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

    if start not in nodes:
        raise KeyError(f"Unknown start node: {start!r}")

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

    # A counter avoids comparing nodes when equal distances are queued.
    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, ()):
            candidate = distance + weight
            if candidate < distances[neighbor]:
                distances[neighbor] = candidate
                predecessors[neighbor] = node
                counter += 1
                heapq.heappush(queue, (candidate, counter, neighbor))

    return distances, predecessors


def shortest_path(
    graph: Graph[Node], start: Node, target: Node
) -> tuple[float, list[Node]]:
    """Return the shortest distance and path, or (inf, []) if unreachable."""
    distances, predecessors = dijkstra(graph, start)
    if target not in distances:
        raise KeyError(f"Unknown target node: {target!r}")
    if distances[target] == inf:
        return inf, []

    path: list[Node] = []
    current: Node | None = target
    while current is not None:
        path.append(current)
        current = predecessors[current]
    path.reverse()
    return distances[target], path


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

    distance, path = shortest_path(graph, "A", "D")
    assert distance == 4
    assert path == ["A", "C", "B", "D"]

    distance, path = shortest_path(graph, "A", "E")
    assert distance == inf
    assert path == []


if __name__ == "__main__":
    main()