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The Bellman-Ford algorithm solves the single-source shortest path problem in a finite directed graph with finite real edge weights, allowing negative weights. It repeatedly applies edge relaxation, so an improved tentative distance can improve its neighbours on a later visit. This makes it a label-correcting shortest-path algorithm.^[ref] If no negative-cost directed cycle is reachable from the source, at most $|V|-1$ complete passes suffice for all distances; a further check detects a reachable negative-cost directed cycle.^[ref]

There are two orientations to keep distinct. The usual single-source problem asks for distances from $s$, extending routes along outgoing directed edges. The Bellman inequalities above instead concern distances to a sink $r$, prepending incoming directed edges to known routes. The All-to-one Bellman-Ford recurrence below preserves that convention. Reversing every directed edge, with its weight unchanged, converts one problem into the other.

Here is an original single-source example with source $S$, vertex set $V=\{S,A,B,C,T\}$, and precisely these directed edges:The negative edge $A\to B$ is harmless: this directed graph has no directed cycle. Its best route to $T$ is $S\to A\to B\to C\to T$, of cost $4-2+3+2=7$. The other routes to $T$ cost $12$ through $S,A,T$ and $10$ through $S,B,C,T$.

Initialize $d(S)=0$ and the other tentative distances to $\infty$. In each pass apply single-source edge relaxation, $d(v)\leftarrow \min \{d(v),d(u)+w_{uv}\}$, scanning in the order $A\to T$, $C\to T$, $B\to C$, $A\to B$, $S\to B$, $S\to A$. Skip a candidate with $d(u)=\infty$. The figure records labels after each complete pass and uses current labels immediately: it is an in-place trace. In pass 2, $B\to C$ still uses $d(B)=5$, giving $8$; only later does $A\to B$ lower $B$ to $2$. Pass 3 carries that improvement to $C$, and pass 4 carries it to $T$. The final labels in order $(S,A,B,C,T)$ are $(0,4,2,5,7)$; the final shortest-path predecessors are $S,A,B,C$ for $A,B,C,T$ respectively.

The scan order explains the delay. Scanning instead as $S\to A$, $S\to B$, $A\to B$, $B\to C$, $C\to T$, $A\to T$ gives those same final labels in just one in-place pass. The Synchronous Bellman-Ford recurrence has a precise bound on the number of edges used by a recorded route; an in-place pass can propagate along several edges and need not have that interpretation. Both implementations have the $|V|-1$ pass guarantee under the stated no-reachable-negative-cycle hypothesis.

With a stored list of edges, complete-pass Bellman-Ford algorithm has time complexity $O(|V||E|+|V|)$ and uses $O(|V|)$ extra space for labels and shortest-path predecessors, using Big O notation and constant-cost arithmetic. The extra $|V|$ accounts for initialization even when $E$ is empty. A pass with no strict decrease permits early termination. A queue variant instead revisits outgoing edges of vertices whose labels improved; the authored implementation provides this alternative and has $O(|V||E|)$ worst-case running time on its usual nontrivial instances.^[ref] For nonnegative weights, the Dijkstra algorithm and its existing distance-frontier figure explain how permanent labels can avoid these repeated passes; the existing edge relaxation figure illustrates one update at a time.

The Bellman inequalities turn local directed edge checks into a global certificate for the shortest path problem. Here the convention is all-to-one: every journey ends at a fixed vertex $r$, and $c_{ij}$ is the cost of the original directed edge $i\to j$. Work in a finite directed graph with finite real costs. Write $L_i$ for candidate labels, normalized by $L_r=0$, on the vertices that can reach $r$. A feasible potential means precisely that these finite labels satisfyfor every directed edge within that set. These labels give lower bounds on costs to $r$. A discovered directed path of cost $U_i$ instead gives an upper bound. The following constructed graph shows both bounds and the exact distances between them; its weights are illustrative costs, not measured travel data.

The arrows in the picture point toward the destination, whereas edge relaxation propagates an improved label in the opposite direction: an improved $d_j$ can lower $d_i$. In the pictured graph, the directed paths from $a$ to $r$ cost $7$, $2+3=5$, and $2-2+4=4$. Thus $\delta (a,r)=4$. The candidate $L=(3,1,3,0)$ in vertex order $(a,b,c,r)$ is feasible, but underestimates the exact distances $(4,2,4,0)$. Merely satisfying the Bellman inequalities does not make a candidate exact.

Why do they give lower bounds? Along any directed path $i=v_0\to v_1\to \cdots \to v_k=r$, successive substitution givesThe same calculation works for a directed walk, with repeated vertices allowed. Taking the infimum of route costs gives $L_i\le \delta (i,r)$. A discovered route gives $\delta (i,r)\le U_i$. If the bounds meet, $L_i=U_i$, they certify an optimum without enumerating routes. Equivalently, check the Bellman inequalities for the proposed distances and exhibit directed paths realizing them. The authored Bellman-Ford algorithm implementation checks the corresponding single-source inequalities and equality on recorded route edges; reversing every directed edge gives this all-to-one convention.^[ref]

For the exact labels in the picture, the slack $c_{ij}+d_j-d_i$ is respectively $0,3,1,0,0$ on $a\to b$, $a\to r$, $b\to r$, $b\to c$, $c\to r$. Zero slack means equality; the zero-slack directed path $a\to b\to c\to r$ realizes the cost $4$. This is the useful beauty of the certificate: a few local inequalities rule out every cheaper journey, while one explicit journey attains the bound.

Keep unreachable vertices separate: $\delta (i,r)=\infty$ when no directed path to $r$ exists, and a finite lower bound alone cannot certify that fact. Extended labels can use $\infty +c=\infty$ for finite $c$, but never compute a slack as $\infty -\infty$. If $i$ can reach a negative-cost directed cycle and then reach $r$, repeatedly traversing the directed cycle makes route costs unbounded below: their infimum is $-\infty$, with no finite shortest directed walk. A finite feasible potential on such a directed cycle is impossible, since summing its Bellman inequalities would give $0\le \sum c_{ij}<0$. A negative-cost directed cycle that cannot reach $r$ does not affect distances to $r$. These are statements about unrestricted directed walks; requiring a directed path with no repeated vertices is a different optimization problem when negative-cost directed cycles exist.^[ref]

For the single-source convention, the same original directed edge $u\to v$ instead gives $L(v)\le L(u)+w_{uv}$ with $L(s)=0$, a lower bound on distance from $s$. The endpoint carrying the label on the left therefore changes with the direction of the problem.

Edge relaxation tests whether extending a recorded route by one directed edge improves a tentative distance. For a single source $s$ and an original directed edge $u\to v$ of weight $w_{uv}$, form the candidate $q=d(u)+w_{uv}$. If $d(u)$ is finite and $q<d(v)$, replace $d(v)$ by $q$ and record the shortest-path predecessor $\pi (v)=u$. Otherwise leave both unchanged. The minimum formula in the header gives the resulting label in either case. The authored implementation uses this strict comparison and records the incoming directed edge.^[ref]

The table lists labels after each specified operation, keeping $d(s)=0$ throughout this constructed acyclic graph. Relaxing $a\to t$ first offers $6+2=8$. Relaxing $b\to a$ then offers $2-1=1$, lowering $a$ from $6$ to $1$; revisiting $a\to t$ now offers $1+2=3$. Finally $b\to t$ offers $2+7=9$, so the label $3$ stays. Changing a label can make a previously checked outgoing directed edge worth checking again. This trace illustrates the operation itself, with a deliberately specified order, rather than a run of the Dijkstra algorithm.

A single edge relaxation does not require nonnegative weights: its candidate is the cost of a real extended directed walk, even when the last directed edge has negative weight. Nonnegative weights are needed by the Dijkstra algorithm to make its selected minimum tentative distance permanent. The Bellman-Ford algorithm permits negative weights and revisits labels; its authored implementation also checks for reachable negative-cost directed cycles.^[ref][ref]

For the all-to-one convention of the Bellman inequalities, relax the same original directed edge $i\to j$ by $d_i\leftarrow \min \{d_i,c_{ij}+d_j\}$. This prepends $i\to j$ to a discovered route from $j$ to $r$, and improves the tail label $d_i$, not the head label $d_j$. Skip the operation when $d_j=\infty$. In exact arithmetic, a full pass over all relevant directed edges with no improvement establishes all their Bellman inequalities; combined with the discovered-route upper bounds, this certifies finite distances. One unsuccessful operation establishes only that this particular candidate did not improve its label at that moment.