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

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

    # Counter avoids comparing nodes when distances tie.
    queue: list[tuple[float, int, Node]] = [(0.0, 0, start)]
    counter = 1

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

        for neighbor, weight in graph[node]:
            if weight < 0:
                raise ValueError("Dijkstra's algorithm requires non-negative weights")
            if neighbor not in graph:
                raise KeyError(f"Unknown neighbor node: {neighbor!r}")

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

    return distances, previous


def shortest_path(
    graph: Graph[Node], start: Node, target: Node
) -> tuple[float, list[Node]]:
    """Return the distance and node sequence for a shortest path."""
    if target not in graph:
        raise KeyError(f"Unknown target node: {target!r}")

    distances, previous = dijkstra(graph, start)
    if distances[target] == inf:
        return inf, []

    path: list[Node] = []
    current: Node | None = target
    while current is not None:
        path.append(current)
        current = previous[current]

    path.reverse()
    return distances[target], path


def main() -> None:
    graph = {
        "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(shortest_path(graph, "A", "D"))


if __name__ == "__main__":
    main()