mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-27 23:11:09 +00:00
bc8df6de31
* [pre-commit.ci] pre-commit autoupdate updates: - [github.com/astral-sh/ruff-pre-commit: v0.2.2 → v0.3.2](https://github.com/astral-sh/ruff-pre-commit/compare/v0.2.2...v0.3.2) - [github.com/pre-commit/mirrors-mypy: v1.8.0 → v1.9.0](https://github.com/pre-commit/mirrors-mypy/compare/v1.8.0...v1.9.0) * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
177 lines
6.3 KiB
Python
177 lines
6.3 KiB
Python
"""Borůvka's algorithm.
|
|
|
|
Determines the minimum spanning tree (MST) of a graph using the Borůvka's algorithm.
|
|
Borůvka's algorithm is a greedy algorithm for finding a minimum spanning tree in a
|
|
connected graph, or a minimum spanning forest if a graph that is not connected.
|
|
|
|
The time complexity of this algorithm is O(ELogV), where E represents the number
|
|
of edges, while V represents the number of nodes.
|
|
O(number_of_edges Log number_of_nodes)
|
|
|
|
The space complexity of this algorithm is O(V + E), since we have to keep a couple
|
|
of lists whose sizes are equal to the number of nodes, as well as keep all the
|
|
edges of a graph inside of the data structure itself.
|
|
|
|
Borůvka's algorithm gives us pretty much the same result as other MST Algorithms -
|
|
they all find the minimum spanning tree, and the time complexity is approximately
|
|
the same.
|
|
|
|
One advantage that Borůvka's algorithm has compared to the alternatives is that it
|
|
doesn't need to presort the edges or maintain a priority queue in order to find the
|
|
minimum spanning tree.
|
|
Even though that doesn't help its complexity, since it still passes the edges logE
|
|
times, it is a bit simpler to code.
|
|
|
|
Details: https://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithm
|
|
"""
|
|
|
|
from __future__ import annotations
|
|
|
|
from typing import Any
|
|
|
|
|
|
class Graph:
|
|
def __init__(self, num_of_nodes: int) -> None:
|
|
"""
|
|
Arguments:
|
|
num_of_nodes - the number of nodes in the graph
|
|
Attributes:
|
|
m_num_of_nodes - the number of nodes in the graph.
|
|
m_edges - the list of edges.
|
|
m_component - the dictionary which stores the index of the component which
|
|
a node belongs to.
|
|
"""
|
|
|
|
self.m_num_of_nodes = num_of_nodes
|
|
self.m_edges: list[list[int]] = []
|
|
self.m_component: dict[int, int] = {}
|
|
|
|
def add_edge(self, u_node: int, v_node: int, weight: int) -> None:
|
|
"""Adds an edge in the format [first, second, edge weight] to graph."""
|
|
|
|
self.m_edges.append([u_node, v_node, weight])
|
|
|
|
def find_component(self, u_node: int) -> int:
|
|
"""Propagates a new component throughout a given component."""
|
|
|
|
if self.m_component[u_node] == u_node:
|
|
return u_node
|
|
return self.find_component(self.m_component[u_node])
|
|
|
|
def set_component(self, u_node: int) -> None:
|
|
"""Finds the component index of a given node"""
|
|
|
|
if self.m_component[u_node] != u_node:
|
|
for k in self.m_component:
|
|
self.m_component[k] = self.find_component(k)
|
|
|
|
def union(self, component_size: list[int], u_node: int, v_node: int) -> None:
|
|
"""Union finds the roots of components for two nodes, compares the components
|
|
in terms of size, and attaches the smaller one to the larger one to form
|
|
single component"""
|
|
|
|
if component_size[u_node] <= component_size[v_node]:
|
|
self.m_component[u_node] = v_node
|
|
component_size[v_node] += component_size[u_node]
|
|
self.set_component(u_node)
|
|
|
|
elif component_size[u_node] >= component_size[v_node]:
|
|
self.m_component[v_node] = self.find_component(u_node)
|
|
component_size[u_node] += component_size[v_node]
|
|
self.set_component(v_node)
|
|
|
|
def boruvka(self) -> None:
|
|
"""Performs Borůvka's algorithm to find MST."""
|
|
|
|
# Initialize additional lists required to algorithm.
|
|
component_size = []
|
|
mst_weight = 0
|
|
|
|
minimum_weight_edge: list[Any] = [-1] * self.m_num_of_nodes
|
|
|
|
# A list of components (initialized to all of the nodes)
|
|
for node in range(self.m_num_of_nodes):
|
|
self.m_component.update({node: node})
|
|
component_size.append(1)
|
|
|
|
num_of_components = self.m_num_of_nodes
|
|
|
|
while num_of_components > 1:
|
|
for edge in self.m_edges:
|
|
u, v, w = edge
|
|
|
|
u_component = self.m_component[u]
|
|
v_component = self.m_component[v]
|
|
|
|
if u_component != v_component:
|
|
"""If the current minimum weight edge of component u doesn't
|
|
exist (is -1), or if it's greater than the edge we're
|
|
observing right now, we will assign the value of the edge
|
|
we're observing to it.
|
|
|
|
If the current minimum weight edge of component v doesn't
|
|
exist (is -1), or if it's greater than the edge we're
|
|
observing right now, we will assign the value of the edge
|
|
we're observing to it"""
|
|
|
|
for component in (u_component, v_component):
|
|
if (
|
|
minimum_weight_edge[component] == -1
|
|
or minimum_weight_edge[component][2] > w
|
|
):
|
|
minimum_weight_edge[component] = [u, v, w]
|
|
|
|
for edge in minimum_weight_edge:
|
|
if isinstance(edge, list):
|
|
u, v, w = edge
|
|
|
|
u_component = self.m_component[u]
|
|
v_component = self.m_component[v]
|
|
|
|
if u_component != v_component:
|
|
mst_weight += w
|
|
self.union(component_size, u_component, v_component)
|
|
print(f"Added edge [{u} - {v}]\nAdded weight: {w}\n")
|
|
num_of_components -= 1
|
|
|
|
minimum_weight_edge = [-1] * self.m_num_of_nodes
|
|
print(f"The total weight of the minimal spanning tree is: {mst_weight}")
|
|
|
|
|
|
def test_vector() -> None:
|
|
"""
|
|
>>> g = Graph(8)
|
|
>>> for u_v_w in ((0, 1, 10), (0, 2, 6), (0, 3, 5), (1, 3, 15), (2, 3, 4),
|
|
... (3, 4, 8), (4, 5, 10), (4, 6, 6), (4, 7, 5), (5, 7, 15), (6, 7, 4)):
|
|
... g.add_edge(*u_v_w)
|
|
>>> g.boruvka()
|
|
Added edge [0 - 3]
|
|
Added weight: 5
|
|
<BLANKLINE>
|
|
Added edge [0 - 1]
|
|
Added weight: 10
|
|
<BLANKLINE>
|
|
Added edge [2 - 3]
|
|
Added weight: 4
|
|
<BLANKLINE>
|
|
Added edge [4 - 7]
|
|
Added weight: 5
|
|
<BLANKLINE>
|
|
Added edge [4 - 5]
|
|
Added weight: 10
|
|
<BLANKLINE>
|
|
Added edge [6 - 7]
|
|
Added weight: 4
|
|
<BLANKLINE>
|
|
Added edge [3 - 4]
|
|
Added weight: 8
|
|
<BLANKLINE>
|
|
The total weight of the minimal spanning tree is: 46
|
|
"""
|
|
|
|
|
|
if __name__ == "__main__":
|
|
import doctest
|
|
|
|
doctest.testmod()
|