how does Dijkstra's shortest path algorithm work time complexity

Core Mechanism

, is:

Dijkstra's algorithm visits the vertices in order of increasing distance from $s$.

That is, at each step the algorithm selects the unvisited vertex with the smallest known distance $d(v)$ from the source:

$$d(v) = \min_{u \in \text{visited}} \bigl( d(u) + w(u, v) \bigr)$$

where $w(u, v)$ is the edge weight between vertices $u$ and $v$.

Historical Significance

The Sorting Bottleneck and Complexity

notes:

"We do not know how to sort in linear time."

The evidence does not state explicit big-O bounds (e.g., with a binary heap or Fibonacci heap), so I cannot cite those figures from the retrieved papers.

An Alternative: Avoiding the Sorting Bottleneck

presented a deterministic linear time and linear space algorithm for the undirected SSSP problem with positive integer weights, which avoids the sorting bottleneck entirely:

"The algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure, identifying vertex pairs that may be visited in any order."

Application to Multi-Criteria Problems

Summary of What the Evidence Supports

Honest scope note: The retrieved evidence does not explicitly state standard complexity expressions such as $O((V + E)\log V)$. Those bounds are not cited above because the grounding rules prohibit stating claims not directly supported by the evidence blocks.