# Variants
Today we'll practice with variants and other user-defined data types,
learn more about pattern matching, define binary tree data structures
with variants, and learn about two more OCaml features, exceptions
and polymorphic variants.
## Data types
##### Exercise: cards [✭✭]
* Define a variant type `suit` that represents the four suits, ♣
♦ ♥ ♠, in a [standard 52-card deck][cards]. All the
constructors of your type should be constant.
* Define a type `rank` that represents the possible ranks of a card: 2,
3, ..., 10, Jack, Queen, King, or Ace. There are many possible
solutions; you are free to choose whatever works for you. One is to make
`rank` be a synonym of `int`, and to assume that Jack=11, Queen=12,
King=13, Ace=1. Another is to use variants.
* Define a type `card` that represents the suit and rank of a single
card. Make it a record with two fields.
* Define a few values of type `card`: the Ace of Clubs, the Queen of Hearts,
the Two of Diamonds, the Seven of Spades.
[cards]: https://en.wikipedia.org/wiki/Standard_52-card_deck
□
## Pattern matching
##### Exercise: matching [✭]
For each pattern in the list below, give a value of type
`int option list` that does *not* match the pattern and is
not the empty list, or explain why that's impossible.
- `(Some x)::tl`
- `[Some 3110; None]`
- `[Some x; _]`
- `h1::h2::tl`
- `h :: tl`
□
##### Exercise: quadrant [✭✭]
Complete the `quadrant` function below, which should return the quadrant
of the given `x, y` point according to the diagram on the right (borrowed from [Wikipedia](https://en.wikipedia.org/wiki/File:Cartesian_coordinates_2D.svg)).
Points that lie on an axis do not belong to any quandrant. *Hints: (a) define a helper function
for the sign of an integer, (b) match against a pair.*
```
type quad = I | II | III | IV
type sign = Neg | Zero | Pos
let sign (x:int) :sign =
...
let quadrant : int*int -> quad option = fun (x,y) ->
match ... with
| ... -> Some I
| ... -> Some II
| ... -> Some III
| ... -> Some IV
| ... -> None
```
□
There is an extended syntax for pattern matching:
```
match e with
| p1 when g1 -> e1
| ...
| pn when gn -> en
```
The `gi` here are Boolean expressions much like the guard of an if expression.
Each of the `when gi` clauses is optional. If they occur, then branch `i`
is chosen only if `pi` matches `e` and `gi` evaluates to true.
##### Exercise: quadrant when [✭✭]
Rewrite the quadrant function to use the `when` syntax. You won't need
your helper function from before.
```
let quadrant' : int*int -> quad option = function
| ... when ... -> Some I
| ... when ... -> Some II
| ... when ... -> Some III
| ... when ... -> Some IV
| ... -> None
```
## Binary trees
Here is a definition for a binary tree data type:
```
type 'a tree =
| Leaf
| Node of 'a * 'a tree * 'a tree
```
A node carries a data item of type `'a` and has a left and right subtree. A leaf
is empty. Compare this definition to the definition of a list and notice how
similar their structure is:
```
type 'a tree = type 'a mylist =
| Leaf | Nil
| Node of 'a * 'a tree * 'a tree | Cons of 'a * 'a mylist
```
The only essential difference is that `Cons` carries one sublist, whereas
`Node` carries two subtrees.
Here is code that constructs a small tree:
```
(* the code below constructs this tree:
4
/ \
2 5
/ \ / \
1 3 6 7
*)
let t =
Node(4,
Node(2,
Node(1,Leaf,Leaf),
Node(3,Leaf,Leaf)
),
Node(5,
Node(1,Leaf,Leaf),
Node(3,Leaf,Leaf)
)
)
```
The *size* of a tree is the number of nodes in it (that is, `Node`s, not `Leaf`s).
For example, the size of tree `t` above is 7. Here is a function
function `size : 'a tree -> int` that returns the number of nodes in
a tree:
```
let rec size = function
| Leaf -> 0
| Node(_,l,r) -> 1 + size l + size r
```
##### Exercise: depth [✭✭]
Write a function `depth : 'a tree -> int` that returns the number of
nodes in any longest path from the root to a leaf. For example, the
depth of an empty tree (simply `Leaf`) is `0`, and the depth of tree `t`
above is `3`. *Hint: there is a library function `max : 'a -> 'a -> 'a`
that returns the maximum of any two values of the same type.*
□
##### Exercise: shape [✭✭✭]
Write a function `same_shape : 'a tree -> -> 'a tree -> bool` that determines whether
two trees have the same shape, regardless of whether the values they carry at each node
are the same. *Hint: use a pattern match with three branches, where the expression
being matched is a pair of trees.*
□
## Exceptions
OCaml has an exception mechanism similar to many other programming languages.
Exceptions are essentially variants. A new type of OCaml exception is defined
with this syntax:
```
exception E of t
```
where `E` is a constructor name and `t` is a type. The `of t` is optional.
Notice how this is similar to defining a constructor of a variant type.
To create an exception value, use the same syntax you would for creating
a variant value. Here, for example, is an exception value whose constructor
is `Failure`, which carries a `string`:
```
Failure "something went wrong"
```
This constructor is [pre-defined in the standard library][stdlib-exn] (scroll down
to "predefined exceptions") and is one of the more common exceptions that OCaml
programmers use.
[stdlib-exn]: http://caml.inria.fr/pub/docs/manual-ocaml/core.html#sec512
To raise an exception value `e`, simply write
```
raise e
```
There is a convenient function `failwith : string -> 'a` in the standard library
that raises `Failure`. That is, `failwith s` is equivalent to `raise (Failure s)`.
##### Exercise: list max exn [✭✭]
Write a function `list_max : int list -> int`
that returns the maximum integer in a list, or raises `Failure` if the list is empty.
□
To catch an exception, use this syntax:
```
try e with
| p1 -> e1
| ...
| pn -> en
```
The expression `e` is what might raise an exception. If it does not, the entire
`try` expression evaluates to whatever `e` does. If `e` does raise an exception value
`v`, that value `v` is that matched against the provide patterns, exactly like
`match` expression.
##### Exercise: list max exn string [✭✭]
Write a function `list_max_string : int list -> string`
that returns a string containing the maximum integer in a list, or
the string `"empty"` if the list is empty. *Hint: `string_of_int` in
the standard library will do what its name suggests.*
□
If it is part of a function's specification that it raises an exception, you
might want to write OUnit tests that check whether the function correctly does so.
Here's how to do that:
```
open OUnit2
let tests = "suite" >:::
[
"empty" >:: (fun _ -> assert_raises (Failure "hd") (fun () -> List.hd []));
"nonempty" >:: (fun _ -> assert_equal 1 (List.hd [1]));
]
let _ = run_test_tt_main tests
```
The expression `assert_raises exc (fun () -> e)` checks to see whether expression `e`
raises exception `exc`. If so, the OUnit test case succeeds, otherwise it fails.
**Note:** a common error is to forget the `(fun () -> ...)` around `e`. If you do,
the OUnit test case will fail, and you will likely be confused as to why. The reason
is that, without the extra anonymous function, the exception is raised before
`assert_raises` ever gets a chance to handle it.
##### Exercise: list max exn ounit [✭]
Write two OUnit tests to determine whether your solution to **list max exn**, above,
correctly raises an exception when its input is the empty list, and whether it
correctly returns the max value of the input list when that list is nonempty.
□
## Dictionaries
Recall from the previous lab that an *association list* is
a list of pairs that implements a *dictionary*, and that insertion
in an association list is constant time, but lookup is linear
time.
A more efficient dictionary implementation can be achieved with *binary
search trees*. Each node of the tree will be a pair of a key and value,
just like in an association list. Furthermore, the tree obeys
the **binary search tree invariant**: for any node n, all of
the values in the left subtree of n have keys that are less than n's
key, and all of the values in the right subtree of n have keys that are
greater than n's key. (We do not allow duplicate keys.)
For example, here is a binary search tree representing the same dictionary
as that in exercise **assoc list** in the previous lab:
```
let d =
Node((2,"two"),
Node((1,"one"),Leaf,Leaf),
Node((3,"three"),Leaf,Leaf)
)
```
The following exercises explore how to implement a binary search tree.
##### Exercise: lookup [✭✭]
Write a function `lookup : 'a -> ('a*'b) tree -> 'b option`
that returns the value associated with the given key, if it is present.
□
##### Exercise: insert [✭✭✭]
Write a function `insert : 'a -> 'b -> ('a*'b) tree -> ('a*'b) tree`
that adds a new key-value mapping to the given binary search tree. If the key
is already mapped in the tree, the output tree should map the key to the
new value.
□
##### Exercise: is_bst [✭✭✭✭]
Write a function `is_bst : ('a*'b) tree -> bool` that returns `true` if
and only if the given tree satisfies the binary search tree invariant.
An efficient version of this function that visits each node at most once
is somewhat tricky to write.
*Hint: write a recursive helper function that takes a tree and either
gives you (i) the minimum and maximum value in the tree, or (ii) tells
you that the tree is empty, or (iii) tells you that the tree does not
satisfy the invariant. Your `is_bst` function will not be recursive, but
will call your helper function and pattern match on the result. You
will need to define a new variant type for the return type of your helper
function.*
□
A truly efficient binary search tree should be *balanced*: the depths of various
paths through the tree should all be about the same. One famous balanced tree
data structure is the [2-3 tree][23tree], invented by Cornell's own
Prof. John Hopcroft in 1970.
[23tree]: https://en.wikipedia.org/wiki/2%E2%80%933_tree
## Polymorphic variants
Thus far, whenever you've wanted to define a variant type, you have had to give
it a name, such as `day`, `shape`, or `'a mylist`:
```
type day = Sun | Mon | Tue | Wed | Thu | Fri | Sat
type shape =
| Point of point
| Circle of point * float
| Rect of point * point
type 'a mylist = Nil | Cons of 'a * 'a mylist
```
Occasionally, you might need a variant type only for the return value of a single
function. For example, here's a function `f` that can either return
an `int` or \\(\infty\\); you are forced to define a variant type to represent
that result:
```
type fin_or_inf = Finite of int | Infinity
let f = function
| 0 -> Infinity
| 1 -> Finite 1
| n -> Finite (-n)
```
The downside of this definition is that you were forced to defined
`fin_or_inf` even though it won't be used throughout much of your program.
There's another kind of variant in OCaml that supports this kind of programming:
*polymorphic variants*. Polymorphic variants are just like variants, except:
1. You don't have declare their type or constructors before using them.
2. There is no name for a polymorphic variant type. (So another name
for this feature could have been "anonymous variants".)
3. The constructors of a polymorphic variant
start with ` ` ` (this the "grave accent", also called
backquote, back tick, and reverse single quote; it is typically found on the
same key as the `~` character, near the escape key).
Using polymorphic variants, we can rewrite `f`:
```
(* note: no type definition *)
let f = function
| 0 -> `Infinity
| 1 -> `Finite 1
| n -> `Finite (-n)
```
With this definition, the type of `f` is
```
val f : int -> [> `Finite of int | `Infinity ]
```
This type says that `f` either returns `` `Finite n`` for some `n:int`
or `` `Infinity``. The square brackets do not denote a list, but rather
a set of possible constructors. The `>` sign means that any code that
pattern matches against a value of that type must *at least* handle the
constructors `` `Finite`` and `` `Infinity``, and possibly more. For
example, we could write:
```
match f 3 with
| `NegInfinity -> "negative infinity"
| `Finite n -> "finite!"
| `Infinity -> "infinite"
```
It's perfectly fine for the pattern match to include constructors other
than `` `Finite`` or `` `Infinity``, because `f` is guaranteed never to
return any constructors other than those.
##### Exercise: quadrant poly [✭✭]
Modify your definition of quadrant to use polymorphic variants. The
types of your functions should become these:
```
val sign : int -> [> `Neg | `Pos | `Zero ]
val quadrant : int * int -> [> `I | `II | `III | `IV ] option
```
□
There are other, more compelling uses for polymorphic variants that we'll
see later in the course. They are particularly useful in libraries.
For now, we generally will steer you away from extensive use of polymorphic
variants, because their types can become difficult to manage.
(You'll also notice our syntax highlighter doesn't do a great job with them.
Apologies for that. If there are any wizard web developers who want
to talk to Prof. Clarkson about improving the syntax highlighting, please
get in touch!)