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== Complexity ==
== Complexity ==


'''Statement:''' Log(n)
'''Statement:''' The asymptotic complexity is in <math>\mathcal{O}(\log n)</math>.


'''Proff:''' Obvious.
'''Proof:''' follows immediately from the variant.

Latest revision as of 06:29, 27 April 2016

Binary search

Algorithmic problem: Finding an element in a sorted array

Type of algorithm: recursion

Abstract view

The subroutine does nothing but calling another, recursive subroutine with the same output and the following input: array [math]\displaystyle{ A }[/math], element [math]\displaystyle{ s }[/math], comparison [math]\displaystyle{ cmp }[/math] and, in addition, two indices of [math]\displaystyle{ A }[/math], [math]\displaystyle{ \ell }[/math] and [math]\displaystyle{ r }[/math], such that [math]\displaystyle{ \ell\lt r }[/math]. In the original call to this recursive subroutine, [math]\displaystyle{ \ell }[/math] is the first and [math]\displaystyle{ r }[/math] the last index of [math]\displaystyle{ A }[/math].

Invariant: For every recursive call: If [math]\displaystyle{ s }[/math] is present at least once in [math]\displaystyle{ A }[/math], then all indices of [math]\displaystyle{ A }[/math] where [math]\displaystyle{ s }[/math] is present are in the interval [math]\displaystyle{ [\ell,\ldots,r] }[/math].

Variant: The value [math]\displaystyle{ r-\ell }[/math] is roughly halved in every descent in the recursion tree.

Break condition: Either an occurrence of [math]\displaystyle{ s }[/math] is found or it is [math]\displaystyle{ \ell\gt r }[/math]

Induction basis

The invariant is trivially fulfilled because [math]\displaystyle{ [\ell,\ldots,r] }[/math] is the entire index range of [math]\displaystyle{ A }[/math].

Induction step

  1. Set [math]\displaystyle{ m:=(\ell+r)/2 }[/math].
  2. If [math]\displaystyle{ A[m]=s }[/math] terminate and return [math]\displaystyle{ m }[/math].
  3. Otherwise, if [math]\displaystyle{ s }[/math] precedes [math]\displaystyle{ A[m] }[/math] with respect to [math]\displaystyle{ cmp }[/math], apply a recursive call with [math]\displaystyle{ A }[/math], [math]\displaystyle{ s }[/math], [math]\displaystyle{ cmp }[/math], [math]\displaystyle{ \ell }[/math] and[math]\displaystyle{ m-1 }[/math].
  4. Otherwise, apply a recursive call with [math]\displaystyle{ A }[/math], [math]\displaystyle{ s }[/math], [math]\displaystyle{ cmp }[/math], [math]\displaystyle{ m+1 }[/math] and [math]\displaystyle{ r }[/math].

Correctness: obvious.

Complexity

Statement: The asymptotic complexity is in [math]\displaystyle{ \mathcal{O}(\log n) }[/math].

Proof: follows immediately from the variant.