality  of reals.

 

Task 3.2  (5%).  Derive recurrence  relations  for an asymptotic estimate  of WgeometricTree(n)

and SgeometricTree(n), then using these recurrences, give a big-O estimate for both WgeometricTree(n)

and SgeometricTree(n).

 

4    Quicksorting a List

 

The quicksort algorithm  for sorting lists of integers can be implemented  in ML as a recursive function

 

quicksort : int list -> int list

 

that  uses a helper function

 

part : int * int list -> int list * int list

 

with the following specification:

 

(* REQUIRES true *)

(* ENSURES part(x, L) = a pair of lists (A,B) such that      *) (*        A consists of the items in L that are less than x *) (*    and B consists of the items in L that are            *) (*          greater than or equal to x.                    *)

 

Another  way to state  this specification is:

 

For all integers x and integer lists L, part(x, L) returns  a pair of lists (A, B) such that  A consists of the items in L that  are less than  x and B consists of the items in L that  are greater  than  or equal to x.

 

The key idea is that  one can sort a non-empty  list x::L by partitioning L into two lists (the  items less than  x, and the  items greater  than  or equal to x), recursively sorting  these two lists.  The final result is obtained  by combining the sorted sublists and x.

 

Task 4.1 (10%). Define an ML function

 

part : int * int list -> int list * int list

 

that  satisfies the above specification.

 

 

We now would like to implement the quicksort algorithm  on trees.  Your implementation should  use the  part function  which you defined in the  previous  task.   Do not  introduce additional  helper functions!  Also, do not mess with the types or specs!

 

Task 4.2 (10%). Using part, define a recursive ML function

 

quicksort : int list -> int list

 

such that  for all int lists L, quicksort(L) returns  a sorted permutation of L.

Recall the following definition (from lectures 5 and 6) of sortedness on lists: A list of integers is <-sorted if each item in the list is ≤ all items that  occur later in the list.  The ML function sorted below checks for this property.

 

(* sorted : int list -> bool

* sorted L = true iff L is <-sorted

*)

fun sorted ([ ] : int list) : bool     = true

| sorted ([x] : int list) : bool     = true

| sorted (x::y::L : int list) : bool = (compare(x,y) <> GREATER) andalso sorted(y::L)

 

 

5    Tree Traversal

 

Recall that  the in-order traversal  of a tree t visits all of the nodes in the left subtree  of t, then the node at t, and then all of the values in the right subtree  of t.

In class we covered the  ML function  trav below is an  implementation of an  in-order traversal  of a tree.

 

(* trav : tree -> int list

* trav t = the in-order traversal of t

*)

fun trav (t:tree) : int list =

case t of

Empty         => [ ]

| Node(t1, x, t2) => trav t1 @ (x :: trav t2)

 

 

Task 5.1 (5%).  Define an ML function

 

traver : tree * int list -> int list

 

that,  when given a tree t and int list A, produces a list which represents the in-order traversal of that  tree appended  onto A. In other words,

 

traver (t, A) = trav (t) @ A

 

Your implementation of traver  should not  use @ (or equivalents)  and  should run  in linear work on the size of the tree (where size is the number  of nodes in the tree).

 

 

6    Tree Size

 

 

Task 6.1 (10%). Prove, by structural induction  on trees, that  for all values t : tree,

 

size(t) = length(trav t).

 

Where the size function is defined as below:

 

fun size(Empty : tree) : int = 0

| size(Node(l, x, r) = size(l) + 1 + size(r)

 

You may use the following lemmas:

 

  1. For all values A : int list, B : int list length (A@B) = length(A) + length(B)
  2. For all values x : int, L : int list

 

length(x :: L) = 1 + length(L)

 

  1. For all values x : int, y : int, z : int x + (y + z) = x + y + z

and you can use the definition of the length function for lists, as given in class, as well

as basic properties  of the list operations  as used in class and the tree functions as defined in this assignment.  If you use any of these, be sure to cite them in your proof.

 

 

7    Heaps and Heaps of Heaps

 

You’ve seen one definition of a sorted tree, where all elements in the left subtree  of a node are less than  or equal to the node’s value, and all elements in the right subtree  of the node are greater  than  or equal  to  the  node’s value,  and  the  subtrees  are sorted.   Consider  the definition of sorted which defines a tree as a minheap

A tree t is a minheap if it obeys the following invariants:

 

  1. t is Empty

 

  1. t is a Node(L, x, R), where R, L are minheaps and value(L), value(R) ≥ x

 

Here, value is the value at the root node of the tree.

In this section you will implement the ML function heapify, which, given an arbitrary

tree t returns  a minheap  with exactly the elements of t.

 

Task 7.1 (5%).  Define an ML function

 

treecompare : tree * tree -> order

 

that,  when given two trees, returns  a value of type order, based on which tree has a larger value at the root node.  For example:

 

val a = Node(Empty, 4, Empty)

val b = Node(Node(Empty, 7, Empty), 3, Empty)

treecompare(a, b) =>* GREATER

 

Task 7.2 (10%). Define a recursive ML function

 

swapDown : tree -> tree

 

with the following specification:

 

(* REQUIRES the subtrees of t are both minheaps

* ENSURES swapDown(t) = if t is Empty or all of t’s immediate children are

*        empty then just return t, otherwise returns a minheap which

*        contains exactly the elements in t.

*)

 

 

Task 7.3 (10%). Define a recursive ML function

 

heapify : tree -> tree

 

which, given an arbitrary tree t, evaluates  to a minheap with exactly the elements of t.

 

Task 7.4 (10%). Assume that  your implementation of swapDown is correct (it works accord- ing to spec).  Using this, prove that your implementation of heapify is correct.

 

Task 7.5  (10%).  Give the  work and span for swapDown and heapify. You should give a recurrence relation  and closed form for each.

Note: Remember that  there are two ways to analyze a tree.