Bellman-Ford

General information

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.