Bellman-Ford: Difference between revisions

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# A [[Paths|distance-valued]] <math>(n \times n)</math> matrix <math>M</math>, where <math>n=|V|</math>. The eventual contents of <math>M</math> will be returned as the result of the algorithm.
# A [[Paths|distance-valued]] <math>(n \times n)</math> matrix <math>M</math>, where <math>n=|V|</math>. The eventual contents of <math>M</math> will be returned as the result of the algorithm.
# A [[Paths|distance-valued]] <math>(n \times n)</math> matrix <math>L</math>, where <math>n=|V|</math>. This matrix represents the graph and the arc lengths and will not be changed throughout the algorithm. It is <math>L=M^1</math> in the terminology of Section "Powers of distance matrices" on [[Paths#Powers of distance matrices|this]] page.
# A [[Paths|distance-valued]] <math>(n \times n)</math> matrix <math>L</math>, where <math>n=|V|</math>. This matrix represents the graph and the arc lengths and will not be changed throughout the algorithm. It is <math>L=M^1</math> in the terminology of Section "Powers of distance matrices" on [[Paths#Powers of distance matrices|this]] page.
==Abstract view==
'''Invariant:''' After <math>i \ge 0</math> iterations, <math>M^i (v,w)</math> contains the length of a shortest <math>(v,w)</math>-path with at most <math>i+1</math> arcs (for all <math>v,w \in V</math>).
'''Variant:''' <math>i</math> increases by 1.
'''Break condition:''' <math>i=n-1</math>.
==Induction basis==
'''Abstract view:''' For all <math>v,w \in V</math>, we set
# <math>M^0 (v,v):= L(v,v) := 0</math>;
# <math>M^0 (v,w):= L(v,w) := \ell (v,w)</math> if <math>(v,w)\in A</math>;
# <math>M^0 (v,w):= L(v,w) := +\infty</math>, if <math>v\neq w</math> and <math>(v,w)\notin A</math>.
==Induction step==
'''Abstract view:''' For all <math>v,w \in V</math>, we set <math>M^{i+1} (v,w) := \min \{ M^i (v,w) + L(u,w) \mid u \in V \}</math>.
(Note that <math>M^i (v,w) + L(u,w) = M^{i+1} (v,w)</math> is one of the arguments over which the minimum is taken, so the right-hand side is identical to <math>\min \{ M^i (v,w), \min \{ M^i (v,u)+ L(u,w) \mid u \in V \} \}</math>.)
'''Implementation:''' Obvious.
'''Correctness:''' Follows from the explanations of [[Paths#Powers of distance matrices|powers of the distance matrix]] and from the fact that a shortest path cannot have more than <math>|V|-1</math> arcs.
==Complexity==
'''Statement:''' The asymptotic complexity is <math>\Theta (n^4)</math> in the best and worst case.
'''Proof:''' The main loop terminates after <math>n-1</math> iterations. In each iteration of this loop, we update all <math>n^2</math> matrix entries, and computing one update value requires <math>\Theta (n)</math> steps.

Revision as of 10:05, 6 October 2014

Algorithmic problem: All pairs shortest paths Prerequisites: Type of algorithm: loop Auxiliary data:

  1. A distance-valued [math]\displaystyle{ (n \times n) }[/math] matrix [math]\displaystyle{ M }[/math], where [math]\displaystyle{ n=|V| }[/math]. The eventual contents of [math]\displaystyle{ M }[/math] will be returned as the result of the algorithm.
  2. A distance-valued [math]\displaystyle{ (n \times n) }[/math] matrix [math]\displaystyle{ L }[/math], where [math]\displaystyle{ n=|V| }[/math]. This matrix represents the graph and the arc lengths and will not be changed throughout the algorithm. It is [math]\displaystyle{ L=M^1 }[/math] in the terminology of Section "Powers of distance matrices" on this page.

Abstract view

Invariant: After [math]\displaystyle{ i \ge 0 }[/math] iterations, [math]\displaystyle{ M^i (v,w) }[/math] contains the length of a shortest [math]\displaystyle{ (v,w) }[/math]-path with at most [math]\displaystyle{ i+1 }[/math] arcs (for all [math]\displaystyle{ v,w \in V }[/math]).

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

Break condition: [math]\displaystyle{ i=n-1 }[/math].

Induction basis

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

  1. [math]\displaystyle{ M^0 (v,v):= L(v,v) := 0 }[/math];
  2. [math]\displaystyle{ M^0 (v,w):= L(v,w) := \ell (v,w) }[/math] if [math]\displaystyle{ (v,w)\in A }[/math];
  3. [math]\displaystyle{ M^0 (v,w):= L(v,w) := +\infty }[/math], if [math]\displaystyle{ v\neq w }[/math] and [math]\displaystyle{ (v,w)\notin A }[/math].

Induction step

Abstract view: For all [math]\displaystyle{ v,w \in V }[/math], we set [math]\displaystyle{ M^{i+1} (v,w) := \min \{ M^i (v,w) + L(u,w) \mid u \in V \} }[/math].

(Note that [math]\displaystyle{ M^i (v,w) + L(u,w) = M^{i+1} (v,w) }[/math] is one of the arguments over which the minimum is taken, so the right-hand side is identical to [math]\displaystyle{ \min \{ M^i (v,w), \min \{ M^i (v,u)+ L(u,w) \mid u \in V \} \} }[/math].)

Implementation: Obvious.

Correctness: Follows from the explanations of powers of the distance matrix and from the fact that a shortest path cannot have more than [math]\displaystyle{ |V|-1 }[/math] arcs.

Complexity

Statement: The asymptotic complexity is [math]\displaystyle{ \Theta (n^4) }[/math] in the best and worst case.

Proof: The main loop terminates after [math]\displaystyle{ n-1 }[/math] iterations. In each iteration of this loop, we update all [math]\displaystyle{ n^2 }[/math] matrix entries, and computing one update value requires [math]\displaystyle{ \Theta (n) }[/math] steps.