Homework #9 Solution

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1    Introduction

 

This  homework  will build  your  experience  with  functors  and  sequences.   As a reminder, functors  allow us  to  produce  more  extensible  code with  less redundancy.    The  first  half of this  homework will guide you through  the  implementation of an extensible  serialization framework.  The second half of this homework stresses analysis of operations  on sequences, and  how they  improve  upon  those  on lists.   As you will see in lecture,  sequences provide similar functionality  to lists, but  with greater  potential  for parallelization.

 

 

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/09 directory  should contain  a file named exactly

hw09.pdf containing  your written  solutions to the homework.

To submit your solutions, run make from the hw/09 directory (that contains a code folder and a file hw09.pdf). This should produce a file hw09.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 hw09.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 PDF is valid; 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  (usually  either  0.0 or 1.0) 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 marshal.sml and  sequences.sml files 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.

 

 

1.3     Due Date

 

 

 

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

 

 

 

1.5     The SML/NJ Build System

 

We will be using several SML files in this assignment.  In order to avoid tedious and error- prone sequences of use commands,  the authors  of the SML/NJ compiler wrote a program that  will load and compile programs whose file names are given in a text  file. The structure CM has a function

 

val make: string -> unit

 

make reads a file usually named sources.cm with the following form:

 

Group is

 

$/basis.sml file1.sml file2.sml file3.sml

 

Loading your code using the REPL  is simple. Launch SML in the directory containing  your work, and then:

 

$ sml

Standard ML of New Jersey v110.69 [built: Wed Apr 29 12:25:34 2009]

– CM.make “sources.cm”; [autoloading]

[library $smlnj/cm/cm.cm is stable]

[library $smlnj/internal/cm-sig-lib.cm is stable]

 

Simply call

 

CM.make “sources.cm”;

 

at  the  REPL  whenever you change your code instead  of a use command  like in previous assignments.  The compilation manager offers a better  interface to the command line. There is less typing and less of an issue with name shadowing between iterations  of your code. In short,  on this assignment,  the development cycle will be:

 

  1. Edit your source files.

 

  1. At the REPL, type

 

 

CM.make “sources.cm”;

 

 

  1. Fix errors and debug.

 

  1. If done, consider doing 251 homework; else go to 1.

 

Be warned that  CM.make will make a directory  in the current working directory  called .cm. This  is populated  with  metadata needed  to  work out  compilation  dependencies,  but  can become quite  large.   The  .cm directory  can  safely be  deleted  at  the  completion  of this assignment.

It’s sometimes the case that  the metadata in the .cm directory gets in to an inconsistent state—if you run CM.make with different versions of SML in the same directory, for example. This often produces bizarre error messages.  When that  happens,  it’s also safe to delete the

.cm directory  and compile again from scratch.

 

 

1.5.1     Emphatic Warning

 

CM will not  return  cleanly if any  of the  files listed  in the  sources have no code in them. Because we want you to learn how to write modules from scratch,  we have handed out a few files that  are empty  except for a few place holder comments.  That  means that  there  are a few files in the sources.cm we handed  out that  are commented  out, so that  when you first get your tarball  CM.make “sources.cm” will work cleanly.

You must uncomment  these lines as  you  progress through the assignment!

If you forget,  it  will look like your code compiles cleanly even though  it  almost  certainly doesn’t.

 

2    Marshalling

 

2.1     Introduction

 

One common programming  task is marshalling : transforming  data  into a form where it can be written  out  and  later  read  back  in.   This  is useful if you want data  to  persist  across different runs  of your program  by storing  it in a file, or if you want to send data  across a network.1     It’s easy to write  a string  to a file or to send a string  over the  network,  so we won’t take this problem any farther  than  producing a string from data  and vice versa.

In this  problem,  you will define a small marshalling  library,  focused on capturing  the notion of data  that  can be marshalled  with the typeclass

 

signature MARSHAL =

sig

type t

 

 

 

 

 

end

(* invariant: For all v, s: read (write v ^ s) == SOME (v , s) *)

val write : t -> string

val read : string -> (t * string) option

 

 

That  is: a type t may be marshalled  only if it supports  read and write operations  that convert it to and from a string, which interact appropriately. “Appropriately” means that if you read from a string whose prefix was constructed by write, then read returns  the value that  was written,  along with any suffix. More formally:

 

 

For all values v:t and   s:string, read (write v ^ s) = SOME (v, s)

 

 

For example, given M : MARSHAL, if a web server sends a string  produced  by M.write v to a browser, and then  the browser calls M.read on that  string,  the  spec says that  the client will recover the value the server intended.  Note that  this spec allows read to have arbitrary behavior when applied to a string that  does not have a prefix produced by write—e.g., we assume that  the string is not corrupted  during transmission.

 

 

2.2     Utility Structure

 

To  minimize  the  amount of tedious  parsing  code you  need  to  write,  we’ve provided  an implementation of the following signature  in the handout  code. Be sure to understand these functions and use them when you can:  they should make your code a lot cleaner.

 

1 Check      out      http://en.wikipedia.org/wiki/Marshalling_(computer_science),       http://en. wikipedia.org/wiki/Serialization for a lot more information.

 

This  signature  lives in  util.sig.  We  have  provided  an  implementation  for you  in util.sml. You should understand the signature  and freely use the functions  the structure ascribing to it provides, but  you do not need to understand their implementation.2

 

signature UTIL =

sig

(* peelOff (s1,s2) = SOME s’ if s2 = s1 ^ s’

*               = NONE otherwise

*

* Ex:

*  peelOff (“a”,”a”) = SOME(“”)

*  peelOff (“a”,”ab”) = SOME(“b”)

*  peelOff (“a”,”c”) = NONE

*)

val peelOff : string * string -> string option

 

(* peelInt s = SOME (i,s’) if the longest non-empty prefix of s

*              comprised only of an optional leading #”~” and following digits

*              parses to the integer i, and s = i ^ s’

*          = NONE otherwise

*

* Ex:

*  peelInt “55hello” = SOME(55,”hello”)

*  peelInt “~55hello” = SOME(~55,”hello”)

*  peelInt “-55hello” = NONE

*  peelInt “hello55” = NONE

*  peelInt “12~100″  = SOME(12,”~100”)

*)

val peelInt : string -> (int * string) option end

 

 

2.3     Marshalling Booleans

 

Here is an example structure that  demonstrates one of many possible ways to marshal  the type bool.3  In write, we choose to marshal  the value true as the string “(TRUE)” and the value false as the string “(FALSE)”.

To try to read back one of these values from a string s, we first try to peel off “(TRUE)”

from s.  If that  succeeds, we return  the value true and whatever  is left over; if it fails, we

 

2 This  signature  is provided  in util.sig and  implemented in a module Util ascribing  to UTIL provided in util.sml.

3 This  particular implementation is provided  in marshal.sml; you should feel free to experiment with it

to use it for testing.

 

try to peel off “(FALSE)”. If this succeeds, return  the value false and any leftovers.  If this fails, having now failed overall, we return  NONE.

 

 

2.4     Integers, Pairs, and Lists—Oh, my!

 

Your task is to write marshallers  for integers, pairs, and lists.

 

 

2.4.1     One  Possible Strategy

 

To marshal  values of a type, consider how many constructors  that  type has and how many arguments  each of them takes.  To keep track of the tree-structure of an expression, you will need to be able to distinguish  between constructors  and between different instances  of the same constructor.

One way to  do this  is to  represent  a value  constructed with  the  constructor con and arguments  A1 through  An as

 

(con A1 A2 … An)

 

If you consider the type bool to be defined as

 

datatype bool = true | false

 

then  this  is exactly  what  we did above:  the type is given by two nullary  constructors  and nothing  else.

 

 

2.4.2     Tasks

 

Submit  your solutions for the following tasks in marshal.sml.

 

Task 2.1  (10%).  Implement a structure MarshalInt that  ascribes to MARSHAL and defines mashalling for the type int.

 

Task 2.2 (10%). Prove the following theorem of correctness for integer marshalling: For all values v:int and   s:string, read (write v ^ s) = SOME (v, s)

You may assume the following lemmas (but  cite them when you use them):

 

 

Lemma 1:  For all i : int, all non-digit c : char, all s : string,

peelInt (Int.toString i ^ “c” ^ s) = SOME(i, “c” ^ s)

 

Lemma 2: For all s1 : string, s2 : string, peelOff s1 (s1 ^ s2) = SOME (s2)

 

Lemma 3:  ^ is associative.

 

You may also assume Int.toString and ^ are total  (but  again, cite this when used).

 

 

Task 2.3 (10%). The signature

 

signature MARSHALPAIR =

sig

structure M1 : MARSHAL

structure M2 : MARSHAL

end

 

packages together  two modules that  can be marshaled.  4  Implement a functor

 

functor MarshalPair (P : MARSHALPAIR) : MARSHAL

 

that  implements marshalling for the type P.S1.t * P.S2.t. You may assume that  P.S1 and P.S2 both  obey the  above invariant, but  you may not  make any other  assumptions  about them.

 

Task 2.4 (10%). Implement a functor

 

functor MarshalList (S : MARSHAL) : MARSHAL

 

that  defines marshaling for the type S.t list. You may assume that  S obeys the marshaling invariant above, but  you may not assume anything  else about  S.

 

Task 2.5 (5%). Define marshalling instances for the following types using only the modules above.

 

  1. int list list with a structure named ILL ascribing to MARSHAL.

 

  1. (int list) * bool with a structure named ILSB ascribing to MARSHAL.

 

  1. (int * bool) list with a structure named ISBL ascribing to MARSHAL.

 

  1. (int * (bool list)) list with a structure named ISBLL ascribing to MARSHAL.

 

 

Task 2.6 (Extra  Credit).Write a structure MarshalString ascribing to MARSHAL that  defines marshalling  for the type string. Hint:

 

 

 

 

 

 

 

 

 

 

5

 

 

4 This signature  is provided  in marshalpair.sig.

http://xkcd.com/327/

 

3    Sequence Basics

 

For the remainder  of the problems in this homework, you will be using the sequence library. We have provided you a sequence implementation which you will be using for this and the remaining homeworks.  For complete documentation see handout/sequencereference.pdf in your git repository.  You should take some time now to read the handout  and familiarize yourself with the functions in the sequence library and their cost bounds.

Now, for the following tasks you will work with and analyze functions on sequences.

 

 

 

Here is an example of how to perform an analysis on functions using sequences:

 

fun sum (s : int seq) : int = reduce (op +) 0 s

fun count (s : int seq seq) : int = sum (map sum s)

 

sum takes  an integer sequence and adds up all the numbers  in it using reduce, just like we did with folds for lists and trees.  count sums up all the numbers  in a sequence of sequences, by (1) summing each individual  sequence and then  (2) summing the sequence that  results.

 

(Note :  We could rewrite  count with mapreduce, so it takes  only one pass, we will be talking more about mapreduce later).

Example Task Suppose all sequences, inner and outer,  have length  n.  Give a tight O-bound for the span of count. Briefly explain why your answer is correct.

Solution :

Let count be on a sequence of n sequences, each of length n. This requires O(log n) span :

 

sum s is implemented using reduce with constant-time arguments,  and thus has O(log n)  span,  where n is the  length  of s.  Each  call to sum inside the  the  map doesn’t contribute anything,  and both the inner and outer sums are on sequences of length  n,  and  therefore  have  O(log n)  span.   The  total  span  is the  sum  of the  inner span and the  outer  span,  because of the  data  dependency:  the  outer additions  happen after the inner sum have been computed.  The sum of log n and log n is still O(log n), so the total  span is O(log n).

 

 

3.1     Append

 

As we know, appending  two lists can be expensive, and is unfortunately non-parallelizable. What  about  appending  two sequences?

 

Task 3.1 (6%).  Provide a non-recursive implementation of Seq.append called myAppend in the  file sequences.sml. Your solution  may not  use Seq.append or Seq.flatten. That’s cheating.

 

Task 3.2  (2%).  Give a tight O-bound for the work of myAppend. Make sure you explicitly state  what  quantities you are  analyzing  the  work in terms  of.   Briefly explain  why your answer is correct.

 

Task 3.3  (2%).  Give a tight O-bound for the span of myAppend. Make sure you explicitly state what quantities you are analyzing the span in terms of. Briefly explain why your answer is correct.

 

 

3.2     Two-Largest

 

One classic problem on lists is finding the largest elements in the list.  In the following tasks, you will implement a function which finds the two largest elements of a sequence.

 

Task 3.4 (3%).  Implement the function

 

max4 : (’a * ’a -> order) -> ((’a * ’a) * (’a * ’a)) -> (’a * ’a)

 

such that  for a comparison function cmp and two pairs of elements (a, b) and (c, d), max4 cmp ((a, b), (c, d)) returns  the two largest elements of the four.  Order does not matter. For example,

 

val (3, 4) = max4 Int.compare ((1, 4), (3, 2))

val (5, 6) = max4 Int.compare ((3, 4), (5, 6))

 

For your convenience, we have implemented  the maxL function which you may find useful for max4. maxL : (’a * ’a -> order) -> ’a list -> (’a * ’a list) takes a compar- ison function cmp and a non-empty  list L and returns  the largest element of L along with a list L’ which is all elements of L excluding the max.

 

Task 3.5 (7%).  Implement the function

 

twoLargest : (’a * ’a -> order) -> (’a * ’a) -> ’a seq -> (’a * ’a)

 

such that  for a comparison  function  cmp, a pair  of elements  (a, b) and  a sequence s, twoLargest cmp (a, b) s returns  the two largest elements amongst  all the elements of s and (a, b). Again, order does not matter. For example:

 

(* s = <1, 5, 8, 2>

s’ = <“b”> where s and s’ are sequences *)

val (5, 8) = twoLargest Int.compare (0, 0) s

val (“b”, “c”) = twoLargest String.compare (“a”, “c”) s’

 

Note: your solution  must have O(n)  work and  O(log n)  span  where n is the  size of the input sequence (you may assume that  the comparison function has O(1) work and span).  To achieve the time bounds, make sure to look at the functions available to you in the sequence library.  You may also find max4 useful for implementing  twoLargest.

 

4    Kittens on  Stairs

 

A cute kitten  is trying  to climb some stairs!6    Every second, it climbs up to a higher stair or tumbles  adorably  down to a lower stair.   You want  to congratulate it for doing so well, so you want to find the farthest  climb that  the kitten  made up the stairs.  More specifically, given a sequence of integers  s, you wish to find the  maximal  si − sj , i > j. Since SML is almost as wonderful as a kitten,  you write the following code:

 

fun suffixes (s : ’a Seq.seq) : (’a Seq.seq) Seq.seq = Seq.tabulate (fn x => Seq.drop (x + 1) s) (Seq.length s)

 

val SOME(minInt) = Int.minInt

val maxS : int Seq.seq -> int = Seq.reduce Int.max minInt fun maxAll (s : int Seq.seq Seq.seq) : int =

maxS (Seq.map maxS s)

 

fun withSuffixes (t : int Seq.seq) : (int * int Seq.seq) Seq.seq = Seq.zip (t, suffixes t)

 

fun mostSteps (s : int Seq.seq) : int =

let fun diff (start, stops) = Seq.map (fn stop => stop – start) stops val diffs = Seq.map diff (withSuffixes s)

in maxAll diffs end

 

After using your new functions and congratulating the kitten,  you realize that  the work of suffixes s, in terms  of the length  of s, is in O(n2 ), where n is the length  of s.  After watching  a few cat  videos, you realize that  the  span of suffixes s is in O(1).  Then  you realize your brain is telling you to get back to your homework.

 

Task 4.1  (2%).   Give a tight O-bound  for the  work of withSuffixes s, in terms  of the length of s. Briefly explain why your answer is correct.

 

Task 4.2  (2%).   Give a tight O-bound  for the  span  of withSuffixes s, in terms  of the length of s. Briefly explain why your answer is correct.

 

Task 4.3 (3%).  Give a tight O-bound for the work of

 

maxAllhhx1 , . . . , x1  i, . . . , hxn , . . . , xn  ii

 

1                k1

1                 kn

 

 

(i.e.  the ith inner sequence has length ki  and the outer  sequence of sequences has length n)

in terms of k1, . . . , kn  and n.  Briefly explain why your answer is correct.

 

 

 

6 Relevant:  http://www.youtube.com/watch?v=D4BJrdrBi0Y

 

Task 4.4 (3%).  Give a tight O-bound for the span of

 

maxAllhhx1 , . . . , x1  i, . . . , hxn , . . . , xn  ii

 

1                k1

1                 kn

 

 

in terms of k1, . . . , kn  and n.  Briefly explain why your answer is correct.

 

Task 4.5  (3%).  Give a tight O-bound for the work of mostSteps s, in terms of the length of s. Briefly explain why your answer is correct.

 

Task 4.6  (3%).  Give a tight O-bound for the span of mostSteps s, in terms of the length of s. Briefly explain why your answer is correct.

 

5    Rainfall

 

Old MacDonald had a farm, and on that  farm he would like to know where to expect rainfall to collect.  He has given you an n × n two-dimensional grid of integers, or an int Seq.seq Seq.seq, of land  elevations  across  his farm.   A position  on a farm  is defined as pair  of indices (i, j) corresponding  to  the  j-th entry  in the  i-th  row.   For  example,  the  sequence hh1, 2, 3i, h4, 0, 5i, h6, 7, 8ii has the entry  1 at position (0, 0) and looks like

 

1 2 3

4 0 5

6 7 8

 

Your task is to produce a sequence of sinks, where we define a sink to be a position with elevation lower than  the four points surrounding  it (up, down, left, right).  You do not need to consider diagonals.  Any points on the edges of Old MacDonald’s input  sequence cannot be sinks. You may assume that  the sequence is made up of unique heights.

 

In  the  above  example,  rainfall <<1, 2, 3>, <4, 0, 5>, <6, 7, 8>> = <(1, 1)> because the only possible sink is at (1, 1) and 0 is less 2, 4, 5, and 7 so the index (1, 1) is a sink.

 

Task 5.1 (13%). Help Old MacDonald by implementing the following function in sequences.sml:

 

fun rainfall : int Seq.seq Seq.seq -> (int * int) Seq.seq

 

which takes an n × n sequence of heights and returns  a sequence of sink positions.  Your sink sequence must  be ordered  by index (e.g.  (1, 1) precedes (2, 0)).  Your solution  should be non-recursive and also use functions in the sequence library.

Feel free to use any of the testing helper functions or farms provided in the SequenceHelper

structure when testing your code!

 

 

Hint 1: You may find it useful to implement the two helper functions atEdge and isSink

in sequences.sml for determining  if an index is an edge or sink, respectively.

 

Hint 2:  Make sure you’ve taken  a good look at the sequence library at your disposal!

 

 

 

Task 5.2 (3%). Give a tight O-bound for the work of rainfall s, in terms of n where s is an n × n sequence. Briefly explain why your answer is correct.

 

Task 5.3 (3%).  Give a tight O-bound for the span of rainfall s, in terms of n where s is

n × n.  Briefly explain why your answer is correct.