From 1fb4f95080045da194c7ef40a1ffb229cb979920 Mon Sep 17 00:00:00 2001 From: Isaac Cheng <47993930+IsaacCheng9@users.noreply.github.com> Date: Tue, 13 Jan 2026 23:55:03 +0000 Subject: [PATCH 1/9] Add `find()` and `union()` methods for union-find implementation --- src/boruvkas_algorithm/boruvka.py | 56 +++++++++++++++++++++++++++++-- 1 file changed, 54 insertions(+), 2 deletions(-) diff --git a/src/boruvkas_algorithm/boruvka.py b/src/boruvkas_algorithm/boruvka.py index 02eed0c..e053d4e 100644 --- a/src/boruvkas_algorithm/boruvka.py +++ b/src/boruvkas_algorithm/boruvka.py @@ -18,9 +18,13 @@ def __init__(self, num_vertices: int): Args: num_vertices: The number of vertices to generate in the graph. """ - self.vertices = list(range(num_vertices)) + self.vertices: list[int] = list(range(num_vertices)) # [(vertex_1, vertex_2, weight)] - self.edges = [] + self.edges: list[tuple[int, int, int]] = [] + # Each node is its own parent initially. + self.parent: list[int] = list(range(num_vertices)) + # Each tree has size 1 (itself) initially. + self.rank: list[int] = [1] * num_vertices def add_edge(self, vertex_1: int, vertex_2: int, weight: int) -> None: """ @@ -47,6 +51,54 @@ def print_graph_info(self) -> None: for edge in sorted(self.edges): print(f" {edge}") + def find(self, node: int) -> int: + """ + Finds the root parent of the node using path compression. + + Args: + node: The node to find the root parent of. + + Returns: + The root parent of the node. + """ + cur_parent = self.parent[node] + while cur_parent != self.parent[cur_parent]: + # Compress the links as we go up the chain of parents to make + # it faster to traverse in the future - amortised O(a(n)) time, + # where a(n) is the inverse Ackermann function. + self.parent[cur_parent] = self.parent[self.parent[cur_parent]] + cur_parent = self.parent[cur_parent] + + return cur_parent + + def union(self, node1: int, node2: int) -> bool: + """ + Combines the two nodes into the larger segment. + + Args: + node1: The first node to combine. + node2: The second node to combine. + + Returns: + True if the nodes were combined, False if they were already in the + same segment. + """ + root1 = self.find(node1) + root2 = self.find(node2) + # If they have the same root parent, a cycle exists. + if root1 == root2: + return False + + # Combine the two nodes into the larger segment based on the rank. + if self.rank[root1] > self.rank[root2]: + self.parent[root2] = root1 + self.rank[root1] += self.rank[root2] + else: + self.parent[root1] = root2 + self.rank[root2] += self.rank[root1] + + return True + def merge_components( self, vertex_to_component: Dict[int, int], From b90a50619cab44daa265e72b3cbbe9268b08dde6 Mon Sep 17 00:00:00 2001 From: Isaac Cheng <47993930+IsaacCheng9@users.noreply.github.com> Date: Wed, 14 Jan 2026 00:00:29 +0000 Subject: [PATCH 2/9] Remove old type hints using typing module --- src/boruvkas_algorithm/boruvka.py | 30 ++++++++++++++---------------- 1 file changed, 14 insertions(+), 16 deletions(-) diff --git a/src/boruvkas_algorithm/boruvka.py b/src/boruvkas_algorithm/boruvka.py index e053d4e..1fde43a 100644 --- a/src/boruvkas_algorithm/boruvka.py +++ b/src/boruvkas_algorithm/boruvka.py @@ -5,8 +5,6 @@ import matplotlib.pyplot as plt import networkx as nx -from typing import Dict, List, Optional, Tuple - class Graph: """ @@ -101,8 +99,8 @@ def union(self, node1: int, node2: int) -> bool: def merge_components( self, - vertex_to_component: Dict[int, int], - component_sizes: List[int], + vertex_to_component: dict[int, int], + component_sizes: list[int], vertex_1: int, vertex_2: int, ) -> None: @@ -139,8 +137,8 @@ def merge_components( def update_min_edge_per_component( self, - vertex_to_component: Dict[int, int], - min_connecting_edge_per_component: List[Optional[Tuple]], + vertex_to_component: dict[int, int], + min_connecting_edge_per_component: list, ): """ Check each edge and update the shortest edge for each vertex if it @@ -176,13 +174,13 @@ def update_min_edge_per_component( def connect_components_with_min_edges( self, - component_sizes: List[int], - min_connecting_edge_per_component: List[Optional[Tuple]], - mst_edges: List[Tuple[int, int, int]], + component_sizes: list[int], + min_connecting_edge_per_component: list, + mst_edges: list[tuple[int, int, int]], mst_weight: int, - vertex_to_component: Dict[int, int], + vertex_to_component: dict[int, int], num_components: int, - ) -> Tuple[int, int]: + ) -> tuple[int, int]: """ Connect components using the minimum connecting edges. @@ -216,7 +214,7 @@ def connect_components_with_min_edges( return mst_weight, num_components - def initialize_components(self) -> Tuple[Dict[int, int], List[int], int]: + def initialize_components(self) -> tuple[dict[int, int], list[int], int]: """ Initialize each vertex as its own component with size 1, and set the initial number of components equal to the number of vertices. @@ -232,10 +230,10 @@ def initialize_components(self) -> Tuple[Dict[int, int], List[int], int]: def perform_iteration( self, - vertex_to_component: Dict[int, int], - component_sizes: List[int], + vertex_to_component: dict[int, int], + component_sizes: list[int], num_components: int, - mst_edges: List[Tuple[int, int, int]], + mst_edges: list[tuple[int, int, int]], mst_weight: int, ): """ @@ -272,7 +270,7 @@ def perform_iteration( return mst_weight, num_components - def draw_mst(self, mst_edges: List[Tuple[int, int, int]]) -> None: + def draw_mst(self, mst_edges: list[tuple[int, int, int]]) -> None: """ Draw the graph with the minimum spanning tree highlighted using networkx. From 4d980414a2041f63dbc3e469c2a903c94153928a Mon Sep 17 00:00:00 2001 From: Isaac Cheng <47993930+IsaacCheng9@users.noreply.github.com> Date: Wed, 14 Jan 2026 08:50:34 +0000 Subject: [PATCH 3/9] Improve efficiency by using union-find to group components --- src/boruvkas_algorithm/boruvka.py | 148 +++++++----------------------- tests/test_boruvka.py | 4 +- 2 files changed, 33 insertions(+), 119 deletions(-) diff --git a/src/boruvkas_algorithm/boruvka.py b/src/boruvkas_algorithm/boruvka.py index 1fde43a..e330fdf 100644 --- a/src/boruvkas_algorithm/boruvka.py +++ b/src/boruvkas_algorithm/boruvka.py @@ -7,45 +7,41 @@ class Graph: - """ - A graph that contains vertices and edges. - """ - def __init__(self, num_vertices: int): """ Args: num_vertices: The number of vertices to generate in the graph. """ self.vertices: list[int] = list(range(num_vertices)) - # [(vertex_1, vertex_2, weight)] + # [(node1, node2, weight)] self.edges: list[tuple[int, int, int]] = [] # Each node is its own parent initially. self.parent: list[int] = list(range(num_vertices)) # Each tree has size 1 (itself) initially. self.rank: list[int] = [1] * num_vertices - def add_edge(self, vertex_1: int, vertex_2: int, weight: int) -> None: + def add_edge(self, node1: int, node2: int, weight: int) -> None: """ Add an edge to the graph. Args: - vertex_1: The first vertex of the edge. - vertex_2: The second vertex of the edge. + node1: The first node of the edge. + node2: The second node of the edge. weight: The weight of the edge. Raises: - ValueError: If either vertex does not exist in the graph. + ValueError: If either node does not exist in the graph. """ - if vertex_1 not in self.vertices or vertex_2 not in self.vertices: + if node1 not in self.vertices or node2 not in self.vertices: raise ValueError("One or both vertices not found in graph.") - self.edges.append((vertex_1, vertex_2, weight)) + self.edges.append((node1, node2, weight)) def print_graph_info(self) -> None: """ Print the graph's vertices and edges. """ print(f"Vertices: {self.vertices}") - print("Edges (vertex_1, vertex_2, weight):") + print("Edges (node1, node2, weight):") for edge in sorted(self.edges): print(f" {edge}") @@ -97,101 +93,52 @@ def union(self, node1: int, node2: int) -> bool: return True - def merge_components( - self, - vertex_to_component: dict[int, int], - component_sizes: list[int], - vertex_1: int, - vertex_2: int, - ) -> None: - """ - Merge two components of the graph into one, ensuring that the smaller - component is merged into the larger one to optimize the merging - process. - - Args: - vertex_to_component: A mapping of vertices to their component - identifiers. - component_sizes: A list where the index represents the component - identifier and the value is the size of the - component. - vertex_1: A vertex in the first component to be merged. - vertex_2: A vertex in the second component to be merged. - """ - # Identify the components of the two vertices. - component_1 = vertex_to_component[vertex_1] - component_2 = vertex_to_component[vertex_2] - - # Determine the smaller and larger components. - if component_sizes[component_1] < component_sizes[component_2]: - smaller, larger = component_1, component_2 - else: - smaller, larger = component_2, component_1 - - # Merge the smaller component into larger component. - for vertex, component in vertex_to_component.items(): - if component == smaller: - vertex_to_component[vertex] = larger - # Update the size of the larger component. - component_sizes[larger] += component_sizes[smaller] - - def update_min_edge_per_component( - self, - vertex_to_component: dict[int, int], - min_connecting_edge_per_component: list, - ): + def update_min_edge_per_component(self, min_connecting_edge_per_component: list): """ - Check each edge and update the shortest edge for each vertex if it + Check each edge and update the shortest edge for each node if it connects two components together. Args: - vertex_to_component: A dictionary containing the component of each - vertex. min_connecting_edge_per_component: A list with the shortest edge - for each vertex that connects to + for each node that connects to a new component. """ for edge in self.edges: - vertex_1, vertex_2, weight = edge - vertex_1_component = vertex_to_component[vertex_1] - vertex_2_component = vertex_to_component[vertex_2] + node1, node2, weight = edge + node1_component = self.find(node1) + node2_component = self.find(node2) # If the vertices are in different components and the edge is # smaller than the current minimum weight edge for either # component, update them. - if vertex_1_component != vertex_2_component: + if node1_component != node2_component: if ( - not min_connecting_edge_per_component[vertex_1_component] - or weight < min_connecting_edge_per_component[vertex_1_component][2] + not min_connecting_edge_per_component[node1_component] + or weight < min_connecting_edge_per_component[node1_component][2] ): - min_connecting_edge_per_component[vertex_1_component] = edge + min_connecting_edge_per_component[node1_component] = edge if ( - not min_connecting_edge_per_component[vertex_2_component] - or weight < min_connecting_edge_per_component[vertex_2_component][2] + not min_connecting_edge_per_component[node2_component] + or weight < min_connecting_edge_per_component[node2_component][2] ): - min_connecting_edge_per_component[vertex_2_component] = edge + min_connecting_edge_per_component[node2_component] = edge def connect_components_with_min_edges( self, - component_sizes: list[int], min_connecting_edge_per_component: list, mst_edges: list[tuple[int, int, int]], mst_weight: int, - vertex_to_component: dict[int, int], num_components: int, ) -> tuple[int, int]: """ Connect components using the minimum connecting edges. Args: - component_sizes: List containing the sizes of each component. min_connecting_edge_per_component: List storing the shortest edge for each component. mst_edges: List of edges in the minimum spanning tree. mst_weight: Total weight of the minimum spanning tree. - vertex_to_component: Dictionary mapping vertices to their - component. num_components: Total number of components in the graph. Returns: @@ -199,39 +146,18 @@ def connect_components_with_min_edges( """ for edge in min_connecting_edge_per_component: if edge is not None: - vertex_1, vertex_2, weight = edge - if vertex_to_component[vertex_1] != vertex_to_component[vertex_2]: - mst_edges.append((vertex_1, vertex_2, weight)) + node1, node2, weight = edge + if self.find(node1) != self.find(node2): + mst_edges.append((node1, node2, weight)) mst_weight += weight - self.merge_components( - vertex_to_component, component_sizes, vertex_1, vertex_2 - ) + self.union(node1, node2) num_components -= 1 - print( - f"Added edge {vertex_1} - {vertex_2} with " - f"weight {weight} to MST." - ) + print(f"Added edge {node1} - {node2} with weight {weight} to MST.") return mst_weight, num_components - def initialize_components(self) -> tuple[dict[int, int], list[int], int]: - """ - Initialize each vertex as its own component with size 1, and set the - initial number of components equal to the number of vertices. - - Returns: - Tuple containing the mapping of vertex to its component, the list - of component sizes, and the initial number of components. - """ - vertex_to_component = {vertex: vertex for vertex in self.vertices} - component_sizes = [1] * len(self.vertices) - num_components = len(self.vertices) - return vertex_to_component, component_sizes, num_components - def perform_iteration( self, - vertex_to_component: dict[int, int], - component_sizes: list[int], num_components: int, mst_edges: list[tuple[int, int, int]], mst_weight: int, @@ -242,8 +168,6 @@ def perform_iteration( these edges. Args: - vertex_to_component: Mapping of vertices to their component. - component_sizes: List containing the sizes of each component. num_components: Total number of components in the graph. mst_edges: List of edges in the minimum spanning tree so far. mst_weight: Total weight of the minimum spanning tree so far. @@ -254,17 +178,13 @@ def perform_iteration( # Initialize list to store minimum connecting edge for each component. min_connecting_edge_per_component = [None] * len(self.vertices) # Update the minimum connecting edge for each component. - self.update_min_edge_per_component( - vertex_to_component, min_connecting_edge_per_component - ) + self.update_min_edge_per_component(min_connecting_edge_per_component) # Connect components using the minimum connecting edges and update MST # weight and number of components. mst_weight, num_components = self.connect_components_with_min_edges( - component_sizes, min_connecting_edge_per_component, mst_edges, mst_weight, - vertex_to_component, num_components, ) @@ -283,8 +203,8 @@ def draw_mst(self, mst_edges: list[tuple[int, int, int]]) -> None: G.add_nodes_from(self.vertices) # Add all edges to the graph with weights. for edge in self.edges: - vertex1, vertex2, weight = edge - G.add_edge(vertex1, vertex2, weight=weight) + node1, node2, weight = edge + G.add_edge(node1, node2, weight=weight) pos = nx.spring_layout(G) # Draw the graph edges and highlight the edges in the MST in red. nx.draw_networkx_edges( @@ -315,11 +235,7 @@ def run_boruvkas_algorithm(self): self.print_graph_info() mst_weight = 0 mst_edges = [] - ( - vertex_to_component, - component_sizes, - num_components, - ) = self.initialize_components() + num_components = len(self.vertices) # Track the number of iterations. num_iterations = 0 @@ -332,8 +248,6 @@ def run_boruvkas_algorithm(self): ) # Perform one iteration of the algorithm. mst_weight, num_components = self.perform_iteration( - vertex_to_component, - component_sizes, num_components, mst_edges, mst_weight, @@ -341,7 +255,7 @@ def run_boruvkas_algorithm(self): # Summarise the MST found. print("\nMST found with Boruvka's algorithm.") - print("MST edges (vertex_1, vertex_2, weight):") + print("MST edges (node1, node2, weight):") for edge in sorted(mst_edges): print(f" {edge}") print(f"MST weight: {mst_weight}") diff --git a/tests/test_boruvka.py b/tests/test_boruvka.py index 7030a70..598067a 100644 --- a/tests/test_boruvka.py +++ b/tests/test_boruvka.py @@ -30,11 +30,11 @@ def test_add_edge_invalid_vertices(setup_graph: Graph): Test the addition of an edge with non-existing vertices. Expects a ValueError to be raised when trying to add an edge with at least - one non-existing vertex. + one non-existing node. """ graph = setup_graph with pytest.raises(ValueError): - # Use vertex indices that do not exist in the graph. + # Use node indices that do not exist in the graph. graph.add_edge(10, 11, 5) From bf4e2c832b24e46b0a0cabe1dfea2808dfff503b Mon Sep 17 00:00:00 2001 From: Isaac Cheng <47993930+IsaacCheng9@users.noreply.github.com> Date: Wed, 14 Jan 2026 08:51:23 +0000 Subject: [PATCH 4/9] Update vertex -> node in README --- README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 8193bbd..5669beb 100644 --- a/README.md +++ b/README.md @@ -16,7 +16,7 @@ An implementation of Boruvka's algorithm to find a minimum spanning tree in a gr ``` Finding MST with Boruvka's algorithm: Vertices: [0, 1, 2, 3, 4, 5, 6, 7, 8] -Edges (vertex_1, vertex_2, weight): +Edges (node1, node2, weight): (0, 1, 4) (0, 6, 7) (1, 2, 9) @@ -50,7 +50,7 @@ Added edge 0 - 6 with weight 7 to MST. Added edge 2 - 3 with weight 6 to MST. 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