## Binary Sort

Binary sort is a variant of insertion sort where binary search is used to find the insertion point.

Auteurs: Jean-Christophe Filliâtre

Catégories: Array Data Structure / Sorting Algorithms / Algorithms

Outils: Why3

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Binary sort

Binary sort is a variant of insertion sort where binary search is used to find the insertion point. This lowers the number of comparisons (from N^2 to N log(N)) and thus is useful when comparisons are costly.

For instance, Binary sort is used as an ingredient in Java 8's TimSort implementation (which is the library sort for Object[]).

Author: Jean-Christophe Filliâtre (CNRS)

```module BinarySort

use int.Int
use int.ComputerDivision
use ref.Ref
use array.Array
use array.ArrayPermut

let lemma occ_shift (m1 m2: int -> 'a) (mid k: int) (x: 'a) : unit
requires { 0 <= mid <= k }
requires { forall i. mid < i <= k -> m2 i = m1 (i - 1) }
requires { m2 mid = m1 k }
ensures  { M.Occ.occ x m1 mid (k+1) = M.Occ.occ x m2 mid (k+1) }
= for i = mid to k - 1 do
invariant { M.Occ.occ x m1 mid i = M.Occ.occ x m2 (mid+1) (i+1) }
()
done;
assert { M.Occ.occ (m1 k) m1 mid (k+1) =
1 + M.Occ.occ (m1 k) m1 mid k };
assert { M.Occ.occ (m1 k) m2 mid (k+1) =
1 + M.Occ.occ (m1 k) m2 (mid+1) (k+1) };
assert { M.Occ.occ x m1 mid (k+1) = M.Occ.occ x m2 mid (k+1)
by x = m1 k \/ x <> m1 k }

let binary_sort (a: array int) : unit
ensures { forall i j. 0 <= i <= j < length a -> a[i] <= a[j] }
ensures { permut_sub (old a) a 0 (length a) }
=
for k = 1 to length a - 1 do
(* a[0..k-1) is sorted; insert a[k] *)
invariant { forall i j. 0 <= i <= j < k -> a[i] <= a[j] }
invariant { permut_sub (old a) a  0 (length a) }
let v = a[k] in
let left = ref 0 in
let right = ref k in
while !left < !right do
invariant { 0 <= !left <= !right <= k }
invariant { forall i. 0      <= i < !left -> a[i] <= v        }
invariant { forall i. !right <= i < k     ->         v < a[i] }
variant   { !right - !left }
let mid = !left + div (!right - !left) 2 in
if v < a[mid] then right := mid else left := mid + 1
done;
(* !left is the place where to insert value v
so we move a[!left..k) one position to the right *)
label L in
self_blit a !left (!left + 1) (k - !left);
a[!left] <- v;
assert { permut_sub (a at L) a !left (k + 1) };
assert { permut_sub (a at L) a 0 (length a) };
done

end
```