Python/graphs/minimum_spanning_tree_prims2.py

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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 Dict, Optional, Tuple, Union
def get_parent_position(position: int) -> int:
"""
heap helper function get the position of the parent of the current node
>>> get_parent_position(1)
0
>>> get_parent_position(2)
0
"""
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)
1
"""
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)
2
"""
return (2 * position) + 2
class MinPriorityQueue:
"""
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())
2
>>> queue.update_key(4, 50)
>>> print(queue.extract_min())
4
>>> print(queue.extract_min())
1
>>> print(queue.extract_min())
3
"""
def __init__(self) -> None:
self.heap = []
self.position_map = {}
self.elements = 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: Union[int, str], 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) -> Union[int, str]:
# 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: Union[int, str], 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: Union[int, str]) -> 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: Union[int, str]) -> 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:
"""
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 = {}
self.nodes = 0
def __repr__(self) -> str:
return str(self.connections)
def __len__(self) -> int:
return self.nodes
def add_node(self, node: Union[int, str]) -> 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: Union[int, str], node2: Union[int, str], 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,
) -> Tuple[Dict[str, int], Dict[str, Optional[str]]]:
"""
>>> 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"])
3
>>> abs(dist["d"] - dist["b"])
15
>>> abs(dist["a"] - dist["c"])
13
"""
# prim's algorithm for minimum spanning tree
dist = {node: maxsize for node in graph.connections}
parent = {node: None for node in graph.connections}
priority_queue = MinPriorityQueue()
[priority_queue.push(node, weight) for node, weight in dist.items()]
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
if __name__ == "__main__":
from doctest import testmod
testmod()