Lab #11 Solution

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1    Introduction

 

In this lab, we will talk about  what sequences are and you’ll get some practice  using them. Please take advantage of this opportunity to practice  writing functions  with the assistance of the  TAs and  your classmates.   You are encouraged  to collaborate  with your classmates and to ask the TAs for help.

 

 

1.1     Getting Started

 

Update your clone of the git repository to get the files for this weeks lab as usual by running

 

git pull

 

from the top level directory  (probably  named 15150).

 

 

1.2     Methodology

 

You should practice  writing  requires and ensures specifications, and tests  on the  functions you write on this assignment.  In particular, every function you write should have both specs and tests.

 

 

1.3     Compiling This Lab

 

As is common with modular code, this lab is distributed across many files and relies on the SML/NJ compilation manager to introduce structures into the environment at the right time. The files that  contain  relevant code are listed in the  file sources.cm, and the  compilation manager  takes  it from there.  When you want to run your code for this lab, at  the  REPL, you will enter

 

CM.make “sources.cm”;

 

As you progress through  the lab, you’ll have to edit the sources.cm file to uncomment the files you’ve filled in. Make sure you’re comfortable with this process! The current homework is organized in the same way, so ask your TA or a neighbour if you can’t get this to work.

 

2    Sequences Cheat-Sheet

 

For your convenience a brief description  of some of the functions on sequences is given here.

 

  • Seq.map : (’a -> ’b) -> ’a Seq.seq -> ’b Seq.seq, which takes a function and a sequence and returns a sequence whose elements are the result of applying the given function to the corresponding element in the given sequence.

 

  • Seq.reduce : (’a * ’a -> ’a) -> ’a -> ’a Seq.seq -> ’a, which combines all the elements of a sequence using a particular function and base case.

 

  • Seq.mapreduce : (’a->’b) -> ’b -> (’b * ’b -> ’b) -> ’a Seq.seq -> ’b, which combines the operations  of  Seq.map and  Seq.reduce by  applying  the  given  func-

tion  of type  ’a -> ’b to each element  of the  sequence before combining them  as in

Seq.reduce.

 

  • Seq.length : ’a Seq.seq -> int, which returns the number  of elements  in the sequence.

 

  • Seq.nth : int -> ’a Seq.seq -> ’a, which returns the element  of the  given se- quence at the indicated  index, assuming it is in bounds.

 

  • Seq.tabulate : (int -> ’a) -> int -> ’a Seq.seq, which computes a sequence of the given length such that the value of each element of the sequence is the result of applying the function to its index.

 

  • Seq.singleton : ’a -> ’a Seq.seq, which takes a value and produces a sequence containing only that value.

 

  • Seq.append : ’a Seq.seq -> ’a Seq.seq -> ’a Seq.seq, which takes two sequences and appends the second to the end of the first

 

3    Mapreduce All  Day

 

mapreduce is intended  to be used with a function f of type t1 -> t2 for some types t1 and t2, an associative  binary  function g of type t2 * t2 -> t2, a value z:t2, and a sequence of items of type t1.

The behavior of mapreduce f z g <x1, . . ., xn> (assuming that  g is associative) is given by:

 

mapreduce f z g hx1 , . . . xni = (f x1 ) g (f x2 ) g . . .  g (f xn) g z

 

The  implementation of mapreduce uses  a  balanced  parenthesization format  for pairwise combinations,  as with  reduce.  Its  work and  span  are  as for reduce, when f and  g are constant time.

Although asymptotically mapping and then reducing has the same complexity as mapreduce, it, mapreduce, is actually  the more efficient of the two because it doesn’t create an interme- diate sequence.

 

Let’s learn how to use mapreduce.  First  things first: it is so easy to construct  lists.

 

Task 3.1   Write  the function

 

seqFromList : ’a list -> ’a Seq.seq

 

and

 

seqToList : ’a Seq.seq -> ’a list

 

Hint:  You should use mapreduce for seqToList.

 

 

 

Recall the function List.exists : (’a -> bool) -> ’a list -> bool, which deter- mines whether an element of the list satisfies the given predicate.  You will write an analogous function for sequences:

 

Task 3.2   Write  the function

 

seqExists : (’a -> bool) -> ’a Seq.seq -> bool

 

to determine  if the  sequence has an element  that  satisfies the  given predicate.   Hint:  You should use mapreduce.

Task 3.3   Write  the function

 

acronym : (char Seq.seq Seq.seq) -> string

 

that,  given a sequence of nonempty  character  sequences, finds the  string  whose characters are the first character  of each sequence in their order of appearance.  Example:

 

 

acronym <<#”S”, #”O”, #”P”, #”R”, #”A”, #”N”, #”O”>,

<#”A”, #”L”, #”T”, #”O”>,

<#”T”, #”E”, #”N”, #”O”, #”R”>,

<#”B”, #”A”, #”S”, #”S”>>

= “SATB”

 

Hint:  you should use mapreduce for acronym. str : char -> string and ^: string *

string -> string (careful, it’s infix) will also be useful.

 

4    Sequence Puzzles

 

The following functions  ask you to become familiar with Seq.tabulate, Seq.length, and

Seq.nth. Add your functions to lab11.sml.

 

 

4.1     Transpose

 

Recall the function transpose from Homework 5:

 

transpose [[1,2,3], [4,5,6]]

==> [[1,4], [2,5], [3,6]]

 

 

Task 4.1 Write

 

fun transpose (s : ’a Seq.seq Seq.seq) : ’a Seq.seq Seq.seq

 

that  transposes  a sequence of sequences.  You may assume that  s is rectangular, with  di- mensions  m × n,  where m, n  > 0.   Your  solution  should  have  O(m × n)  work and  O(1) span.

 

 

4.2     Filter

 

We’ve used the function filter, which takes a predicate  and a list, and evaluates  to a list with only the items that  satisfy the given predicate,  before. One of your tasks for this lab is to write an analogous function for sequences.

 

Task 4.2 Write

 

fun filter’ (p: ’a -> bool) (s: ’a Seq.seq) : ’a Seq.seq

 

such that  filter’ p s evaluates  to a sequence that  includes only the  elements  x of s for which p x = true. Your implementation must not use  Seq.filter.

 

 

4.3     Reduce

 

Contraction is an algorithmic  technique  in which we take a problem, reduce it to a smaller problem and then  recurse on the smaller problem.  It’s similar to Divide and Conquer (the technique  behind merge sort) but differs in a key way. Divide and Conquer is based around the idea that  we take our problem, divide it up, perform a recursive call on each part,  then combine the results  together.  In contraction, we take the input,  make it smaller, and then

 

perform  a single recursive  call on the  smaller  input.   Note  that  this  difference can result in contraction algorithms  having much better  runtime  then  divide and conquer algorithms. Contraction is a very powerful technique  that  you will explore more in 15-210 (if you go on to take it).  For our purposes we will use use contraction to implement the sequence function reduce with  O(n)  work and  O(log n) span.   The  idea is to do pairwise reduction,  and  is illustrated by the following trace:

 

reduce op+ 0 <1,2,3,4,4,3,2,1>

==> reduce op+ 0 <3, 7, 7, 3 >

==> reduce op+ 0 <10,    10   >

==> reduce op+ 0 <20>

==> 20

 

 

 

Task 4.3 Write

 

fun reduce (f : ’a * ’a -> ’a)(b : ’a)(s : ’a Seq.seq) : ’a Seq.seq

 

such  that  reduce f b s functions  the  same  as Seq.reduce and  runs  in O(n)  work and O(log n) span.  You may assume that  Seq.length s is a power of 2. Your implementation must not use  Seq.reduce or  Seq.mapreduce.

 

Have a TA check your code before proceeding!

 

5    Finitely Branching Parallel Trees

 

A Finitely  Branching  Parallel  Tree  is similar to the  binary  trees  introduced  earlier  in the class,  with  the  exception  that  each  node  can  now have  an  arbitrary  number  of children (rather than  exactly 2). To represent this, we define each node to be a sequence of fbtrees (this allows us to evaluate  children of nodes in parallel).  The datatype for these fbtrees is as such:

 

datatype ’a fbtree = Leaf of ’a | Node of ’a fbtree seq

 

All of the functions we’ve previously defined for binary trees have analogs for fbtrees as well. For example, here is the size function on fbtrees.

 

fun size (Leaf x) = 1

| size (Node s) = Seq.reduce (op +) 0 (Seq.map size s);

 

 

Task 5.1 Write  the function

 

depth : ’a fbtree -> int

 

Depth  for fbtrees is defined as the  longest path  from the  root to a leaf in the  tree.  A leaf should have depth  1.

 

Task 5.2 Write  the function

 

trav : ’a fbtree -> ’a list

 

such that  trav T will evaluate to the inorder traversal  of the tree T. This function might be useful when testing  your code. Hint:  the function @ may be useful.

Higher order functions,  like map and  reduce, can be implemented  for fbtrees as well. Here is the implementation for map:

 

(* fbmap : (’a -> ’b) -> ’a fbtree -> ’b fbtree *) (* REQUIRES: f is a total function *)

(* ENSURES: the output of map f T is equivalent to applying f to all the leaves of T *)

fun fbmap f (Leaf x) = Leaf (f x)

| fbmap f (Node s) = Node (Seq.map (fbmap f) s)

 

Task 5.3 Write  the function

 

fbreduce : (’a * ’a -> ’a) -> ’a -> ’a fbtree -> ’a

 

When g is associative  and z is a zero for g, fbreduce g z T will evaluate  to the  result  of pairwise combining the items in trav T using g. g is associative when, for all a,b,c, we have g(a,g(b,c)) = g(g(a,b),c)).  z is a zero for g when, for all x, we have g(x,z) = x.