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Prim's (also known as Jarník's) algorithm is a greedy algorithm that finds a minimum
spanning tree for a weighted undirected graph. This means it finds a subset of the
edges that forms a tree that includes every vertex, where the total weight of all the
edges in the tree is minimized. The algorithm operates by building this tree one vertex
at a time, from an arbitrary starting vertex, at each step adding the cheapest possible
connection from the tree to another vertex.
from sys import maxsize
from typing import Generic, Optional, TypeVar
T = TypeVar("T")
def get_parent_position(position: int) -> int:
    heap helper function get the position of the parent of the current node
    >>> get_parent_position(1)
    >>> get_parent_position(2)
    return (position - 1) // 2
def get_child_left_position(position: int) -> int:
    heap helper function get the position of the left child of the current node
    >>> get_child_left_position(0)
    return (2 * position) + 1
def get_child_right_position(position: int) -> int:
    heap helper function get the position of the right child of the current node
    >>> get_child_right_position(0)
    return (2 * position) + 2
class MinPriorityQueue(Generic[T]):
    Minimum Priority Queue Class
    Functions:
    is_empty: function to check if the priority queue is empty
    push: function to add an element with given priority to the queue
    extract_min: function to remove and return the element with lowest weight (highest
                 priority)
    update_key: function to update the weight of the given key
    _bubble_up: helper function to place a node at the proper position (upward
                movement)
    _bubble_down: helper function to place a node at the proper position (downward
                movement)
    _swap_nodes: helper function to swap the nodes at the given positions
    >>> queue = MinPriorityQueue()
    >>> queue.push(1, 1000)
    >>> queue.push(2, 100)
    >>> queue.push(3, 4000)
    >>> queue.push(4, 3000)
    >>> print(queue.extract_min())
    >>> queue.update_key(4, 50)
    >>> print(queue.extract_min())
    >>> print(queue.extract_min())
    >>> print(queue.extract_min())
    def __init__(self) -> None:
        self.heap: list[tuple[T, int]] = []
        self.position_map: dict[T, int] = {}
        self.elements: int = 0
    def __len__(self) -> int:
        return self.elements
    def __repr__(self) -> str:
        return str(self.heap)
    def is_empty(self) -> bool:
        # Check if the priority queue is empty
        return self.elements == 0
    def push(self, elem: T, weight: int) -> None:
        # Add an element with given priority to the queue
        self.heap.append((elem, weight))
        self.position_map[elem] = self.elements
        self.elements += 1
        self._bubble_up(elem)
    def extract_min(self) -> T:
        # Remove and return the element with lowest weight (highest priority)
        if self.elements > 1:
            self._swap_nodes(0, self.elements - 1)
        elem, _ = self.heap.pop()
        del self.position_map[elem]
        self.elements -= 1
        if self.elements > 0:
            bubble_down_elem, _ = self.heap[0]
            self._bubble_down(bubble_down_elem)
        return elem
    def update_key(self, elem: T, weight: int) -> None:
        # Update the weight of the given key
        position = self.position_map[elem]
        self.heap[position] = (elem, weight)
        if position > 0:
            parent_position = get_parent_position(position)
            _, parent_weight = self.heap[parent_position]
            if parent_weight > weight:
                self._bubble_up(elem)
            else:
                self._bubble_down(elem)
        else:
            self._bubble_down(elem)
    def _bubble_up(self, elem: T) -> None:
        # Place a node at the proper position (upward movement) [to be used internally
        # only]
        curr_pos = self.position_map[elem]
        if curr_pos == 0:
            return
        parent_position = get_parent_position(curr_pos)
        _, weight = self.heap[curr_pos]
        _, parent_weight = self.heap[parent_position]
        if parent_weight > weight:
            self._swap_nodes(parent_position, curr_pos)
            return self._bubble_up(elem)
        return
    def _bubble_down(self, elem: T) -> None:
        # Place a node at the proper position (downward movement) [to be used
        # internally only]
        curr_pos = self.position_map[elem]
        _, weight = self.heap[curr_pos]
        child_left_position = get_child_left_position(curr_pos)
        child_right_position = get_child_right_position(curr_pos)
        if child_left_position < self.elements and child_right_position < self.elements:
            _, child_left_weight = self.heap[child_left_position]
            _, child_right_weight = self.heap[child_right_position]
            if child_right_weight < child_left_weight:
                if child_right_weight < weight:
                    self._swap_nodes(child_right_position, curr_pos)
                    return self._bubble_down(elem)
        if child_left_position < self.elements:
            _, child_left_weight = self.heap[child_left_position]
            if child_left_weight < weight:
                self._swap_nodes(child_left_position, curr_pos)
                return self._bubble_down(elem)
        else:
            return
        if child_right_position < self.elements:
            _, child_right_weight = self.heap[child_right_position]
            if child_right_weight < weight:
                self._swap_nodes(child_right_position, curr_pos)
                return self._bubble_down(elem)
        else:
            return
    def _swap_nodes(self, node1_pos: int, node2_pos: int) -> None:
        # Swap the nodes at the given positions
        node1_elem = self.heap[node1_pos][0]
        node2_elem = self.heap[node2_pos][0]
        self.heap[node1_pos], self.heap[node2_pos] = (
            self.heap[node2_pos],
            self.heap[node1_pos],
        self.position_map[node1_elem] = node2_pos
        self.position_map[node2_elem] = node1_pos
class GraphUndirectedWeighted(Generic[T]):
    Graph Undirected Weighted Class
    Functions:
    add_node: function to add a node in the graph
    add_edge: function to add an edge between 2 nodes in the graph
    def __init__(self) -> None:
        self.connections: dict[T, dict[T, int]] = {}
        self.nodes: int = 0
    def __repr__(self) -> str:
        return str(self.connections)
    def __len__(self) -> int:
        return self.nodes
    def add_node(self, node: T) -> None:
        # Add a node in the graph if it is not in the graph
        if node not in self.connections:
            self.connections[node] = {}
            self.nodes += 1
    def add_edge(self, node1: T, node2: T, weight: int) -> None:
        # Add an edge between 2 nodes in the graph
        self.add_node(node1)
        self.add_node(node2)
        self.connections[node1][node2] = weight
        self.connections[node2][node1] = weight
def prims_algo(
    graph: GraphUndirectedWeighted[T],
) -> tuple[dict[T, int], dict[T, Optional[T]]]:
    >>> graph = GraphUndirectedWeighted()
    >>> graph.add_edge("a", "b", 3)
    >>> graph.add_edge("b", "c", 10)
    >>> graph.add_edge("c", "d", 5)
    >>> graph.add_edge("a", "c", 15)
    >>> graph.add_edge("b", "d", 100)
    >>> dist, parent = prims_algo(graph)
    >>> abs(dist["a"] - dist["b"])
    >>> abs(dist["d"] - dist["b"])
    >>> abs(dist["a"] - dist["c"])
    # prim's algorithm for minimum spanning tree
    dist: dict[T, int] = {node: maxsize for node in graph.connections}
    parent: dict[T, Optional[T]] = {node: None for node in graph.connections}
    priority_queue: MinPriorityQueue[T] = MinPriorityQueue()
    for node, weight in dist.items():
        priority_queue.push(node, weight)
    if priority_queue.is_empty():
        return dist, parent
    # initialization
    node = priority_queue.extract_min()
    dist[node] = 0
    for neighbour in graph.connections[node]:
        if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
            dist[neighbour] = dist[node] + graph.connections[node][neighbour]
            priority_queue.update_key(neighbour, dist[neighbour])
            parent[neighbour] = node
    # running prim's algorithm
    while not priority_queue.is_empty():
        node = priority_queue.extract_min()
        for neighbour in graph.connections[node]:
            if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
                dist[neighbour] = dist[node] + graph.connections[node][neighbour]
                priority_queue.update_key(neighbour, dist[neighbour])
                parent[neighbour] = node
    return dist, parent
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