Pivot partitioning by scanning: Difference between revisions

From Algowiki
Jump to navigation Jump to search
No edit summary
No edit summary
Line 12: Line 12:
'''Invariant:'''  
'''Invariant:'''  
# The values of <math>m_1</math> and <math>m_2</math> only depend on the input sequence and are, hence, constant throughout the loop. More specifically, in  there are
# The values of <math>m_1</math> and <math>m_2</math> only depend on the input sequence and are, hence, constant throughout the loop. More specifically, in  there are
  ## <math>m_1 - 1</math> elements less than <math>p</math>,
  ## <math>m_1-1</math> elements less than <math>p</math>,
  ## <math>m_1 - m_2</math> elements equal to <math>p</math>, and
  ## <math>m_1-m_2</math> elements equal to <math>p</math>, and
  ## <math>n - m_2</math>greater than <math>p</math>.
  ## <math>n - m_2</math>greater than <math>p</math>.
# Before and after each iteration, it is:
# Before and after each iteration, it is:

Revision as of 15:19, 13 January 2015

Algorithmic problem: Pivot partioning by scanning

Prerequisites: None

Type of algorithm: loop

Auxiliary data: Five index pointers, [math]\displaystyle{ m_1,m_2,i_1,i_2,i_3 \in \mathbb N. }[/math]


Abstract View

Invariant:

  1. The values of [math]\displaystyle{ m_1 }[/math] and [math]\displaystyle{ m_2 }[/math] only depend on the input sequence and are, hence, constant throughout the loop. More specifically, in there are
## [math]\displaystyle{ m_1-1 }[/math] elements less than [math]\displaystyle{ p }[/math],
## [math]\displaystyle{ m_1-m_2 }[/math] elements equal to [math]\displaystyle{ p }[/math], and
## [math]\displaystyle{ n - m_2 }[/math]greater than [math]\displaystyle{ p }[/math].
  1. Before and after each iteration, it is:
## [math]\displaystyle{ 1 &le i_2 &le m_1 &le m_2 &lei3 &le i_3 &le n+1\lt math\gt ;
 ## \lt math\gt S[j] }[/math]for ; if , it is ;
for ; if , it is ;
for ; if , it is .

Variant: At least one of , and is increased by at least , none of them is decreased. Break condition: It is , and .

Varian: [math]\displaystyle{ i }[/math] increases by [math]\displaystyle{ 1 }[/math].

Break condition: It is [math]\displaystyle{ i=n }[/math]

Induction basis

Abstract view: [math]\displaystyle{ \forall v,w \in V }[/math] we set

  1. [math]\displaystyle{ M(v,v):=0, \forall v \in V }[/math]
  2. [math]\displaystyle{ M(v,w):=l(v,w), (v,w) \in A }[/math]
  3. [math]\displaystyle{ M(v,w):= +\infty }[/math], otherwise

Implementation: Obvious.

Proof: Nothing to show.

Induction step

Abstract view:

Implementation:

Correctness: Let [math]\displaystyle{ p }[/math] denote a shoortest [math]\displaystyle{ (v,w }[/math]-path subject to the constraint that all internal nodes are taken from [math]\displaystyle{ \{u_1, \ldots, u_n \} }[/math].

  1. If [math]\displaystyle{ p }[/math] does not contain [math]\displaystyle{ u_i }[/math], [math]\displaystyle{ p }[/math] is even a shortest [math]\displaystyle{ (v,w) }[/math]-path such that all internal nodes are taken from [math]\displaystyle{ \{u_1, \ldots , u_{i-1} \} }[/math]. In this case, the induction hypothesis implies that the value of [math]\displaystyle{ M(v,w) }[/math] immediately before the i-th iteration equals the length of [math]\displaystyle{ p }[/math]. Clearly, the i-th iteration does not change the value of [math]\displaystyle{ M(v,w) }[/math] in this case.
  2. On the other hand, suppose [math]\displaystyle{ p }[/math] does contain [math]\displaystyle{ u_i }[/math]. Due to the prefix property, the segment of [math]\displaystyle{ p }[/math] from [math]\displaystyle{ v }[/math] to [math]\displaystyle{ u_i }[/math] and from [math]\displaystyle{ u_i }[/math] to [math]\displaystyle{ w }[/math] are a shortest [math]\displaystyle{ (v,u_i) }[/math]-path and a shortest [math]\displaystyle{ (u_i,w) }[/math]-path, respectively, subject to the constraint that all internal nodes are from [math]\displaystyle{ \{u_1, \ldots , u_{i-1} \} }[/math]. The induction hypothesis implies that tehese lengths equal the values [math]\displaystyle{ M(v,u_i) }[/math] and [math]\displaystyle{ M(u_i, w) }[/math], respectively.

Complexity

Statement: [math]\displaystyle{ \mathcal{O}(n^3) }[/math]

Proof: The overall loop has [math]\displaystyle{ n }[/math] iterations. In each iteration, we update all [math]\displaystyle{ n^2 }[/math] matrix entities. Each update requires a constant number of steps.