from __future__ import annotations

import heapq
from math import inf
from typing import Hashable, TypeVar

Node = TypeVar("Node", bound=Hashable)
Graph = dict[Node, list[tuple[Node, float]]]


def dijkstra(graph: Graph[Node], start: Node) -> tuple[dict[Node, float], dict[Node, Node]]:
    """Return shortest distances and predecessors from start."""
    nodes = set(graph)
    for edges in graph.values():
        nodes.update(neighbor for neighbor, _ in edges)

    distances = {node: inf for node in nodes}
    predecessors: dict[Node, Node] = {}
    distances[start] = 0.0
    queue: list[tuple[float, Node]] = [(0.0, 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
                predecessors[neighbor] = node
                heapq.heappush(queue, (candidate, neighbor))

    return distances, predecessors


def shortest_path(graph: Graph[Node], start: Node, target: Node) -> tuple[float, list[Node]]:
    """Return the distance and node sequence for the shortest path."""
    distances, predecessors = dijkstra(graph, start)
    if distances.get(target, inf) == inf:
        return inf, []

    path = [target]
    while path[-1] != start:
        path.append(predecessors[path[-1]])
    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": [],
    }

    assert shortest_path(graph, "A", "D") == (4.0, ["A", "C", "B", "D"])
    assert shortest_path(graph, "A", "E") == (inf, [])
    print("All tests passed.")


if __name__ == "__main__":
    main()