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271 lines
8.8 KiB
Python
271 lines
8.8 KiB
Python
"""
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Prim's (also known as Jarník's) algorithm is a greedy algorithm that finds a minimum
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spanning tree for a weighted undirected graph. This means it finds a subset of the
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edges that forms a tree that includes every vertex, where the total weight of all the
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edges in the tree is minimized. The algorithm operates by building this tree one vertex
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at a time, from an arbitrary starting vertex, at each step adding the cheapest possible
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connection from the tree to another vertex.
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"""
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from __future__ import annotations
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from sys import maxsize
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from typing import Generic, TypeVar
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T = TypeVar("T")
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def get_parent_position(position: int) -> int:
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"""
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heap helper function get the position of the parent of the current node
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>>> get_parent_position(1)
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0
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>>> get_parent_position(2)
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0
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"""
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return (position - 1) // 2
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def get_child_left_position(position: int) -> int:
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"""
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heap helper function get the position of the left child of the current node
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>>> get_child_left_position(0)
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1
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"""
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return (2 * position) + 1
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def get_child_right_position(position: int) -> int:
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"""
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heap helper function get the position of the right child of the current node
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>>> get_child_right_position(0)
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2
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"""
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return (2 * position) + 2
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class MinPriorityQueue(Generic[T]):
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"""
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Minimum Priority Queue Class
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Functions:
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is_empty: function to check if the priority queue is empty
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push: function to add an element with given priority to the queue
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extract_min: function to remove and return the element with lowest weight (highest
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priority)
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update_key: function to update the weight of the given key
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_bubble_up: helper function to place a node at the proper position (upward
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movement)
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_bubble_down: helper function to place a node at the proper position (downward
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movement)
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_swap_nodes: helper function to swap the nodes at the given positions
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>>> queue = MinPriorityQueue()
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>>> queue.push(1, 1000)
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>>> queue.push(2, 100)
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>>> queue.push(3, 4000)
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>>> queue.push(4, 3000)
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>>> queue.extract_min()
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2
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>>> queue.update_key(4, 50)
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>>> queue.extract_min()
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4
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>>> queue.extract_min()
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1
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>>> queue.extract_min()
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3
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"""
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def __init__(self) -> None:
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self.heap: list[tuple[T, int]] = []
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self.position_map: dict[T, int] = {}
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self.elements: int = 0
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def __len__(self) -> int:
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return self.elements
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def __repr__(self) -> str:
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return str(self.heap)
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def is_empty(self) -> bool:
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# Check if the priority queue is empty
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return self.elements == 0
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def push(self, elem: T, weight: int) -> None:
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# Add an element with given priority to the queue
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self.heap.append((elem, weight))
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self.position_map[elem] = self.elements
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self.elements += 1
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self._bubble_up(elem)
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def extract_min(self) -> T:
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# Remove and return the element with lowest weight (highest priority)
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if self.elements > 1:
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self._swap_nodes(0, self.elements - 1)
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elem, _ = self.heap.pop()
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del self.position_map[elem]
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self.elements -= 1
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if self.elements > 0:
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bubble_down_elem, _ = self.heap[0]
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self._bubble_down(bubble_down_elem)
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return elem
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def update_key(self, elem: T, weight: int) -> None:
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# Update the weight of the given key
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position = self.position_map[elem]
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self.heap[position] = (elem, weight)
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if position > 0:
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parent_position = get_parent_position(position)
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_, parent_weight = self.heap[parent_position]
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if parent_weight > weight:
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self._bubble_up(elem)
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else:
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self._bubble_down(elem)
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else:
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self._bubble_down(elem)
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def _bubble_up(self, elem: T) -> None:
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# Place a node at the proper position (upward movement) [to be used internally
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# only]
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curr_pos = self.position_map[elem]
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if curr_pos == 0:
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return
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parent_position = get_parent_position(curr_pos)
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_, weight = self.heap[curr_pos]
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_, parent_weight = self.heap[parent_position]
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if parent_weight > weight:
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self._swap_nodes(parent_position, curr_pos)
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return self._bubble_up(elem)
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return
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def _bubble_down(self, elem: T) -> None:
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# Place a node at the proper position (downward movement) [to be used
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# internally only]
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curr_pos = self.position_map[elem]
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_, weight = self.heap[curr_pos]
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child_left_position = get_child_left_position(curr_pos)
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child_right_position = get_child_right_position(curr_pos)
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if child_left_position < self.elements and child_right_position < self.elements:
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_, child_left_weight = self.heap[child_left_position]
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_, child_right_weight = self.heap[child_right_position]
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if child_right_weight < child_left_weight:
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if child_right_weight < weight:
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self._swap_nodes(child_right_position, curr_pos)
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return self._bubble_down(elem)
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if child_left_position < self.elements:
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_, child_left_weight = self.heap[child_left_position]
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if child_left_weight < weight:
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self._swap_nodes(child_left_position, curr_pos)
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return self._bubble_down(elem)
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else:
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return
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if child_right_position < self.elements:
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_, child_right_weight = self.heap[child_right_position]
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if child_right_weight < weight:
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self._swap_nodes(child_right_position, curr_pos)
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return self._bubble_down(elem)
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else:
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return
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def _swap_nodes(self, node1_pos: int, node2_pos: int) -> None:
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# Swap the nodes at the given positions
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node1_elem = self.heap[node1_pos][0]
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node2_elem = self.heap[node2_pos][0]
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self.heap[node1_pos], self.heap[node2_pos] = (
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self.heap[node2_pos],
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self.heap[node1_pos],
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)
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self.position_map[node1_elem] = node2_pos
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self.position_map[node2_elem] = node1_pos
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class GraphUndirectedWeighted(Generic[T]):
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"""
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Graph Undirected Weighted Class
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Functions:
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add_node: function to add a node in the graph
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add_edge: function to add an edge between 2 nodes in the graph
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"""
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def __init__(self) -> None:
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self.connections: dict[T, dict[T, int]] = {}
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self.nodes: int = 0
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def __repr__(self) -> str:
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return str(self.connections)
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def __len__(self) -> int:
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return self.nodes
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def add_node(self, node: T) -> None:
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# Add a node in the graph if it is not in the graph
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if node not in self.connections:
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self.connections[node] = {}
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self.nodes += 1
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def add_edge(self, node1: T, node2: T, weight: int) -> None:
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# Add an edge between 2 nodes in the graph
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self.add_node(node1)
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self.add_node(node2)
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self.connections[node1][node2] = weight
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self.connections[node2][node1] = weight
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def prims_algo(
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graph: GraphUndirectedWeighted[T],
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) -> tuple[dict[T, int], dict[T, T | None]]:
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"""
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>>> graph = GraphUndirectedWeighted()
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>>> graph.add_edge("a", "b", 3)
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>>> graph.add_edge("b", "c", 10)
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>>> graph.add_edge("c", "d", 5)
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>>> graph.add_edge("a", "c", 15)
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>>> graph.add_edge("b", "d", 100)
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>>> dist, parent = prims_algo(graph)
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>>> abs(dist["a"] - dist["b"])
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3
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>>> abs(dist["d"] - dist["b"])
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15
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>>> abs(dist["a"] - dist["c"])
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13
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"""
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# prim's algorithm for minimum spanning tree
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dist: dict[T, int] = {node: maxsize for node in graph.connections}
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parent: dict[T, T | None] = {node: None for node in graph.connections}
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priority_queue: MinPriorityQueue[T] = MinPriorityQueue()
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for node, weight in dist.items():
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priority_queue.push(node, weight)
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if priority_queue.is_empty():
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return dist, parent
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# initialization
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node = priority_queue.extract_min()
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dist[node] = 0
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for neighbour in graph.connections[node]:
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if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
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dist[neighbour] = dist[node] + graph.connections[node][neighbour]
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priority_queue.update_key(neighbour, dist[neighbour])
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parent[neighbour] = node
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# running prim's algorithm
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while not priority_queue.is_empty():
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node = priority_queue.extract_min()
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for neighbour in graph.connections[node]:
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if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
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dist[neighbour] = dist[node] + graph.connections[node][neighbour]
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priority_queue.update_key(neighbour, dist[neighbour])
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parent[neighbour] = node
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return dist, parent
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