Added Borůvka's algorithm.

graphs/boruvka.py

@@ -0,0 +1,198 @@
+"""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
+    graph,or a minimum spanning forest in the case of 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.
+
+    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 more simple to code.
+
+    Details: https://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithm
+"""
+
+
+class Graph:
+    def __init__(self, num_of_nodes: int) -> None:
+        """
+        Arguments:
+            num_of_nodes - the number of nodes in the graph
+        Attributes:
+            m_v - 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_v = num_of_nodes
+        self.m_edges = []
+        self.m_component = {}
+
+    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.keys():
+                self.m_component[k] = self.find_component(k)
+
+    def union(self, component_size: list, 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 = [-1] * self.m_v
+
+        # A list of components (initialized to all of the nodes)
+        for node in range(self.m_v):
+            self.m_component.update({node: node})
+            component_size.append(1)
+
+        num_of_components = self.m_v
+
+        while num_of_components > 1:
+            l_edges = len(self.m_edges)
+            for i in range(l_edges):
+
+                u = self.m_edges[i][0]
+                v = self.m_edges[i][1]
+                w = self.m_edges[i][2]
+
+                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"""
+
+                    if (
+                        minimum_weight_edge[u_component] == -1
+                        or minimum_weight_edge[u_component][2] > w
+                    ):
+                        minimum_weight_edge[u_component] = [u, v, w]
+                    if (
+                        minimum_weight_edge[v_component] == -1
+                        or minimum_weight_edge[v_component][2] > w
+                    ):
+                        minimum_weight_edge[v_component] = [u, v, w]
+
+            for node in range(self.m_v):
+                if minimum_weight_edge[node] != -1:
+                    u = minimum_weight_edge[node][0]
+                    v = minimum_weight_edge[node][1]
+                    w = minimum_weight_edge[node][2]
+
+                    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(
+                            "Added edge ["
+                            + str(u)
+                            + " - "
+                            + str(v)
+                            + "]\n"
+                            + "Added weight: "
+                            + str(w)
+                            + "\n"
+                        )
+                        num_of_components -= 1
+
+            minimum_weight_edge = [-1] * self.m_v
+        print("The total weight of the minimal spanning tree is: " + str(mst_weight))
+
+
+def test_vector() -> None:
+    """
+    >>> g=Graph(8)
+    >>> g.add_edge(0, 1, 10)
+    >>> g.add_edge(0, 2, 6)
+    >>> g.add_edge(0, 3, 5)
+    >>> g.add_edge(1, 3, 15)
+    >>> g.add_edge(2, 3, 4)
+    >>> g.add_edge(3, 4, 8)
+    >>> g.add_edge(4, 5, 10)
+    >>> g.add_edge(4, 6, 6)
+    >>> g.add_edge(4, 7, 5)
+    >>> g.add_edge(5, 7, 15)
+    >>> g.add_edge(6, 7, 4)
+    >>> 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()