# Homework 1

This homework is written in literate Haskell; you can download the raw source to fill in yourself. You’re welcome to submit literate Haskell yourself, or to start fresh in a new file, literate or not.

Please submit homeworks via the new submission page.

Let’s learn some Haskell! We’ll be going over some rudiments in class, and there’s excellent documentation online.

In most places where I’d like you to fill in a definition, I’ve used the convenient Haskell term `undefined`, which let’s you compile an incomplete program. (Running undefined parts of your program is an error, and your program will crash.)

Please leave the following line in. (If you take it out, the grader will reject your program.) We’ll talk more about Haskell’s module system later in the semester.

``module Hw01 where``

You can test this program by running `ghci` on it. If you edit your code, you can use the `:reload` command to load in your new definitions.

If your program has type errors, it won’t compile. If you change the types of any functions, it won’t compile with my tester. If you take things out, like type definitions, your program won’t compile. If your submitted program doesn’t compile, you will get no points. If you’re unsure, ask!

The following imports are needed for Problem 9.

``````import qualified Data.Map as Map
import Data.Map (Map, (!))

import qualified Data.Set as Set
import Data.Set (Set)``````

Problem 1: natural recursion

Please don’t use any Prelude functions to implement these—just write natural recursion, like we did in class.

Write a function called `sumUp` that sums a list of numbers.

``````sumUp :: [Int] -> Int
sumUp []     = undefined
sumUp (x:xs) = undefined``````

Write a function called `evens` that selects out the even numbers from a list. For example, `evens [1,2,3,4,5]` should yield `[2,4]`. You can use the library function `even`.

``````evens :: [Int] -> [Int]
evens []     = undefined
evens (x:xs) = undefined``````

Write a function called `incAll` that increments a list of numbers by one. You’ll have to fill in the arguments and write the cases yourself.

``````incAll :: [Int] -> [Int]
incAll = undefined``````

Now write a function called `incBy` that takes a number and increments a list of numbers by that number.

``````incBy :: Int -> [Int] -> [Int]
incBy = undefined``````

Write a function `append` that takes two lists and appends them. For example, `append [1,2] [3,4] == [1,2,3,4]`. (This function is called `(++)` in the standard library… but don’t use that to define your version!)

``````append :: [Int] -> [Int] -> [Int]
append = undefined``````

Problem 2: data types

Haskell (and functional programming in general) is centered around datatype definitions. Here’s a definition for a simple tree:

``data IntTree = Empty | Node IntTree Int IntTree deriving (Eq,Show)``

Write a function `isLeaf` that determines whether a given node is a leaf, i.e., both its children are `Empty`.

``````isLeaf :: IntTree -> Bool
isLeaf Empty = undefined
isLeaf (Node l x r) = undefined``````

Write a function `sumTree` that sums up all of the values in an `IntTree`.

``````sumTree :: IntTree -> Int
sumTree = undefined``````

Write a function `fringe` that yields the fringe of the tree from left to right, i.e., the list of values in the leaves of the tree, reading left to right.

For example, the fringe of `Node (Node Empty 1 (Node Empty 2 Empty)) 5 (Node (Node Empty 7 Empty) 10 Empty)` is `[2,7]`.

``````fringe :: IntTree -> [Int]
fringe = undefined``````

Problem 3: insertion sort

Write a function `insertionSort` that takes a list of `Int`s and produces one in sorted order. Use the insertion sort algorithm. You might want to write a helper function.

``````insertionSort :: [Int] -> [Int]
insertionSort = undefined``````

Problem 4: binary search trees

Write a function `isBST` to determine whether or not a given tree is a strict binary search tree, i.e., the tree is either empty, or it is node such that:

• all values in the left branch are less than the value of the node, and
• all values in the right branch are greater than the value of the node,
• both children are BSTs.

I’ve given you a helper function `maybeBounded` that checks whether a given `Int` is bounded. It uses the Haskell `Maybe` type, which is essentially defined as:

``data Maybe Int = Nothing | Just Int``

`Maybe` makes a type nullable. In Java, every non-primitive type is nullable—the `null` object can have any class. In Haskell, you must explicitly ask for nullability, and nullness and non-nullness are both explicit: `Nothing` is null, and the non-null `Just x` holds a value `x`. We’ll look at this more deeply in the next assignment, when we talk about datatypes.

``````maybeBounded :: Maybe Int -> Maybe Int -> Int -> Bool
maybeBounded Nothing Nothing x = True
maybeBounded Nothing (Just upper) x = x < upper
maybeBounded (Just lower) Nothing x = lower < x
maybeBounded (Just lower) (Just upper) x = lower < x && x < upper``````
``````isBST :: IntTree -> Bool
isBST = undefined``````

Write a function `insertBST` that performs BST insert. You may assume your input is a BST.

``````insertBST :: Int -> IntTree -> IntTree
insertBST = undefined``````

Write a function `deleteBST` that removes a given value from a BST. You may assume your input is a BST. Feel free to look up the algorithm… I had to!

It doesn’t really matter which algorithm you use, so long as the function works correctly, i.e., for all BSTs `t`:

• `deleteBST x t` is a BST,
• `deleteBST x t` runs in O(log n) time in expectation,
• `x` doesn’t appear in `deleteBST x t`,
• for all `y` in `t`, if `y /= x`, then `y` appears in `deleteBST y t`.

You are, as always, free to introduce any helper functions you might need.

``````deleteBST :: Int -> IntTree -> IntTree
deleteBST = undefined``````

Problem 5: maps and folds

We’re going to define each of the functions we defined in Problem 1, but we’re going to do it using higher-order functions that are built into the Prelude. In particular, we’re going to use `map`, `filter`, and the two folds, `foldr` and `foldl`. To avoid name conflicts, we’ll name all of the new versions with a prime, `'`.

Define a function `sumUp'` that sums up a list of numbers.

``````sumUp' :: [Int] -> Int
sumUp' l = undefined``````

Define a function `evens'` that selects out the even numbers from a list.

``````evens' :: [Int] -> [Int]
evens' l = undefined``````

Define a function `incAll'` that increments a list of numbers by one.

``````incAll' :: [Int] -> [Int]
incAll' l = undefined``````

Define a function `incBy'` that takes a number and then increments a list of numbers by that number.

``````incBy' :: Int -> [Int] -> [Int]
incBy' n l = undefined``````

Define a function `rev'` that reverses a list. Don’t use anything but a folding function (your choice), the list constructors, and lambdas/higher-order functions.

``````rev' :: [Int] -> [Int]
rev' l = undefined``````

Define two versions of the function `append'` that appends two lists. One, `appendr`, should use `foldr`; the other, `appendl`, should use `foldl`. You can use the list constructors, higher-order functions, and `rev'`.

``````appendr :: [Int] -> [Int] -> [Int]
appendr l1 l2 = undefined

appendl :: [Int] -> [Int] -> [Int]
appendl l1 l2 = undefined``````

Problem 6: defining higher-order functions

We’re going to define several versions of the `map` and `filter` functions manually, using only natural recursion and folds—no using the Prelude or list comprehensions. Note that I’ve written the polymorphic types for you.

Define `map1` using natural recursion.

``````map1 :: (a -> b) -> [a] -> [b]
map1 = undefined``````

Define `map2` using a folding function.

``````map2 :: (a -> b) -> [a] -> [b]
map2 f l = undefined``````

Define `filter1` using natural recursion.

``````filter1 :: (a -> Bool) -> [a] -> [a]
filter1 = undefined``````

Define `filter2` using a folding function.

``````filter2 :: (a -> Bool) -> [a] -> [a]
filter2 p l = undefined``````

Problem 7: polymorphic datatypes

We’ve already briefly seen the `Maybe` type in the first homework. In the next two problems, we’ll look at `Maybe`, pairs, and `Either` in more detail.

Haskell’s type system is rigid compared to most other languages. In time, you will come to view this as a feature—languages that let you ‘cheat’ their safety mechanisms end up making you pay for it with complexity elsewhere. But for now, let’s get familiar with the structures and strictures of types.

The `Maybe` datatype introduces nullability in a controlled fashion—values of the type `Maybe a` can be `Nothing` or `Just x`, where `x` is a value of type `a`. Note that `Maybe` is polymorphpic: we can choose whatever type we want for `a`, e.g., `Just 5 :: Maybe Int`, or we can leave `a` abstract, e.g., `Just x :: Maybe a` iff `x :: a`.

Write a function `mapMaybe` that behaves like `map` when its higher-order function argument returns `Just x`, but filters out results where the function returns `Nothing`.

``````mapMaybe :: (a -> Maybe b) -> [a] -> [b]
mapMaybe = undefined``````

The pair datatype allows us to aggregate values: values of type `(a,b)` will have the form `(x,y)`, where `x` has type `a` and `y` has type `b`.

Write a function `swap` that takes a pair of type `(a,b)` and returns a pair of type `(b,a)`.

``````swap :: (a,b) -> (b,a)
swap = undefined``````

Write a function `pairUp` that takes two lists and returns a list of paired elements. If the lists have different lengths, return a list of the shorter length. (This is called `zip` in the prelude. Don’t define this function using `zip`!)

``````pairUp :: [a] -> [b] -> [(a,b)]
pairUp = undefined``````

Write a function `splitUp` that takes a list of pairs and returns a pair of lists. (This is called `unzip` in the prelude. Don’t define this function using `unzip`!)

``````splitUp :: [(a,b)] -> ([a],[b])
splitUp = undefined``````

Write a function `sumAndLength` that simultaneously sums a list and computes its length. You can define it using natural recursion or as a fold, but—traverse the list only once!

``````sumAndLength :: [Int] -> (Int,Int)
sumAndLength l = undefined``````

Problem 8: defining polymorphic datatypes

The `Either` datatype introduces choice in a controlled fashion—values of the type `Either a b` can be either `Left x` (where `x` is an `a`) or `Right y` (where `y` is a `b`).

Define a datatype `EitherList` that embeds the `Either` type into a list. (This isn’t a good idea, but it’s a good exercise!)

To see what I mean, let’s combine lists and the `Maybe` datatype. Here’s Haskell’s list datatype:

``data [a] = [] | a:[a]``

Here’s the Maybe datatype:

``data Maybe a = Nothing | Just a``

What kinds of values inhabit the type `[Maybe a]`? There are two cases:

• `[]`, the empty list
• `a:as`, where `a` has type `Maybe a` and `as` is a list of type `[Maybe a]`

But we can really split it into three cases:

• `[]`, the empty list
• `a:as`, where `as` is a list of type `[Maybe a]`, and:
• `a` is `Nothing`
• `a` is `Just a'`, where `a'` has type `a`

Put another way:

• `[]`, the empty list
• `Nothing:as`, where `as` is a list of type `[Maybe a]`
• `Just a:as`, where `a` has type `a` and `as` has type `[Maybe a]`

To define MaybeList, we’ll write a data structure that has those constructors expliclty.

``````data MaybeList a =
Nil
| ConsNothing (MaybeList a)
| ConsJust a (MaybeList a)``````

Note that these match up exactly with the last itemized list of cases.

Okay: do it for `Either`! Fill in the functions below—they should behave like the Prelude functions. You’ll also have to fill in the type. We’ve given you the constructors’ names. Make sure your `Cons` constructors takes arguments in the correct order, or we won’t be able to give you credit for any of this problem.

``````data EitherList a b =
Nil
| ConsLeft {- fill in -}
| ConsRight {- fill in -}
deriving (Eq, Show)

toEither :: [Either a b] -> EitherList a b
toEither = undefined

fromEither :: EitherList a b -> [Either a b]
fromEither = undefined

mapLeft :: (a -> c) -> EitherList a b -> EitherList c b
mapLeft = undefined

mapRight :: (b -> c) -> EitherList a b -> EitherList a c
mapRight = undefined

foldrEither :: (a -> c -> c) -> (b -> c -> c) -> c -> EitherList a b -> c
foldrEither = undefined

foldlEither :: (c -> a -> c) -> (c -> b -> c) -> c -> EitherList a b -> c
foldlEither = undefined``````

Problem 9: maps and sets

Haskell has many convenient data structures in its standard library. We’ll be playing with sets and maps today. Data.Map and Data.set are well documented on-line.

In this problem, we’ll use maps and sets to reason about graphs (in the network/graph theory sense, not in the statistical plotting sense).

We can start by defining what we mean by the nodes of the graph: we’ll have them just be strings. We can achieve this by using a type synonym.

``type Node = String``

To create a `Node`, we can use the constructor, like so:

``````a = "a"
b = "b"
c = "c"
d = "d"
e = "e"``````

We can define a graph now as a map from `Node`s to sets of `Node`s. The `Map` type takes two arguments: the type of the map’s key and the type of the map’s value. Here the keys will be `Node`s and the values will be sets of nodes. The `Set` type takes just one argument, like lists: the type of the set’s elements.

``type Graph = Map Node (Set Node)``

We don’t need to use `newtype` here, because we’re less worried about confusing graphs with other kinds of maps.

Let’s start by building a simple graph, `g1`:

``````    - b -
/     \
a -       - d
\     /
- c -``````
``````g1 = Map.fromList [(a, Set.fromList [b,c]),
(b, Set.fromList [a,d]),
(c, Set.fromList [a,d]),
(d, Set.fromList [b,c])]``````

Note that we’ve been careful to make sure the links are bidirectional: if the `b` is in the value mapped by `a`, then `a` is in the value mapped by `b`.

We can see what `a` has edges to by looking it up in `g1`:

``aEdges = g1 ! a``

Write a function `isBidi` that checks whether a mapping is bidirectional. Feel free to use any function in `Data.Map`, `Data.Set`, or the Prelude, and write as many helper functions as you need.

You can assume that if you find a node in a set, then that node has a (possibly empty) entry in the graph. That is, I won’t give you a graph like:

``badGraph = Map.fromList [(a, Set.singleton b)]``
``````isBidi :: Graph -> Bool
isBidi = undefined``````

Write a function `bidify` that takes an arbitrary graph and makes it bidirectional by adding edges, i.e., if the node `a` points to `b` but not vice versa in a graph `g`, then `a` points to `b` and `b` points to `a` in the graph `bidify g`.

``````bidify :: Graph -> Graph
bidify = undefined``````

Be sure to test your code!