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