from __future__ import annotations

import heapq
from collections.abc import Hashable, Mapping, Sequence
from math import inf
from typing import TypeVar

Node = TypeVar("Node", bound=Hashable)
Graph = Mapping[Node, Sequence[tuple[Node, float]]]


def dijkstra(
    graph: Graph[Node], start: Node
) -> tuple[dict[Node, float], dict[Node, Node | None]]:
    """Return shortest distances and predecessors from start."""
    nodes = set(graph)
    nodes.add(start)
    nodes.update(neighbor for edges in graph.values() for neighbor, _ in edges)

    distances = {node: inf for node in nodes}
    previous: dict[Node, Node | None] = {node: None for node in nodes}
    distances[start] = 0.0

    # The counter prevents comparisons between nodes when distances tie.
    counter = 0
    queue: list[tuple[float, int, Node]] = [(0.0, counter, start)]

    while queue:
        distance, _, node = heapq.heappop(queue)
        if distance != distances[node]:
            continue  # Ignore stale queue entries.

        for neighbor, weight in graph.get(node, ()):
            if weight < 0:
                raise ValueError("Dijkstra's algorithm requires non-negative weights")

            candidate = distance + weight
            if candidate < distances[neighbor]:
                distances[neighbor] = candidate
                previous[neighbor] = node
                counter += 1
                heapq.heappush(queue, (candidate, counter, neighbor))

    return distances, previous


def shortest_path(
    previous: Mapping[Node, Node | None], start: Node, target: Node
) -> list[Node]:
    """Reconstruct a path; return an empty list when target is unreachable."""
    path: list[Node] = []
    current: Node | None = target

    while current is not None:
        path.append(current)
        if current == start:
            return list(reversed(path))
        current = previous.get(current)

    return []


def main() -> None:
    graph = {
        "A": [("B", 4), ("C", 1)],
        "B": [("D", 1)],
        "C": [("B", 2), ("D", 5)],
        "D": [],
        "E": [],
    }

    distances, previous = dijkstra(graph, "A")
    assert distances["D"] == 4
    assert shortest_path(previous, "A", "D") == ["A", "C", "B", "D"]
    assert distances["E"] == inf
    assert shortest_path(previous, "A", "E") == []


if __name__ == "__main__":
    main()