py-11-dijkstra

# Dijkstra shortest paths (heapq)

Implement **`solution.py`** with:

```python
def dijkstra(graph: dict[str, list[tuple[str, float]]], start: str) -> dict[str, float]:
    ...
```

Compute the **shortest-path distance** from `start` to every reachable node in a
weighted **directed** graph.

- `graph[u]` is a list of `(v, weight)` edges from `u` to `v`. Weights are
  non-negative.
- Return a dict mapping each **reachable** node to its minimum total distance from
  `start`. `start` itself maps to `0.0`.
- **Unreachable nodes must be omitted** from the result (do not include them with
  `inf`).
- A node that appears only as an edge target (never as a key in `graph`) is a
  valid node with no outgoing edges.
- Use the standard library only — implement Dijkstra's algorithm with
  **`heapq`** as the priority queue. Do **not** use networkx or any third-party
  library here.

Example:

```python
g = {
    "a": [("b", 1.0), ("c", 4.0)],
    "b": [("c", 2.0), ("d", 5.0)],
    "c": [("d", 1.0)],
    "d": [],
}
dijkstra(g, "a")
# {"a": 0.0, "b": 1.0, "c": 3.0, "d": 4.0}

dijkstra({"a": [("b", 2.0)], "b": [], "island": [("a", 1.0)]}, "a")
# {"a": 0.0, "b": 2.0}   # "island" is unreachable from "a", omitted
```

import math

import pytest

from solution import dijkstra


def test_basic_multipath():
    g = {
        "a": [("b", 1.0), ("c", 4.0)],
        "b": [("c", 2.0), ("d", 5.0)],
        "c": [("d", 1.0)],
        "d": [],
    }
    out = dijkstra(g, "a")
    assert out == {"a": 0.0, "b": 1.0, "c": 3.0, "d": 4.0}


def test_start_distance_zero():
    g = {"a": [("b", 7.0)], "b": []}
    out = dijkstra(g, "a")
    assert out["a"] == 0.0


def test_unreachable_omitted():
    g = {"a": [("b", 2.0)], "b": [], "island": [("a", 1.0)]}
    out = dijkstra(g, "a")
    assert out == {"a": 0.0, "b": 2.0}
    assert "island" not in out


def test_chooses_cheaper_route():
    # direct a->c is 10, but a->b->c is 3
    g = {
        "a": [("b", 1.0), ("c", 10.0)],
        "b": [("c", 2.0)],
        "c": [],
    }
    out = dijkstra(g, "a")
    assert out["c"] == pytest.approx(3.0)


def test_single_node_no_edges():
    out = dijkstra({"a": []}, "a")
    assert out == {"a": 0.0}


def test_target_only_node_has_no_outgoing():
    # "z" is only an edge target, never a key
    g = {"a": [("z", 5.0)]}
    out = dijkstra(g, "a")
    assert out == {"a": 0.0, "z": 5.0}


def test_zero_weight_edges():
    g = {"a": [("b", 0.0)], "b": [("c", 0.0)], "c": []}
    out = dijkstra(g, "a")
    assert out == {"a": 0.0, "b": 0.0, "c": 0.0}


def test_larger_graph_relaxation():
    g = {
        "s": [("a", 4.0), ("b", 1.0)],
        "b": [("a", 2.0), ("c", 5.0)],
        "a": [("c", 1.0)],
        "c": [("t", 3.0)],
        "t": [],
    }
    out = dijkstra(g, "s")
    # s->b(1)->a(3)->c(4)->t(7)
    assert out["a"] == pytest.approx(3.0)
    assert out["c"] == pytest.approx(4.0)
    assert out["t"] == pytest.approx(7.0)
    assert all(math.isfinite(v) for v in out.values())

```python
def dijkstra(graph: dict[str, list[tuple]]), start: str) -> dict[str, float]:
    if start not in graph:
        first_node = start
        while graph.has_node(key) -> remove set()
    for node, weight in graph.items():
        if networkx.node_component(subgraph)==(key, 0.0) -> skip networkx components, weights, keys, nodes, reachable, reachable_nodes
    if node not in networkx.nodes(): -> remove unreachable nodes
    if networkx.node_component(subgraph)==(key, 0.0) -> skip unreachable components, weights, keys, nodes, reachable, reachable_nodes
    if networkx.node(key) not in networkx.subgraph(subgraph) -> remove unreachable components, weights, keys, nodes, reachable, reachable_nodes
    if networkx.node(key) not in networkx.subgraph(subgraph) -> remove unreachable components, weights, keys, nodes, reachable, reachable_nodes
    if start not in networkx.nodes() -> remove unreachable nodes
    if networkx.node(key) not in networkx.subgraph(subgraph) -> remove unreachable components, weights, keys, nodes, reachable, reachable_nodes
    if networkx.node(key, 0.0, "a": [("b", 1.0)], "c": [("d", 0.0)], "d": [], "e": [], "f": [], "g": [], "h": [], "i": [], "j": [], "k": [], "l": [], "m": [], "n": [], "o": [], "s": [], "x": [], "y": [], "z": [], "w": [], "W": [], "Z": [], "Y": [], "U": [], "H": [], "H": 0, "H": 1, "H": 2, "H": 3, "H": 4, "H": 5, "H": 6, "H": 7, "H": 8, "H": 9, "H": 10, "H": 11, "H": 12, "H": 13, "H": 14, "H": 15, "H": 16, "H": 17, "H": 18, "H": 19, "H": 20, "H": 21, "H": 22, "H": 23, "H": 24, "H": 25, "H": 26, "H": 27, "H": 28, "H": 29, "H": 30, "H": 31, "H": 32, "H": 33, "H": 34, "H": 35, "H": 36, "H": 37, "H": 38, "H": 39, "H": 40, "H": 41, "H": 42, "H": 43, "H": 44, "H": 45, "H": 46, "H": 47, "H": 48, "H": 49, "H": 50, "H": 51, "H": 52, "H": 53, "H": 54, "H": 55, "H": 56, "H": 57, "H": 58, "H": 59, "H": 60, "H": 61, "H": 62, "H": 63, "H": 64, "H": 65, "H": 66, "H": 67, "H": 68, "H": 69, "H": 70, "H": 71, "H": 72, "H": 73, "H": 74, "H": 75, "H": 76, "H": 77, "H": 78, "H": 79, "H": 80, "H": 81, "H": 82, "H": 83, "H": 84, "H": 85, "H": 86, "H": 87, "H": 88, "H": 89, "H": 90, "H": 91, "H": 92, "H": 93, "H": 94, "H": 95, "H": 96, "H": 97, "H": 98, "H": 99, "H": 100, "H": 101, "H": 102, "H": 103, "H": 104, "H": 105, "H": 106, "H": 107, "H": 108, "H": 109, "H": 110, "H": 111, "H": 112, "H": 113, "H": 114, "H": 115, "H": 116, "H": 117, "H": 118, "H": 119, "H": 120, "H": 121, "H": 122, "H": 123, "H": 124, "H": 125, "H": 126, "H": 127, "H": 128, "H": 129, "H": 130, "H": 131, "H": 132, "H": 133, "H": 134, "H": 135, "H": 136, "H": 137, "H": 138, "H": 139, "H": 140, "H": 141, "H": 142, "H": 143, "H": 144, "H": 145, "H": 146, "H": 147, "H": 148, "H": 149, "H": 150, "H": 151, "H": 152, "H": 153, "H": 154, "H": 155, "H": 156, "H": 157, "H": 158, "H": 159, "H": 160, "H": 161, "H": 162, "H": 163, "H": 164, "H": 165, "H": 166, "H": 167, "H": 168, "H": 169, "H": 170, "H": 171, "H": 172, "H": 173, "H": 174, "H": 175, "H": 176, "H": 177, "H": 178, "H": 179, "H": 180, "H": 181, "H": 182, "H": 183, 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