Dijkstra's algorithm

Algorithm.
1. Set $d(r)=0$, $d(v)=+\infty$ for $v\neq r$.
2. Priority queue with all vertices.
3. Extract vertex $u$ with minimum $d(u)$; mark as finalized.
4. For each neighbor $v$ of $u$: relax $d(v)=\min(d(v), d(u)+w(u,v))$.

Correctness. Nonnegative weights guarantee that the extracted vertex's distance is final.

Complexity. $O((|V|+|E|)\log|V|)$.

Conclusion. Dijkstra solves SSSP in weighted nonnegative graphs.