HomeTutorials99 Problems (solved) in OCaml [En]
Many of the solutions below have been written by Victor Nicollet. Please contribute more solutions or improve the existing ones.

# 99 Problems (solved) in OCaml

This section is inspired by Ninety-Nine Lisp Problems which in turn was based on “ Prolog problem list”. For each of these questions, some simple tests are shown—they may also serve to make the question clearer if needed. To work on these problems, we recommend you first install OCaml or use it inside your browser. The source of the following problems is available on GitHub.

## Working with lists

Write a function ```last : 'a list -> 'a option``` that returns the last element of a list.
```# let rec last = function
| [] -> None
| [x] -> Some x
| _ :: t -> last t;;val last : 'a list -> 'a option = <fun>
```
```# last [ `a ; `b ; `c ; `d ] = Some `d;;- : bool = true
# last [] = None;;- : bool = true
```
Find the last but one (last and penultimate) elements of a list.
```# let rec last_two = function
| [] | [_] -> None
| [x;y] -> Some (x,y)
| _::t -> last_two t;;val last_two : 'a list -> ('a * 'a) option = <fun>
```
```# last_two [ `a ; `b ; `c ; `d ] = Some (`c,`d);;- : bool = true
# last_two [ `a ] = None;;- : bool = true
```
Find the `k`'th element of a list.
```# let rec at k = function
| [] -> None
| h :: t -> if k = 1 then Some h else at (k-1) t;;val at : int -> 'a list -> 'a option = <fun>
```
```# at 3 [ `a ; `b ; `c ; `d ; `e ] = Some `c;;- : bool = true
# at 3 [ `a ] = None;;- : bool = true
```
Find the number of elements of a list.

OCaml standard library has `List.length` but we ask that you reimplement it. Bonus for a tail recursive solution.

```# (* This function is tail-recursive: it uses a constant amount of
stack memory regardless of list size. *)
let length list =
let rec aux n = function
| [] -> n
| _::t -> aux (n+1) t
in aux 0 list;;val length : 'a list -> int = <fun>
```
```# length [ `a ; `b ; `c] = 3;;- : bool = true
# length [] = 0;;- : bool = true
```
Reverse a list.

OCaml standard library has `List.rev` but we ask that you reimplement it.

```# let rev list =
let rec aux acc = function
| [] -> acc
| h::t -> aux (h::acc) t in
aux [] list;;val rev : 'a list -> 'a list = <fun>
```
```# rev [`a ; `b ; `c] = [`c ; `b ; `a];;- : bool = true
```
Find out whether a list is a palindrome.

HINT: a palindrome is its own reverse.

```# let is_palindrome list =
list = List.rev list
(* One can use either the rev function from the previous problem, or the
built-in List.rev *);;val is_palindrome : 'a list -> bool = <fun>
```
```# is_palindrome [ `x ; `a ; `m ; `a ; `x ];;- : bool = true
# not (is_palindrome [ `a ; `b ]);;- : bool = true
```
Flatten a nested list structure.
```# (* There is no nested list type in OCaml, so we need to define one
first. A node of a nested list is either an element, or a list of
nodes. *)
type 'a node =
| One of 'a
| Many of 'a node list;;```
```# (* This function traverses the list, prepending any encountered elements
to an accumulator, which flattens the list in inverse order. It can
then be reversed to obtain the actual flattened list. *)

let flatten list =
let rec aux acc = function
| [] -> acc
| One x :: t -> aux (x :: acc) t
| Many l :: t -> aux (aux acc l) t in
List.rev (aux [] list);;val flatten : 'a node list -> 'a list = <fun>
```
```# flatten [ One `a ; Many [ One `b ; Many [ One `c ; One `d ] ; One `e ] ]
= [ `a ; `b ; `c ; `d ; `e ];;- : bool = true
```
Eliminate consecutive duplicates of list elements.
```# let rec compress = function
| a :: (b :: _ as t) -> if a = b then compress t else a :: compress t
| smaller -> smaller;;val compress : 'a list -> 'a list = <fun>
```
```# compress [`a;`a;`a;`a;`b;`c;`c;`a;`a;`d;`e;`e;`e;`e] = [`a;`b;`c;`a;`d;`e];;- : bool = true
```
Pack consecutive duplicates of list elements into sublists.
```# let pack list =
let rec aux current acc = function
| [] -> []    (* Can only be reached if original list is empty *)
| [x] -> (x :: current) :: acc
| a :: (b :: _ as t) ->
if a = b then aux (a :: current) acc t
else aux [] ((a :: current) :: acc) t  in
List.rev (aux [] [] list);;val pack : 'a list -> 'a list list = <fun>
```
```# pack [`a;`a;`a;`a;`b;`c;`c;`a;`a;`d;`d;`e;`e;`e;`e]
= [[`a;`a;`a;`a]; [`b]; [`c;`c]; [`a;`a]; [`d;`d]; [`e;`e;`e;`e]];;- : bool = true
```
```# let encode list =
let rec aux count acc = function
| [] -> [] (* Can only be reached if original list is empty *)
| [x] -> (count+1, x) :: acc
| a :: (b :: _ as t) -> if a = b then aux (count + 1) acc t
else aux 0 ((count+1,a) :: acc) t   in
List.rev (aux 0 [] list);;val encode : 'a list -> (int * 'a) list = <fun>
```
```# encode [`a;`a;`a;`a;`b;`c;`c;`a;`a;`d;`e;`e;`e;`e]
= [4,`a ; 1,`b ; 2,`c ; 2,`a ; 1,`d ; 4,`e];;- : bool = true
```
Modified run-length encoding.

Modify the result of the previous problem in such a way that if an element has no duplicates it is simply copied into the result list. Only elements with duplicates are transferred as (N E) lists.

Since OCaml lists are homogeneous, one needs to define a type to hold both single elements and sub-lists.

```# type 'a rle =
| One of 'a
| Many of (int * 'a);;type 'a rle = One of 'a | Many of (int * 'a)
```
```# let pack list =
let rec aux current acc = function
| [] -> [] (* Can only be reached if original list is empty *)
| [x] -> (x :: current) :: acc
| a :: (b :: _ as t) -> if a = b then aux (a :: current) acc t
else aux [] ((a :: current) :: acc) t  in
List.rev (aux [] [] list)
let encode list =
let rec aux = function
| [] -> []
| [] :: t -> aux t
| [x] :: t -> One x :: aux t
| (x :: l) :: t -> Many (1 + List.length l , x) :: aux t  in
aux (pack list);;val pack : 'a list -> 'a list list = <fun>
val encode : 'a list -> 'a rle list = <fun>
```
```# encode [`a;`a;`a;`a;`b;`c;`c;`a;`a;`d;`e;`e;`e;`e]
= [Many (4,`a) ; One `b ; Many (2,`c) ; Many (2,`a) ; One `d ; Many (4,`e)];;- : bool = true
```
Decode a run-length encoded list.

Given a run-length code list generated as specified in the previous problem, construct its uncompressed version.

```# let decode list =
let rec many acc n x =
if n = 0 then acc else many (x :: acc) (n-1) x in
let rec aux acc = function
| [] -> acc
| One x :: t -> aux (x :: acc) t
| Many (n,x) :: t -> aux (many acc n x) t  in
aux [] (List.rev list);;val decode : 'a rle list -> 'a list = <fun>
```
```# decode [Many (4,`a); One `b; Many (2,`c); Many (2,`a); One `d; Many (4,`e)]
= [`a;`a;`a;`a;`b;`c;`c;`a;`a;`d;`e;`e;`e;`e];;- : bool = true
```
Run-length encoding of a list (direct solution).

Implement the so-called run-length encoding data compression method directly. I.e. don't explicitly create the sublists containing the duplicates, as in problem “Pack consecutive duplicates of list elements into sublists”, but only count them. As in problem “Modified run-length encoding”, simplify the result list by replacing the singleton lists (1 X) by X.

```# let encode list =
let rle count x = if count = 0 then One x else Many (count + 1, x) in
let rec aux count acc = function
| [] -> [] (* Can only be reached if original list is empty *)
| [x] -> rle count x :: acc
| a :: (b :: _ as t) -> if a = b then aux (count + 1) acc t
else aux 0 (rle count a :: acc) t   in
List.rev (aux 0 [] list);;val encode : 'a list -> 'a rle list = <fun>
```
```# encode [`a;`a;`a;`a;`b;`c;`c;`a;`a;`d;`e;`e;`e;`e] =
[Many (4,`a); One `b; Many (2,`c); Many (2,`a); One `d; Many (4,`e)];;- : bool = true
```
Duplicate the elements of a list.
```# let rec duplicate = function
| [] -> []
| h :: t -> h :: h :: duplicate t;;val duplicate : 'a list -> 'a list = <fun>
```
```# duplicate [`a;`b;`c;`c;`d] = [`a;`a;`b;`b;`c;`c;`c;`c;`d;`d];;- : bool = true
```
Replicate the elements of a list a given number of times.
```# let replicate list n =
let rec prepend n acc x =
if n = 0 then acc else prepend (n-1) (x :: acc) x in
let rec aux acc = function
| [] -> acc
| h :: t -> aux (prepend n acc h) t  in
(* This could also be written as:
List.fold_left (prepend n) [] (List.rev list) *)
aux [] (List.rev list);;val replicate : 'a list -> int -> 'a list = <fun>
```
```# replicate [`a;`b;`c] 3 = [`a;`a;`a;`b;`b;`b;`c;`c;`c];;- : bool = true
```
Drop every N'th element from a list.
```# let drop list n =
let rec aux i = function
| [] -> []
| h :: t -> if i = n then aux 1 t else h :: aux (i+1) t  in
aux 1 list;;val drop : 'a list -> int -> 'a list = <fun>
```
```# drop [`a;`b;`c;`d;`e;`f;`g;`h;`i;`j] 3 = [`a;`b;`d;`e;`g;`h;`j];;- : bool = true
```
Split a list into two parts; the length of the first part is given.

If the length of the first part is longer than the entire list, then the first part is the list and the second part is empty.

```# let split list n =
let rec aux i acc = function
| [] -> List.rev acc, []
| h :: t as l -> if i = 0 then List.rev acc, l
else aux (i-1) (h :: acc) t  in
aux n [] list;;val split : 'a list -> int -> 'a list * 'a list = <fun>
```
```# split [`a;`b;`c;`d;`e;`f;`g;`h;`i;`j] 3
= ([`a;`b;`c] , [`d;`e;`f;`g;`h;`i;`j]);;- : bool = true
# split [`a;`b;`c;`d] 5 = ([`a; `b; `c; `d], []);;- : bool = true
```
Extract a slice from a list.

Given two indices, `i` and `k`, the slice is the list containing the elements between the `i`'th and `k`'th element of the original list (both limits included). Start counting the elements with 0 (this is the way the `List` module numbers elements).

```# let slice list i k =
let rec take n = function
| [] -> []
| h :: t -> if n = 0 then [] else h :: take (n-1) t
in
let rec drop n = function
| [] -> []
| h :: t as l -> if n = 0 then l else drop (n-1) t
in
take (k - i + 1) (drop i list);;val slice : 'a list -> int -> int -> 'a list = <fun>
```
```# slice [`a;`b;`c;`d;`e;`f;`g;`h;`i;`j] 2 6 = [`c;`d;`e;`f;`g];;- : bool = true
```
Rotate a list N places to the left.
```# let split list n =
let rec aux i acc = function
| [] -> List.rev acc, []
| h :: t as l -> if i = 0 then List.rev acc, l
else aux (i-1) (h :: acc) t    in
aux n [] list

let rotate list n =
let len = List.length list in
(* Compute a rotation value between 0 and len-1 *)
let n = if len = 0 then 0 else (n mod len + len) mod len in
if n = 0 then list
else let a, b = split list n in b @ a;;val split : 'a list -> int -> 'a list * 'a list = <fun>
val rotate : 'a list -> int -> 'a list = <fun>
```
```# rotate [`a;`b;`c;`d;`e;`f;`g;`h] 3 = [`d;`e;`f;`g;`h;`a;`b;`c];;- : bool = true
# rotate [`a;`b;`c;`d;`e;`f;`g;`h] (-2) = [`g;`h;`a;`b;`c;`d;`e;`f];;- : bool = true
```
Remove the K'th element from a list.

The first element of the list is numbered 0, the second 1,...

```# let rec remove_at n = function
| [] -> []
| h :: t -> if n = 0 then t else h :: remove_at (n-1) t;;val remove_at : int -> 'a list -> 'a list = <fun>
```
```# remove_at 1 [`a;`b;`c;`d] = [`a;`c;`d];;- : bool = true
```
Insert an element at a given position into a list.

Start counting list elements with 0.

```# let rec insert_at x n = function
| [] -> []
| h :: t as l -> if n = 0 then x :: l else h :: insert_at x (n-1) t;;val insert_at : 'a -> int -> 'a list -> 'a list = <fun>
```
```# insert_at `alfa 1 [`a;`b;`c;`d] = [`a;`alfa;`b;`c;`d];;- : bool = true
```
Create a list containing all integers within a given range.

If first argument is smaller than second, produce a list in decreasing order.

```# let range a b =
let rec aux a b =
if a > b then [] else a :: aux (a+1) b  in
if a > b then List.rev (aux b a) else aux a b;;val range : int -> int -> int list = <fun>
```
```# range 4 9 = [4;5;6;7;8;9];;- : bool = true
# range 9 4 = [9;8;7;6;5;4];;- : bool = true
```
Extract a given number of randomly selected elements from a list.

The selected items shall be returned in a list. We use the `Random` module but do not initialize it with `Random.self_init` for reproducibility.

```# let rec rand_select list n =
let rec extract acc n = function
| [] -> raise Not_found
| h :: t -> if n = 0 then (h, acc @ t) else extract (h::acc) (n-1) t
in
let extract_rand list len =
extract [] (Random.int len) list
in
let rec aux n acc list len =
if n = 0 then acc else
let picked, rest = extract_rand list len in
aux (n-1) (picked :: acc) rest (len-1)
in
let len = List.length list in
aux (min n len) [] list len;;val rand_select : 'a list -> int -> 'a list = <fun>
```
```# rand_select [`a;`b;`c;`d;`e;`f;`g;`h] 3;;- : [> `a | `b | `c | `d | `e | `f | `g | `h ] list = [`g; `d; `a]
```
Lotto: Draw N different random numbers from the set 1..M.

The selected numbers shall be returned in a list.

```# (* [range] and [rand_select] defined in problems above *)
let lotto_select n m = rand_select (range 1 m) n;;val lotto_select : int -> int -> int list = <fun>
```
```# lotto_select 6 49;;- : int list = [10; 20; 44; 22; 41; 2]
```
Generate a random permutation of the elements of a list.
```# let rec permutation list =
let rec extract acc n = function
| [] -> raise Not_found
| h :: t -> if n = 0 then (h, acc @ t) else extract (h::acc) (n-1) t
in
let extract_rand list len =
extract [] (Random.int len) list
in
let rec aux acc list len =
if len = 0 then acc else
let picked, rest = extract_rand list len in
aux (picked :: acc) rest (len-1)
in
aux [] list (List.length list);;val permutation : 'a list -> 'a list = <fun>
```
```# permutation ['a';'b';'c';'d';'e';'f'];;- : char list = ['a'; 'e'; 'f'; 'b'; 'd'; 'c']
```
Generate the combinations of K distinct objects chosen from the N elements of a list

In how many ways can a committee of 3 be chosen from a group of 12 people? We all know that there are C(12,3) = 220 possibilities (C(N,K) denotes the well-known binomial coefficients). For pure mathematicians, this result may be great. But we want to really generate all the possibilities in a list.

```# let extract k list =
let rec aux k acc emit = function
| [] -> acc
| h :: t ->
if k = 1 then aux k (emit [h] acc) emit t else
let new_emit x = emit (h :: x) in
aux k (aux (k-1) acc new_emit t) emit t
in
let emit x acc = x :: acc in
aux k [] emit list;;val extract : int -> 'a list -> 'a list list = <fun>
```
```# extract 2 [`a;`b;`c;`d]
= [[`c;`d]; [`b;`d]; [`b;`c]; [`a;`d]; [`a;`c]; [`a;`b]];;- : bool = true
```
Group the elements of a set into disjoint subsets.

a) In how many ways can a group of 9 people work in 3 disjoint subgroups of 2, 3 and 4 persons? Write a function that generates all the possibilities and returns them in a list.

b) Generalize the above function in a way that we can specify a list of group sizes and the function will return a list of groups.

```# (* This implementation is less streamlined than the one-extraction
version, because more work is done on the lists after each transform
to prepend the actual items. The end result is cleaner in terms of
code, though. *)

let group list sizes =
let initial = List.map (fun size -> size, []) sizes in
(* The core of the function. Prepend accepts a list of groups, each with
the number of items that should be added, and prepends the item to every
group that can support it, thus turning [1,a ; 2,b ; 0,c] into
[ [0,x::a ; 2,b ; 0,c ] ; [1,a ; 1,x::b ; 0,c] ; [ 1,a ; 2,b ; 0,c ]]

Again, in the prolog language (for which these questions are
originally intended), this function is a whole lot simpler.  *)
let prepend p list =
let emit l acc = l :: acc in
let rec aux emit acc = function
| [] -> emit [] acc
| (n,l) as h :: t ->
let acc = if n > 0 then emit ((n-1, p::l) :: t) acc else acc in
aux (fun l acc -> emit (h :: l) acc) acc t
in
aux emit [] list
in
let rec aux = function
| [] -> [ initial ]
| h :: t -> List.concat (List.map (prepend h) (aux t))
in
let all = aux list in
(* Don't forget to eliminate all group sets that have non-full groups *)
let complete = List.filter (List.for_all (fun (x,_) -> x = 0)) all in
List.map (List.map snd) complete;;val group : 'a list -> int list -> 'a list list list = <fun>
```
```# group [`a;`b;`c;`d] [2;1]
= [[[`a; `b]; [`c]]; [[`a; `c]; [`b]]; [[`b; `c]; [`a]]; [[`a; `b]; [`d]];
[[`a; `c]; [`d]]; [[`b; `c]; [`d]]; [[`a; `d]; [`b]]; [[`b; `d]; [`a]];
[[`a; `d]; [`c]]; [[`b; `d]; [`c]]; [[`c; `d]; [`a]]; [[`c; `d]; [`b]]];;- : bool = true
```
Sorting a list of lists according to length of sublists

a) We suppose that a list contains elements that are lists themselves. The objective is to sort the elements of this list according to their length. E.g. short lists first, longer lists later, or vice versa.

b) Again, we suppose that a list contains elements that are lists themselves. But this time the objective is to sort the elements of this list according to their length frequency; i.e., in the default, where sorting is done ascendingly, lists with rare lengths are placed first, others with a more frequent length come later.

```# (* We might not be allowed to use built-in List.sort, so here's an
eight-line implementation of insertion sort - O(n²) time complexity. *)
let rec insert cmp e = function
| [] -> [e]
| h :: t as l -> if cmp e h <= 0 then e :: l else h :: insert cmp e t

let rec sort cmp = function
| [] -> []
| h :: t -> insert cmp h (sort cmp t)

(* Sorting according to length : prepend length, sort, remove length *)
let length_sort lists =
let lists = List.map (fun list -> List.length list, list) lists in
let lists = sort (fun a b -> compare (fst a) (fst b)) lists in
List.map snd lists;;val insert : ('a -> 'a -> int) -> 'a -> 'a list -> 'a list = <fun>
val sort : ('a -> 'a -> int) -> 'a list -> 'a list = <fun>
val length_sort : 'a list list -> 'a list list = <fun>
# (* Sorting according to length frequency : prepend frequency, sort,
remove frequency. Frequencies are extracted by sorting lengths
and applying RLE to count occurences of each length (see problem
"Run-length encoding of a list.") *)
let rle list =
let rec aux count acc = function
| [] -> [] (* Can only be reached if original list is empty *)
| [x] -> (x, count + 1) :: acc
| a :: (b :: _ as t) ->
if a = b then aux (count + 1) acc t
else aux 0 ((a, count + 1) :: acc) t in
aux 0 [] list

let frequency_sort lists =
let lengths = List.map List.length lists in
let freq = rle (sort compare lengths) in
let by_freq =
List.map (fun list -> List.assoc (List.length list) freq , list) lists in
let sorted = sort (fun a b -> compare (fst a) (fst b)) by_freq in
List.map snd sorted;;val rle : 'a list -> ('a * int) list = <fun>
val frequency_sort : 'a list list -> 'a list list = <fun>
```
```# length_sort [ [`a;`b;`c]; [`d;`e]; [`f;`g;`h]; [`d;`e];
[`i;`j;`k;`l]; [`m;`n]; [`o] ]
= [[`o]; [`d; `e]; [`d; `e]; [`m; `n]; [`a; `b; `c]; [`f; `g; `h];
[`i; `j; `k; `l]];;- : bool = true
# frequency_sort [ [`a;`b;`c]; [`d;`e]; [`f;`g;`h]; [`d;`e];
[`i;`j;`k;`l]; [`m;`n]; [`o] ]
= [[`i; `j; `k; `l]; [`o]; [`a; `b; `c]; [`f; `g; `h]; [`d; `e]; [`d; `e];
[`m; `n]];;- : bool = true
```

## Arithmetic

Determine whether a given integer number is prime.

Recall that `d` divides `n` iff ```n mod d = 0```. This is a naive solution. See the Sieve of Eratosthenes for a more clever one.

```# let is_prime n =
let n = abs n in
let rec is_not_divisor d =
d * d > n || (n mod d <> 0 && is_not_divisor (d+1)) in
n <> 1 && is_not_divisor 2;;val is_prime : int -> bool = <fun>
```
```# not(is_prime 1);;- : bool = true
# is_prime 7;;- : bool = true
# not (is_prime 12);;- : bool = true
```
Determine the greatest common divisor of two positive integer numbers.

Use Euclid's algorithm.

```# let rec gcd a b =
if b = 0 then a else gcd b (a mod b);;val gcd : int -> int -> int = <fun>
```
```# gcd 13 27 = 1;;- : bool = true
# gcd 20536 7826 = 2;;- : bool = true
```
Determine whether two positive integer numbers are coprime.

Two numbers are coprime if their greatest common divisor equals 1.

```# (* [gcd] is defined in the previous question *)
let coprime a b = gcd a b = 1;;val coprime : int -> int -> bool = <fun>
```
```# coprime 13 27;;- : bool = true
# not (coprime 20536 7826);;- : bool = true
```
Calculate Euler's totient function phi(m).

Euler's so-called totient function φ(m) is defined as the number of positive integers r (1 ≤ r < m) that are coprime to m. We let φ(1) = 1.

Find out what the value of phi(m) is if m is a prime number. Euler's totient function plays an important role in one of the most widely used public key cryptography methods (RSA). In this exercise you should use the most primitive method to calculate this function (there are smarter ways that we shall discuss later).

```# (* [coprime] is defined in the previous question *)
let phi n =
let rec count_coprime acc d =
if d < n then
count_coprime (if coprime n d then acc + 1 else acc) (d + 1)
else acc
in
if n = 1 then 1 else count_coprime 0 1;;val phi : int -> int = <fun>
```
```# phi 10 = 4;;- : bool = true
# phi 13 = 12;;- : bool = true
```
Determine the prime factors of a given positive integer.

Construct a flat list containing the prime factors in ascending order.

```# (* Recall that d divides n iff [n mod d = 0] *)
let factors n =
let rec aux d n =
if n = 1 then [] else
if n mod d = 0 then d :: aux d (n / d) else aux (d+1) n
in
aux 2 n;;val factors : int -> int list = <fun>
```
```# factors 315 = [3;3;5;7];;- : bool = true
```
Determine the prime factors of a given positive integer (2).

Construct a list containing the prime factors and their multiplicity. Hint: The problem is similar to problem Run-length encoding of a list (direct solution).

```# let factors n =
let rec aux d n =
if n = 1 then [] else
if n mod d = 0 then
match aux d (n / d) with
| (h,n) :: t when h = d -> (h,n+1) :: t
| l -> (d,1) :: l
else aux (d+1) n
in
aux 2 n;;val factors : int -> (int * int) list = <fun>
```
```# factors 315 = [3,2 ; 5,1 ; 7,1];;- : bool = true
```
Calculate Euler's totient function phi(m) (improved).

See problem “Calculate Euler's totient function phi(m)” for the definition of Euler's totient function. If the list of the prime factors of a number m is known in the form of the previous problem then the function phi(m) can be efficiently calculated as follows: Let ```[(p1, m1); (p2, m2); (p3, m3); ...]``` be the list of prime factors (and their multiplicities) of a given number m. Then φ(m) can be calculated with the following formula:

φ(m) = (p1 - 1) × p1m1 - 1 × (p2 - 1) × p2m2 - 1 × (p3 - 1) × p3m3 - 1 × ...
```# (* Naive power function. *)
let rec pow n p = if p < 1 then 1 else n * pow n (p-1);;val pow : int -> int -> int = <fun>
# (* [factors] is defined in the previous question. *)
let phi_improved n =
let rec aux acc = function
| [] -> acc
| (p,m) :: t -> aux ((p - 1) * pow p (m - 1) * acc) t in
aux 1 (factors n);;val phi_improved : int -> int = <fun>
```
```# phi_improved 10 = 4;;- : bool = true
# phi_improved 13 = 12;;- : bool = true
```
Compare the two methods of calculating Euler's totient function.

Use the solutions of problems “Calculate Euler's totient function phi(m)” and “Calculate Euler's totient function phi(m) (improved)” to compare the algorithms. Take the number of logical inferences as a measure for efficiency. Try to calculate φ(10090) as an example.

```# (* Naive [timeit] function.  It requires the [Unix] module to be loaded. *)
let timeit f a =
let t0 = Unix.gettimeofday() in
ignore(f a);
let t1 = Unix.gettimeofday() in
t1 -. t0;;val timeit : ('a -> 'b) -> 'a -> float = <fun>
```
```# timeit phi 10090;;- : float = 0.00298786163330078125
# timeit phi_improved 10090;;- : float = 5.41210174560546875e-05
```
A list of prime numbers.

Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range.

```# let is_prime n =
let n = max n (-n) in
let rec is_not_divisor d =
d * d > n || (n mod d <> 0 && is_not_divisor (d+1)) in
is_not_divisor 2

let rec all_primes a b =
if a > b then [] else
let rest = all_primes (a + 1) b in
if is_prime a then a :: rest else rest;;val is_prime : int -> bool = <fun>
val all_primes : int -> int -> int list = <fun>
```
```# List.length (all_primes 2 7920) = 1000;;- : bool = true
```
Goldbach's conjecture.

Goldbach's conjecture says that every positive even number greater than 2 is the sum of two prime numbers. Example: 28 = 5 + 23. It is one of the most famous facts in number theory that has not been proved to be correct in the general case. It has been numerically confirmed up to very large numbers. Write a function to find the two prime numbers that sum up to a given even integer.

```# (* [is_prime] is defined in the previous solution *)
let goldbach n =
let rec aux d =
if is_prime d && is_prime (n - d) then (d, n-d)
else aux (d+1) in
aux 2;;val goldbach : int -> int * int = <fun>
```
```# goldbach 28 = (5,23);;- : bool = true
```
A list of Goldbach compositions.

Given a range of integers by its lower and upper limit, print a list of all even numbers and their Goldbach composition.

In most cases, if an even number is written as the sum of two prime numbers, one of them is very small. Very rarely, the primes are both bigger than say 50. Try to find out how many such cases there are in the range 2..3000.

```# (* [goldbach] is defined in the previous question. *)
let rec goldbach_list a b =
if a > b then [] else
if a mod 2 = 1 then goldbach_list (a+1) b
else (a, goldbach a) :: goldbach_list (a+2) b

let goldbach_limit a b lim =
List.filter (fun (_,(a,b)) -> a > lim && b > lim) (goldbach_list a b);;val goldbach_list : int -> int -> (int * (int * int)) list = <fun>
val goldbach_limit : int -> int -> int -> (int * (int * int)) list = <fun>
```
```# goldbach_list 9 20
= [(10, (3, 7)); (12, (5, 7)); (14, (3, 11)); (16, (3, 13)); (18, (5, 13));
(20, (3, 17))];;- : bool = true
# goldbach_limit 1 2000 50
= [(992, (73, 919)); (1382, (61, 1321)); (1856, (67, 1789));
(1928, (61, 1867))];;- : bool = true
```

## Logic and Codes

Let us define a small “language” for boolean expressions containing variables:
```# type bool_expr =
| Var of string
| Not of bool_expr
| And of bool_expr * bool_expr
| Or of bool_expr * bool_expr;;type bool_expr =
Var of string
| Not of bool_expr
| And of bool_expr * bool_expr
| Or of bool_expr * bool_expr
```
A logical expression in two variables can then be written in prefix notation, as in the following example:
```# And(Or(Var "a", Var "b"), And(Var "a", Var "b"));;- : bool_expr = And (Or (Var "a", Var "b"), And (Var "a", Var "b"))
```
Truth tables for logical expressions (2 variables).

Define a function, `table2` which returns the truth table of a given logical expression in two variables (specified as arguments). The return value must be a list of triples containing ```(value_of_a, balue_of_b, value_of_expr)```.

```# let rec eval2 a val_a b val_b = function
| Var x -> if x = a then val_a
else if x = b then val_b
else failwith "The expression contains an invalid variable"
| Not e -> not(eval2 a val_a b val_b e)
| And(e1, e2) -> eval2 a val_a b val_b e1 && eval2 a val_a b val_b e2
| Or(e1, e2) -> eval2 a val_a b val_b e1 || eval2 a val_a b val_b e2
let table2 a b expr =
[(true,  true,  eval2 a true  b true  expr);
(true,  false, eval2 a true  b false expr);
(false, true,  eval2 a false b true  expr);
(false, false, eval2 a false b false expr) ];;val eval2 : string -> bool -> string -> bool -> bool_expr -> bool = <fun>
val table2 : string -> string -> bool_expr -> (bool * bool * bool) list =
<fun>
```
```# table2 "a" "b" (And(Var "a", Or(Var "a", Var "b")))
= [(true, true, true);
(true, false, true);
(false, true, false);
(false, false, false) ];;- : bool = true
```
Truth tables for logical expressions.

Generalize the previous problem in such a way that the logical expression may contain any number of logical variables. Define `table` in a way that ```table variables expr``` returns the truth table for the expression `expr`, which contains the logical variables enumerated in `variables`.

```# (* [val_vars] is an associative list containing the truth value of
each variable.  For efficiency, a Map or a Hashtlb should be preferred. *)
let rec eval val_vars = function
| Var x -> List.assoc x val_vars
| Not e -> not(eval val_vars e)
| And(e1, e2) -> eval val_vars e1 && eval val_vars e2
| Or(e1, e2) -> eval val_vars e1 || eval val_vars e2

(* Again, this is an easy and short implementation rather than an
efficient one. *)
let rec table_make val_vars vars expr =
match vars with
| [] -> [(List.rev val_vars, eval val_vars expr)]
| v :: tl ->
table_make ((v, true) :: val_vars) tl expr
@ table_make ((v, false) :: val_vars) tl expr

let table vars expr = table_make [] vars expr;;val eval : (string * bool) list -> bool_expr -> bool = <fun>
val table_make :
(string * bool) list ->
string list -> bool_expr -> ((string * bool) list * bool) list = <fun>
val table : string list -> bool_expr -> ((string * bool) list * bool) list =
<fun>
```
```# table ["a"; "b"] (And(Var "a", Or(Var "a", Var "b")))
= [["a", true; "b", true], true;
["a", true; "b", false], true;
["a", false; "b", true], false;
["a", false; "b", false], false ];;- : bool = true
# let a = Var "a" and b = Var "b" and c = Var "c" in
table ["a"; "b"; "c"] (Or(And(a, Or(b,c)), Or(And(a,b), And(a,c))));;- : ((string * bool) list * bool) list =
[([("a", true); ("b", true); ("c", true)], true);
([("a", true); ("b", ...); ...], ...); ...]
```
Gray code.

An n-bit Gray code is a sequence of n-bit strings constructed according to certain rules. For example,

```n = 1: C(1) = ['0','1'].
n = 2: C(2) = ['00','01','11','10'].
n = 3: C(3) = ['000','001','011','010',´110´,´111´,´101´,´100´].
```

Find out the construction rules and write a function with the following specification: `gray n` returns the `n`-bit Gray code.

```# let prepend c s =
(* Prepend the char [c] to the string [s]. *)
let s' = String.create (String.length s + 1) in
s'.[0] <- c;
String.blit s 0 s' 1 (String.length s);
s'

let rec gray n =
if n <= 1 then ["0"; "1"]
else let g = gray (n - 1) in
List.map (prepend '0') g @ List.map (prepend '1') g;;val prepend : char -> string -> string = <fun>
val gray : int -> string list = <fun>
```
```# gray 1 = ["0"; "1"];;- : bool = true
# gray 2 = ["00"; "01"; "11"; "10"];;- : bool = false
# gray 3 = ["000"; "001"; "011"; "010"; "110"; "111"; "101"; "100"];;- : bool = false
```
Huffman code.

First of all, consult a good book on discrete mathematics or algorithms for a detailed description of Huffman codes (you can start with the Wikipedia page)!

We suppose a set of symbols with their frequencies, given as a list of `Fr(S,F)` terms. Example: ```fs = [Fr(a,45); Fr(b,13); Fr(c,12); Fr(d,16); Fr(e,9); Fr(f,5)]```. Our objective is to construct a list `Hc(S,C)` terms, where `C` is the Huffman code word for the symbol `S`. In our example, the result could be ```hs = [Hc(a,'0'); Hc(b,'101'); Hc(c,'100'); Hc(d,'111'); Hc(e,'1101'); hc(f,'1100')]``` [hc(a,'01'),...etc.]. The task shall be performed by the function `huffman` defined as follows: `huffman(fs)` returns the Huffman code table for the frequency table `fs`

`# ;;`
`# ;;`

## Binary Trees

A binary tree is either empty or it is composed of a root element and two successors, which are binary trees themselves.

In OCaml, one can define a new type `binary_tree` that carries an arbitrary value of type `'a` at each node.

```# type 'a binary_tree =
| Empty
| Node of 'a * 'a binary_tree * 'a binary_tree;;type 'a binary_tree = Empty | Node of 'a * 'a binary_tree * 'a binary_tree
```

An example of tree carrying `char` data is:

```# let example_tree =
Node('a', Node('b', Node('d', Empty, Empty), Node('e', Empty, Empty)),
Node('c', Empty, Node('f', Node('g', Empty, Empty), Empty)));;val example_tree : char binary_tree =
Node ('a', Node ('b', Node ('d', Empty, Empty), Node ('e', Empty, Empty)),
Node ('c', Empty, Node ('f', Node ('g', Empty, ...), ...)))
```

In OCaml, the strict type discipline guarantees that, if you get a value of type `binary_tree`, then it must have been created with the two constructors `Empty` and `Node`.

Construct completely balanced binary trees

In a completely balanced binary tree, the following property holds for every node: The number of nodes in its left subtree and the number of nodes in its right subtree are almost equal, which means their difference is not greater than one.

Write a function `cbal_tree` to construct completely balanced binary trees for a given number of nodes. The function should generate all solutions via backtracking. Put the letter `'x'` as information into all nodes of the tree.

```# (* Build all trees with given [left] and [right] subtrees. *)
let add_trees_with left right all =
List.fold_left (fun a r -> Node('x', l, r) :: a) all right in

let rec cbal_tree n =
if n = 0 then [Empty]
else if n mod 2 = 1 then
let t = cbal_tree (n / 2) in
else (* n even: n-1 nodes for the left & right subtrees altogether. *)
let t1 = cbal_tree (n / 2 - 1) in
let t2 = cbal_tree (n / 2) in
char binary_tree list ->
char binary_tree list -> char binary_tree list -> char binary_tree list =
<fun>
val cbal_tree : int -> char binary_tree list = <fun>
```
```# cbal_tree 4
= [Node('x', Node('x', Empty, Empty),
Node('x', Node('x', Empty, Empty), Empty));
Node('x', Node('x', Empty, Empty),
Node('x', Empty, Node('x', Empty, Empty)));
Node('x', Node('x', Node('x', Empty, Empty), Empty),
Node('x', Empty, Empty));
Node('x', Node('x', Empty, Node('x', Empty, Empty)),
Node('x', Empty, Empty)); ];;- : bool = true
# List.length(cbal_tree 40);;- : int = 524288
```
Symmetric binary trees

Let us call a binary tree symmetric if you can draw a vertical line through the root node and then the right subtree is the mirror image of the left subtree. Write a function `is_symmetric` to check whether a given binary tree is symmetric.

Hint: Write a function `is_mirror` first to check whether one tree is the mirror image of another. We are only interested in the structure, not in the contents of the nodes.

```# let rec is_mirror t1 t2 =
match t1, t2 with
| Empty, Empty -> true
| Node(_, l1, r1), Node(_, l2, r2) ->
is_mirror l1 r2 && is_mirror r1 l2
| _ -> false

let is_symmetric = function
| Empty -> true
| Node(_, l, r) -> is_mirror l r;;val is_mirror : 'a binary_tree -> 'b binary_tree -> bool = <fun>
val is_symmetric : 'a binary_tree -> bool = <fun>
```
Binary search trees (dictionaries)

Construct a binary search tree from a list of integer numbers.

```# let rec insert tree x = match tree with
| Empty -> Node(x, Empty, Empty)
| Node(y, l, r) ->
if x = y then tree
else if x < y then Node(y, insert l x, r)
else Node(y, l, insert r x)

let construct l = List.fold_left insert Empty l;;val insert : 'a binary_tree -> 'a -> 'a binary_tree = <fun>
val construct : 'a list -> 'a binary_tree = <fun>
```
```# construct [3;2;5;7;1]
= Node(3, Node(2, Node(1, Empty, Empty), Empty),
Node(5, Empty, Node(7, Empty, Empty)));;- : bool = true
```

Then use this function to test the solution of the previous problem.

```# is_symmetric(construct [5;3;18;1;4;12;21]);;- : bool = true
# not(is_symmetric(construct [3;2;5;7;4]));;- : bool = true
```

Apply the generate-and-test paradigm to construct all symmetric, completely balanced binary trees with a given number of nodes.

```# let sym_cbal_trees n =
List.filter is_symmetric (cbal_tree n);;val sym_cbal_trees : int -> char binary_tree list = <fun>
```
```# sym_cbal_trees 5
= [Node('x', Node('x', Node('x', Empty, Empty), Empty),
Node('x', Empty, Node('x', Empty, Empty)));
Node('x', Node('x', Empty, Node('x', Empty, Empty)),
Node('x', Node('x', Empty, Empty), Empty)) ];;- : bool = true
```

How many such trees are there with 57 nodes? Investigate about how many solutions there are for a given number of nodes? What if the number is even? Write an appropriate function.

```# List.length (sym_cbal_trees 57);;- : int = 256
# List.map (fun n -> n, List.length(sym_cbal_trees n)) (range 10 20)
= [(10, 0); (11, 4); (12, 0); (13, 4); (14, 0); (15, 1);
(16, 0); (17, 8); (18, 0); (19, 16); (20, 0)];;- : bool = true
```
Construct height-balanced binary trees

In a height-balanced binary tree, the following property holds for every node: The height of its left subtree and the height of its right subtree are almost equal, which means their difference is not greater than one.

Write a function `hbal_tree` to construct height-balanced binary trees for a given height. The function should generate all solutions via backtracking. Put the letter `'x'` as information into all nodes of the tree.

```# let rec hbal_tree n =
if n = 0 then [Empty]
else if n = 1 then [Node('x', Empty, Empty)]
else
(* [add_trees_with left right trees] is defined in a question above. *)
let t1 = hbal_tree (n - 1)
and t2 = hbal_tree (n - 2) in
add_trees_with t1 t1 (add_trees_with t1 t2 (add_trees_with t2 t1 []));;val hbal_tree : int -> char binary_tree list = <fun>
```
```# let t = hbal_tree 3;;val t : char binary_tree list =
[Node ('x', Node ('x', Empty, Node ('x', Empty, Empty)),
Node ('x', Empty, Node ('x', Empty, Empty)));
Node ('x', Node (...), ...); ...]
# let x = 'x';;val x : char = 'x'
# List.mem (Node(x, Node(x, Node(x, Empty, Empty), Node(x, Empty, Empty)),
Node(x, Node(x, Empty, Empty), Node(x, Empty, Empty)))) t;;- : bool = true
# List.mem (Node(x, Node(x, Node(x, Empty, Empty), Node(x, Empty, Empty)),
Node(x, Node(x, Empty, Empty), Empty))) t;;- : bool = true
# List.length t = 15;;- : bool = true
```
Construct height-balanced binary trees with a given number of nodes

Consider a height-balanced binary tree of height `h`. What is the maximum number of nodes it can contain? Clearly, maxN = 2`h` - 1. However, what is the minimum number minN? This question is more difficult. Try to find a recursive statement and turn it into a function `minNodes` defined as follows: `minNodes h` returns the minimum number of nodes in a height-balanced binary tree of height `h`.

`# ;;`

On the other hand, we might ask: what is the maximum height H a height-balanced binary tree with N nodes can have? `maxHeight n` returns the maximum height of a height-balanced binary tree with `n` nodes.

`# ;;`

Now, we can attack the main problem: construct all the height-balanced binary trees with a given nuber of nodes. `hbal_tree_nodes n` returns a list of all height-balanced binary tree with `n` nodes.

`# ;;`

Find out how many height-balanced trees exist for ```n = 15```.

```# List.length (hbal_tree_nodes 15);;File "", line 1, characters 13-28:
Error: Unbound value hbal_tree_nodes
```
Count the leaves of a binary tree

A leaf is a node with no successors. Write a function `count_leaves` to count them.

```# let rec count_leaves = function
| Empty -> 0
| Node(_, Empty, Empty) -> 1
| Node(_, l, r) -> count_leaves l + count_leaves r;;val count_leaves : 'a binary_tree -> int = <fun>
```
```# count_leaves Empty = 0;;- : bool = true
# count_leaves example_tree = 3;;- : bool = true
```
Collect the leaves of a binary tree in a list

A leaf is a node with no successors. Write a function `leaves` to collect them in a list.

```# let rec leaves = function
| Empty -> []
| Node(c, Empty, Empty) -> [c]
| Node(_, l, r) -> leaves l @ leaves r;;val leaves : 'a binary_tree -> 'a list = <fun>
```
```# leaves Empty = [];;- : bool = true
# leaves example_tree = ['d'; 'e'; 'g'];;- : bool = true
```
Collect the internal nodes of a binary tree in a list

An internal node of a binary tree has either one or two non-empty successors. Write a function `internals` to collect them in a list.

```# let rec internals = function
| Empty | Node(_, Empty, Empty) -> []
| Node(c, l, r) -> internals l @ (c :: internals r);;val internals : 'a binary_tree -> 'a list = <fun>
```
```# internals (Node('a', Empty, Empty)) = [];;- : bool = true
# internals example_tree = ['b'; 'a'; 'c'; 'f'];;- : bool = true
```
Collect the nodes at a given level in a list.

A node of a binary tree is at level N if the path from the root to the node has length N-1. The root node is at level 1. Write a function `at_level t l` to collect all nodes of the tree `t` at level `l` in a list.

```# let rec at_level t l = match t with
| Empty -> []
| Node(c, left, right) ->
if l = 1 then [c]
else at_level left (l - 1) @ at_level right (l - 1);;val at_level : 'a binary_tree -> int -> 'a list = <fun>
```
```# at_level example_tree 2 = ['b'; 'c'];;- : bool = true
# at_level example_tree 2 = ['b'; 'c'];;- : bool = true
# at_level example_tree 5 = [];;- : bool = true
```

Using `at_level` it is easy to construct a function `levelorder` which creates the level-order sequence of the nodes. However, there are more efficient ways to do that.

Construct a complete binary tree

A complete binary tree with height H is defined as follows: The levels 1,2,3,...,H-1 contain the maximum number of nodes (i.e 2i-1 at the level i, note that we start counting the levels from 1 at the root). In level H, which may contain less than the maximum possible number of nodes, all the nodes are "left-adjusted". This means that in a levelorder tree traversal all internal nodes come first, the leaves come second, and empty successors (the nil's which are not really nodes!) come last.

Particularly, complete binary trees are used as data structures (or addressing schemes) for heaps.

We can assign an address number to each node in a complete binary tree by enumerating the nodes in levelorder, starting at the root with number 1. In doing so, we realize that for every node X with address A the following property holds: The address of X's left and right successors are 2*A and 2*A+1, respectively, supposed the successors do exist. This fact can be used to elegantly construct a complete binary tree structure. Write a function `is_complete_binary_tree` with the following specification: `is_complete_binary_tree n t` returns `true` iff `t` is a complete binary tree with `n` nodes.

`# ;;`
`# ;;`
Layout a binary tree (1)

As a preparation for drawing the tree, a layout algorithm is required to determine the position of each node in a rectangular grid. Several layout methods are conceivable, one of them is shown in the illustration below.

In this layout strategy, the position of a node v is obtained by the following two rules:

• x(v) is equal to the position of the node v in the inorder sequence;
• y(v) is equal to the depth of the node v in the tree.

In order to store the position of the nodes, we redefine the OCaml type representing a node (and its successors) as follows:

```# type 'a pos_binary_tree =
| E (* represents the empty tree *)
| N of 'a * int * int * 'a pos_binary_tree * 'a pos_binary_tree;;type 'a pos_binary_tree =
E
| N of 'a * int * int * 'a pos_binary_tree * 'a pos_binary_tree
```

`N(w,x,y,l,r)` represents a (non-empty) binary tree with root `w` "positioned" at `(x,y)`, and subtrees `l` and `r`. Write a function `layout_binary_tree` with the following specification: `layout_binary_tree t` returns the "positioned" binary tree obtained from the binary tree `t`.

`# ;;`
`# ;;`
Layout a binary tree (2)

An alternative layout method is depicted in the above illustration. Find out the rules and write the corresponding OCaml function.

Hint: On a given level, the horizontal distance between neighboring nodes is constant.

`# ;;`
`# ;;`
Layout a binary tree (3)

Yet another layout strategy is shown in the above illustration. The method yields a very compact layout while maintaining a certain symmetry in every node. Find out the rules and write the corresponding Prolog predicate.

Hint: Consider the horizontal distance between a node and its successor nodes. How tight can you pack together two subtrees to construct the combined binary tree? This is a difficult problem. Don't give up too early!

`# ;;`
`# ;;`

Which layout do you like most?

A string representation of binary trees

Somebody represents binary trees as strings of the following type (see example): `"a(b(d,e),c(,f(g,)))"`.

• Write an OCaml function which generates this string representation, if the tree is given as usual (as `Empty` or `Node(x,l,r)` term). Then write a function which does this inverse; i.e. given the string representation, construct the tree in the usual form. Finally, combine the two predicates in a single function `tree_string` which can be used in both directions.
• Write the same predicate `tree_string` using difference lists and a single predicate `tree_dlist` which does the conversion between a tree and a difference list in both directions.

For simplicity, suppose the information in the nodes is a single letter and there are no spaces in the string.

`# ;;`
`# ;;`
Preorder and inorder sequences of binary trees

We consider binary trees with nodes that are identified by single lower-case letters, as in the example of the previous problem.

1. Write functions `preorder` and `inorder` that construct the preorder and inorder sequence of a given binary tree, respectively. The results should be atoms, e.g. 'abdecfg' for the preorder sequence of the example in the previous problem.
2. Can you use `preorder` from problem part 1. in the reverse direction; i.e. given a preorder sequence, construct a corresponding tree? If not, make the necessary arrangements.
3. If both the preorder sequence and the inorder sequence of the nodes of a binary tree are given, then the tree is determined unambiguously. Write a function `pre_in_tree` that does the job.
4. Solve problems 1. to 3. using difference lists. Cool! Use the function `timeit` (defined in problem “Compare the two methods of calculating Euler's totient function.”) to compare the solutions.

What happens if the same character appears in more than one node. Try for instance `pre_in_tree "aba" "baa"`.

`# ;;`
`# ;;`
Dotstring representation of binary trees

We consider again binary trees with nodes that are identified by single lower-case letters, as in the example of problem “A string representation of binary trees”. Such a tree can be represented by the preorder sequence of its nodes in which dots (.) are inserted where an empty subtree (nil) is encountered during the tree traversal. For example, the tree shown in problem “A string representation of binary trees” is represented as 'abd..e..c.fg...'. First, try to establish a syntax (BNF or syntax diagrams) and then write a function `tree_dotstring` which does the conversion in both directions. Use difference lists.

`# ;;`
`# ;;`

## Multiway Trees

A multiway tree is composed of a root element and a (possibly empty) set of successors which are multiway trees themselves. A multiway tree is never empty. The set of successor trees is sometimes called a forest.

To represent multiway trees, we will use the following type which is a direct translation of the definition:

```# type 'a mult_tree = T of 'a * 'a mult_tree list;;type 'a mult_tree = T of 'a * 'a mult_tree list
```

The example tree depicted opposite is therefore represented by the following OCaml expression:

```# T('a', [T('f',[T('g',[])]); T('c',[]); T('b',[T('d',[]); T('e',[])])]);;- : char mult_tree =
T ('a',
[T ('f', [T ('g', [])]); T ('c', []);
T ('b', [T ('d', []); T ('e', ...); ...]); ...])
```
Count the nodes of a multiway tree
```# let rec count_nodes (T(_, sub)) =
List.fold_left (fun n t -> n + count_nodes t) 1 sub;;val count_nodes : 'a mult_tree -> int = <fun>
```
```# count_nodes (T('a', [T('f',[]) ])) = 2;;- : bool = true
```
Tree construction from a node string

We suppose that the nodes of a multiway tree contain single characters. In the depth-first order sequence of its nodes, a special character `^` has been inserted whenever, during the tree traversal, the move is a backtrack to the previous level.

By this rule, the tree in the figure opposite is represented as: `afg^^c^bd^e^^^`.

Write functions `string_of_tree : char mult_tree -> string` to construct the string representing the tree and `tree_of_string : string -> char mult_tree` to construct the tree when the string is given.

```# (* We could build the final string by string concatenation but this is
expensive due to the number of operations.  We use a buffer instead. *)
let rec add_string_of_tree buf (T(c, sub)) =
let string_of_tree t =
let buf = Buffer.create 128 in
Buffer.contents buf;;val add_string_of_tree : Buffer.t -> char mult_tree -> unit = <fun>
val string_of_tree : char mult_tree -> string = <fun>
# let rec tree_of_substring t s i len =
if i >= len || s.[i] = '^' then List.rev t, i + 1
else
let sub, j = tree_of_substring [] s (i+1) len in
tree_of_substring (T(s.[i], sub) ::t) s j len
let tree_of_string s =
match tree_of_substring [] s 0 (String.length s) with
| [t], _ -> t
| _ -> failwith "tree_of_string";;val tree_of_substring :
char mult_tree list -> string -> int -> int -> char mult_tree list * int =
<fun>
val tree_of_string : string -> char mult_tree = <fun>
```
```# let t =
T('a', [T('f',[T('g',[])]); T('c',[]); T('b',[T('d',[]); T('e',[])])]);;val t : char mult_tree =
T ('a',
[T ('f', [T ('g', [])]); T ('c', []);
T ('b', [T ('d', []); T ('e', ...); ...]); ...])
# string_of_tree t = "afg^^c^bd^e^^^";;- : bool = true
# tree_of_string "afg^^c^bd^e^^^" = t;;- : bool = true
```
Determine the internal path length of a tree

We define the internal path length of a multiway tree as the total sum of the path lengths from the root to all nodes of the tree. By this definition, the tree `t` in the figure of the previous problem has an internal path length of 9. Write a function `ipl tree` that returns the internal path length of `tree`.

```# let rec ipl_sub len (T(_, sub)) =
(* [len] is the distance of the current node to the root.  Add the
distance of all sub-nodes. *)
List.fold_left (fun sum t -> sum + ipl_sub (len + 1) t) len sub
let ipl t = ipl_sub 0 t;;val ipl_sub : int -> 'a mult_tree -> int = <fun>
val ipl : 'a mult_tree -> int = <fun>
```
```# ipl t = 9;;- : bool = true
```
Construct the bottom-up order sequence of the tree nodes

Write a function `bottom_up t` which constructs the bottom-up sequence of the nodes of the multiway tree `t`.

```# let rec prepend_bottom_up (T(c, sub)) l =
List.fold_right (fun t l -> prepend_bottom_up t l) sub (c :: l)
let bottom_up t = prepend_bottom_up t [];;val prepend_bottom_up : 'a mult_tree -> 'a list -> 'a list = <fun>
val bottom_up : 'a mult_tree -> 'a list = <fun>
```
```# bottom_up (T('a', [T('b', [])])) = ['b'; 'a'];;- : bool = true
# bottom_up t = ['g'; 'f'; 'c'; 'd'; 'e'; 'b'; 'a'];;- : bool = true
```
Lisp-like tree representation

There is a particular notation for multiway trees in Lisp. The following pictures show how multiway tree structures are represented in Lisp.

Note that in the "lispy" notation a node with successors (children) in the tree is always the first element in a list, followed by its children. The "lispy" representation of a multiway tree is a sequence of atoms and parentheses '(' and ')'. This is very close to the way trees are represented in OCaml, except that no constructor `T` is used. Write a function ```lispy : char mult_tree -> string``` that returns the lispy notation of the tree.

```# let rec add_lispy buf = function
| T(c, []) -> Buffer.add_char buf c
| T(c, sub) ->
List.iter (fun t -> Buffer.add_char buf ' '; add_lispy buf t) sub;
let lispy t =
let buf = Buffer.create 128 in
Buffer.contents buf;;val add_lispy : Buffer.t -> char mult_tree -> unit = <fun>
val lispy : char mult_tree -> string = <fun>
```
```# lispy (T('a', [])) = "a";;- : bool = true
# lispy (T('a', [T('b', [])])) = "(a b)";;- : bool = true
# lispy t = "(a (f g) c (b d e))";;- : bool = true
```

## Graphs

A graph is defined as a set of nodes and a set of edges, where each edge is a pair of different nodes.

There are several ways to represent graphs in OCaml.

• One method is to list all edges, an edge being a pair of nodes. In this form, the graph depicted opposite is represented as the following expression:
```# ['h', 'g';  'k', 'f';  'f', 'b';  'f', 'c';  'c', 'b'];;- : (char * char) list =
[('h', 'g'); ('k', 'f'); ('f', 'b'); ('f', 'c'); ('c', 'b')]
```
We call this edge-clause form. Obviously, isolated nodes cannot be represented.
• Another method is to represent the whole graph as one data object. According to the definition of the graph as a pair of two sets (nodes and edges), we may use the following OCaml type:
```# type 'a graph_term = { nodes : 'a list;  edges : ('a * 'a) list };;type 'a graph_term = { nodes : 'a list; edges : ('a * 'a) list; }
```
Then, the above example graph is represented by:
```# let example_graph =
{ nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k'];
edges = ['h', 'g';  'k', 'f';  'f', 'b';  'f', 'c';  'c', 'b'] };;val example_graph : char graph_term =
{nodes = ['b'; 'c'; 'd'; 'f'; 'g'; 'h'; 'k'];
edges = [('h', 'g'); ('k', 'f'); ('f', 'b'); (...); ...]}
```

We call this graph-term form. Note, that the lists are kept sorted, they are really sets, without duplicated elements. Each edge appears only once in the edge list; i.e. an edge from a node x to another node y is represented as `(x,y)`, the couple `(y,x)` is not present. The graph-term form is our default representation. You may want to define a similar type using sets instead of lists.

• A third representation method is to associate with each node the set of nodes that are adjacent to that node. We call this the adjacency-list form. In our example:
`# ;;`
• The representations we introduced so far well suited for automated processing, but their syntax is not very user-friendly. Typing the terms by hand is cumbersome and error-prone. We can define a more compact and "human-friendly" notation as follows: A graph (with char labelled nodes) is represented by a string of atoms and terms of the type X-Y. The atoms stand for isolated nodes, the X-Y terms describe edges. If an X appears as an endpoint of an edge, it is automatically defined as a node. Our example could be written as:
```# "b-c f-c g-h d f-b k-f h-g";;- : string = "b-c f-c g-h d f-b k-f h-g"
```

We call this the human-friendly form. As the example shows, the list does not have to be sorted and may even contain the same edge multiple times. Notice the isolated node `d`.

Conversions

Write functions to convert between the different graph representations. With these functions, all representations are equivalent; i.e. for the following problems you can always pick freely the most convenient form. This problem is not particularly difficult, but it's a lot of work to deal with all the special cases.

`# ;;`
`# ;;`
Path from one node to another one

Write a function `paths g a b` that returns all acyclic path `p` from node `a` to node `b ≠ a` in the graph `g`. The function should return the list of all paths via backtracking.

```# (* The datastructures used here are far from the most efficient ones
but allow for a straightforward implementation. *)
(* Returns all neighbors satisfying the condition. *)
let neighbors g a cond =
let edge l (b,c) = if b = a && cond c then c :: l
else if c = a && cond b then b :: l
else l in
List.fold_left edge [] g.edges
let rec list_path g a to_b = match to_b with
| [] -> assert false (* [to_b] contains the path to [b]. *)
| a' :: _ ->
if a' = a then [to_b]
else
let n = neighbors g a' (fun c -> not(List.mem c to_b)) in
List.concat(List.map (fun c -> list_path g a (c :: to_b)) n)

let paths g a b =
assert(a <> b);
list_path g a [b];;val neighbors : 'a graph_term -> 'a -> ('a -> bool) -> 'a list = <fun>
val list_path : 'a graph_term -> 'a -> 'a list -> 'a list list = <fun>
val paths : 'a graph_term -> 'a -> 'a -> 'a list list = <fun>
```
```# paths example_graph 'f' 'b' = [['f'; 'c'; 'b']; ['f'; 'b']];;- : bool = true
```
Cycle from a given node

Write a functions `cycle g a` that returns a closed path (cycle) `p` starting at a given node `a` in the graph `g`. The predicate should return the list of all cycles via backtracking.

```# let cycles g a =
let n = neighbors g a (fun _ -> true) in
let p = List.concat(List.map (fun c -> list_path g a [c]) n) in
List.map (fun p -> p @ [a]) p;;val cycles : 'a graph_term -> 'a -> 'a list list = <fun>
```
```# cycles example_graph 'f'
= [['f'; 'b'; 'c'; 'f']; ['f'; 'c'; 'f']; ['f'; 'c'; 'b'; 'f'];
['f'; 'b'; 'f']; ['f'; 'k'; 'f']];;- : bool = true
```
Construct all spanning trees

Write a function `s_tree g` to construct (by backtracking) all spanning trees of a given graph `g`. With this predicate, find out how many spanning trees there are for the graph depicted to the left. The data of this example graph can be found in the test below. When you have a correct solution for the `s_tree` function, use it to define two other useful functions: `is_tree graph` and ```is_connected Graph```. Both are five-minutes tasks!

`# ;;`
```# let g = { nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h'];
edges = [('a', 'b'); ('a', 'd'); ('b', 'c'); ('b', 'e');
('c', 'e'); ('d', 'e'); ('d', 'f'); ('d', 'g');
('e', 'h'); ('f', 'g'); ('g', 'h')] };;val g : char graph_term =
{nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h'];
edges = [('a', 'b'); ('a', 'd'); ('b', 'c'); ...]}
```
Construct the minimal spanning tree

Write a function `ms_tree graph` to construct the minimal spanning tree of a given labelled graph. A labelled graph will be represented as follows:

```# type ('a, 'b) labeled_graph = { nodes : 'a list;
edges : ('a * 'a * 'b) list };;type ('a, 'b) labeled_graph = {
nodes : 'a list;
edges : ('a * 'a * 'b) list;
}
```

(Beware that from now on `nodes` and `edges` mask the previous fields of the same name.)

Hint: Use the algorithm of Prim. A small modification of the solution of P83 does the trick. The data of the example graph to the right can be found below.

`# ;;`
```# let g = { nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h'];
edges = [('a', 'b', 5); ('a', 'd', 3); ('b', 'c', 2); ('b', 'e', 4);
('c', 'e', 6); ('d', 'e', 7); ('d', 'f', 4); ('d', 'g', 3);
('e', 'h', 5); ('f', 'g', 4); ('g', 'h', 1)] };;val g : (char, int) labeled_graph =
{nodes = ['a'; 'b'; 'c'; 'd'; 'e'; 'f'; 'g'; 'h'];
edges = [('a', 'b', 5); ('a', 'd', 3); (...); ...]}
```
Graph isomorphism

Two graphs G1(N1,E1) and G2(N2,E2) are isomorphic if there is a bijection f: N1 → N2 such that for any nodes X,Y of N1, X and Y are adjacent if and only if f(X) and f(Y) are adjacent.

Write a function that determines whether two graphs are isomorphic. Hint: Use an open-ended list to represent the function f.

`# ;;`
`# ;;`
Node degree and graph coloration
• Write a function `degree graph node` that determines the degree of a given node.
• Write a function that generates a list of all nodes of a graph sorted according to decreasing degree.
• Use Welsh-Powell's algorithm to paint the nodes of a graph in such a way that adjacent nodes have different colors.
`# ;;`
`# ;;`
Depth-first order graph traversal

Write a function that generates a depth-first order graph traversal sequence. The starting point should be specified, and the output should be a list of nodes that are reachable from this starting point (in depth-first order).

`# ;;`
`# ;;`
Connected components

Write a predicate that splits a graph into its connected components.

`# ;;`
`# ;;`
Bipartite graphs

Write a predicate that finds out whether a given graph is bipartite.

`# ;;`
`# ;;`
Generate K-regular simple graphs with N nodes

In a K-regular graph all nodes have a degree of K; i.e. the number of edges incident in each node is K. How many (non-isomorphic!) 3-regular graphs with 6 nodes are there?

`# ;;`
`# ;;`

## Miscellaneous Problems

Eight queens problem

This is a classical problem in computer science. The objective is to place eight queens on a chessboard so that no two queens are attacking each other; i.e., no two queens are in the same row, the same column, or on the same diagonal.

Hint: Represent the positions of the queens as a list of numbers 1..N. Example: `[4;2;7;3;6;8;5;1]` means that the queen in the first column is in row 4, the queen in the second column is in row 2, etc. Use the generate-and-test paradigm.

`# ;;`
`# ;;`
Knight's tour

Another famous problem is this one: How can a knight jump on an N×N chessboard in such a way that it visits every square exactly once?

Hints: Represent the squares by pairs of their coordinates `(x,y)`, where both `x` and `y` are integers between 1 and N. Define the function ```jump n (x,y)``` that returns all coordinates `(u,v)` to which a knight can jump from `(x,y)` to on a `n`×`n` chessboard. And finally, represent the solution of our problem as a list knight positions (the knight's tour).

`# ;;`
`# ;;`
Von Koch's conjecture

Several years ago I met a mathematician who was intrigued by a problem for which he didn't know a solution. His name was Von Koch, and I don't know whether the problem has been solved since.

Anyway, the puzzle goes like this: Given a tree with N nodes (and hence N-1 edges). Find a way to enumerate the nodes from 1 to N and, accordingly, the edges from 1 to N-1 in such a way, that for each edge K the difference of its node numbers equals to K. The conjecture is that this is always possible.

For small trees the problem is easy to solve by hand. However, for larger trees, and 14 is already very large, it is extremely difficult to find a solution. And remember, we don't know for sure whether there is always a solution!

Write a function that calculates a numbering scheme for a given tree. What is the solution for the larger tree pictured above?

`# ;;`
`# ;;`
An arithmetic puzzle

Given a list of integer numbers, find a correct way of inserting arithmetic signs (operators) such that the result is a correct equation. Example: With the list of numbers `[2;3;5;7;11]` we can form the equations 2-3+5+7 = 11 or 2 = (3*5+7)/11 (and ten others!).

`# ;;`
`# ;;`
English number words

On financial documents, like cheques, numbers must sometimes be written in full words. Example: 175 must be written as one-seven-five. Write a function `full_words` to print (non-negative) integer numbers in full words.

`# ;;`
`# ;;`
Syntax checker

In a certain programming language (Ada) identifiers are defined by the syntax diagram (railroad chart) opposite. Transform the syntax diagram into a system of syntax diagrams which do not contain loops; i.e. which are purely recursive. Using these modified diagrams, write a function `identifier : string -> bool` that can check whether or not a given string is a legal identifier.

`# ;;`
`# ;;`
Sudoku

Sudoku puzzles go like this:

```   Problem statement                 Solution

.  .  4 | 8  .  . | .  1  7	     9  3  4 | 8  2  5 | 6  1  7
|         |                      |         |
6  7  . | 9  .  . | .  .  .	     6  7  2 | 9  1  4 | 8  5  3
|         |                      |         |
5  .  8 | .  3  . | .  .  4      5  1  8 | 6  3  7 | 9  2  4
--------+---------+--------      --------+---------+--------
3  .  . | 7  4  . | 1  .  .      3  2  5 | 7  4  8 | 1  6  9
|         |                      |         |
.  6  9 | .  .  . | 7  8  .      4  6  9 | 1  5  3 | 7  8  2
|         |                      |         |
.  .  1 | .  6  9 | .  .  5      7  8  1 | 2  6  9 | 4  3  5
--------+---------+--------      --------+---------+--------
1  .  . | .  8  . | 3  .  6	     1  9  7 | 5  8  2 | 3  4  6
|         |                      |         |
.  .  . | .  .  6 | .  9  1	     8  5  3 | 4  7  6 | 2  9  1
|         |                      |         |
2  4  . | .  .  1 | 5  .  .      2  4  6 | 3  9  1 | 5  7  8
```

Every spot in the puzzle belongs to a (horizontal) row and a (vertical) column, as well as to one single 3x3 square (which we call "square" for short). At the beginning, some of the spots carry a single-digit number between 1 and 9. The problem is to fill the missing spots with digits in such a way that every number between 1 and 9 appears exactly once in each row, in each column, and in each square.

`# ;;`
`# ;;`
Nonograms

Around 1994, a certain kind of puzzles was very popular in England. The "Sunday Telegraph" newspaper wrote: "Nonograms are puzzles from Japan and are currently published each week only in The Sunday Telegraph. Simply use your logic and skill to complete the grid and reveal a picture or diagram." As an OCaml programmer, you are in a better situation: you can have your computer do the work!

The puzzle goes like this: Essentially, each row and column of a rectangular bitmap is annotated with the respective lengths of its distinct strings of occupied cells. The person who solves the puzzle must complete the bitmap given only these lengths.

```          Problem statement:          Solution:

|_|_|_|_|_|_|_|_| 3         |_|X|X|X|_|_|_|_| 3
|_|_|_|_|_|_|_|_| 2 1       |X|X|_|X|_|_|_|_| 2 1
|_|_|_|_|_|_|_|_| 3 2       |_|X|X|X|_|_|X|X| 3 2
|_|_|_|_|_|_|_|_| 2 2       |_|_|X|X|_|_|X|X| 2 2
|_|_|_|_|_|_|_|_| 6         |_|_|X|X|X|X|X|X| 6
|_|_|_|_|_|_|_|_| 1 5       |X|_|X|X|X|X|X|_| 1 5
|_|_|_|_|_|_|_|_| 6         |X|X|X|X|X|X|_|_| 6
|_|_|_|_|_|_|_|_| 1         |_|_|_|_|X|_|_|_| 1
|_|_|_|_|_|_|_|_| 2         |_|_|_|X|X|_|_|_| 2
1 3 1 7 5 3 4 3             1 3 1 7 5 3 4 3
2 1 5 1                     2 1 5 1
```

For the example above, the problem can be stated as the two lists `[[3];[2;1];[3;2];[2;2];[6];[1;5];[6];[1];[2]]` and `[[1;2];[3;1];[1;5];[7;1];[5];[3];[4];[3]]` which give the "solid" lengths of the rows and columns, top-to-bottom and left-to-right, respectively. Published puzzles are larger than this example, e.g. 25×20, and apparently always have unique solutions.

`# ;;`
`# ;;`
Crossword puzzle

Given an empty (or almost empty) framework of a crossword puzzle and a set of words. The problem is to place the words into the framework.

The particular crossword puzzle is specified in a text file which first lists the words (one word per line) in an arbitrary order. Then, after an empty line, the crossword framework is defined. In this framework specification, an empty character location is represented by a dot (.). In order to make the solution easier, character locations can also contain predefined character values. The puzzle above is defined in the file p7_09a.dat, other examples are p7_09b.dat and p7_09d.dat. There is also an example of a puzzle (p7_09c.dat) which does not have a solution.

Words are strings (character lists) of at least two characters. A horizontal or vertical sequence of character places in the crossword puzzle framework is called a site. Our problem is to find a compatible way of placing words onto sites.

Hints:

1. The problem is not easy. You will need some time to thoroughly understand it. So, don't give up too early! And remember that the objective is a clean solution, not just a quick-and-dirty hack!
2. For efficiency reasons it is important, at least for larger puzzles, to sort the words and the sites in a particular order.
`# ;;`
`# ;;`