# 1.6. Exercises¶

## 1.6.1. Exercise 1¶

Give an example of a real-life application that requires an implementation of and algorithm (or several algorithms) as its part, and discuss the algorithms involved: how do they interact, what are they inputs and outputs.

## 1.6.2. Exercise 2¶

Programming in OCaml in Emacs is much more pleasant with instant navigation, auto-completion and type information available. Install all the necessary sofwtare following the provided Software Prerequisites.

## 1.6.3. Exercise 3¶

What is the termination measure of walk within find_min? Define it as a function f : 'a list -> int -> int and change the implementation of walk annotating it with outputs to check that the measure indeed decreases.

• Hint: use OCaml’s Printf.printf utility to output results of the termination measure mid-execution.

## 1.6.4. Exercise 4¶

• Implement the function find_min2, similar to find_min (also using the auxiliary walk, but without relying on any other auxiliary functions, e.g., sorting) that finds not the minimal element, but the second minimal element. For instance, it should bive the following output on a list [2; 6; 78; 2; 5; 3; 1]:

# find_min2  [2; 6; 78; 2; 5; 3; 1];;
- : int option = Some 2


Hint: walk is easier to implement if it takes both the “absolute” minimum m1 and the second minimum m2, i.e., has the type int list -> int -> int -> int.

• Write its specification (a relation between its input/output).

Hint: the following definition might be helpful:

let is_min2 ls m1 m2 =
m1 < m2 &&
List.for_all (fun e -> e == m1 || m2 <= e) ls &&
List.mem m2 ls

• Write the precondition for walk and annotate the function with the assertions, enforcing the pre- and postconditions.

Hint: you might want to start from devising the second disjunct of find_min2_walk_pre ls xs m1 m2 to state that “a list has an element that is its second minimum, positioned appropriately with respect to m1 and m2”.

• Test your annotated function find_min2_with_invariant.

## 1.6.5. Exercise 5¶

Give an example of an interesting non-tail recursive function in OCaml and show, if possible, an equivalent tail-recursive function.

## 1.6.6. Exercise 6¶

Implement an imperative version of the function find_min2 from Exercise 4, annotate its with loop invariants and run the tests.

## 1.6.7. Exercise 7¶

Implement a version of an insertion sort that sorts the elements in the descending order and test it.

## 1.6.8. Exercise 8¶

It is possible to implement (quite unelegantly and not very efficiently) the insertion sort on lists, so it would be tail-recursive. For this, we will have to rewrite it, so insert would use the boolean flug run in order to indicate whether the insertion has already taken place, or the iteration should continue:

let insert_sort_tail ls =
let rec walk xs prefix =
match xs with
| [] -> prefix
| h :: t ->
let rec insert elem acc remaining run =
if not run then acc
else match remaining with
| [] -> acc @ [elem]
| h :: t as l ->
if h < elem
then
let run' = true in
let acc' = acc @ [h] in
insert elem acc' t run'
else
let run' = false in
let acc' = acc @ (elem :: l) in
insert elem acc' t run'
in

let acc' = insert h [] prefix true in
walk t acc'
in
walk ls []

• Define the invariants for auxiliary functions:

let insert_inv prefix elem acc remaining run = (* ... *)
let insert_sort_tail_walk_inv ls xs acc = (* ... *)


Annotate the implementation above with them and test it.

• Transform insert_sort_tail into an imperative version, which uses (nested) loops instead of recursion.