minimum_spanning_tree.py

# Minimum spanning tree (MST) is going to use an undirected graph
#
# The disjoint set is represented with an list <n> of integers where 
# <n[i]> is the parent of the node at position <i>.
# If <n[i]> = <i>, <i> it's a root, or a head, of a set


class Edge:
    def __init__(self, u, v, weight):
        self.u = u
        self.v = v
        self.weight = weight


class DisjointSet:
    def __init__(self, n):
        # Args:
        #   n (int): Number of vertices in the graph

        self.parent = [None] * n # Contains wich node is the parent of the node at poisition <i>
        self.size = [1] * n # Contains size of node at index <i>, used to optimize merge
        for i in range(n):
            self.parent[i] = i # Make all nodes his own parent, creating n sets.

    def merge_set(self, a, b):
        # Args:
        #   a, b (int): Indexes of nodes whose sets will be merged.

        # Get the set of nodes at position <a> and <b>
        # If <a> and <b> are the roots, this will be constant O(1)
        a = self.findSet(a)
        b = self.findSet(b)

        # Join the shortest node to the longest, minimizing tree size (faster find)
        if self.size[a] < self.size[b]:
            self.parent[a] = b # Merge set(a) and set(b)
            self.size[b] += self.size[a] # Add size of old set(a) to set(b)
        else:
            self.parent[b] = a # Merge set(b) and set(a)
            self.size[a] += self.size[b] # Add size of old set(b) to set(a)

    def find_set(self, a):
        if self.parent[a] != a: 
            # Very important, memoize result of the 
            # recursion in the list to optimize next
            # calls and make this operation practically constant, O(1)
            self.parent[a] = self.find_set(self.parent[a])

        # node <a> it's the set root, so we can return that index
        return self.parent[a]


def kruskal(n, edges, ds):
    # Args:
    #   n (int): Number of vertices in the graph
    #   edges (list of Edge): Edges of the graph
    #   ds (DisjointSet): DisjointSet of the vertices
    # Returns:
    #   int: sum of weights of the minnimum spanning tree 
    #
    # Kruskal algorithm:
    #   This algorithm will find the optimal graph with less edges and less
    #   total weight to connect all vertices (MST), the MST will always contain
    #   n-1 edges because it's the minimum required to connect n vertices.
    #
    # Procedure:
    #   Sort the edges (criteria: less weight).
    #   Only take edges of nodes in different sets.
    #   If we take a edge, we need to merge the sets to discard these.
    #   After repeat this until select n-1 edges, we will have the complete MST.
    edges.sort(key=lambda edge: edge.weight)

    mst = [] # List of edges taken, minimum spanning tree

    for edge in edges:
        set_u = ds.findSet(edge.u) # Set of the node <u>
        set_v = ds.findSet(edge.v) # Set of the node <v>
        if set_u != set_v:
            ds.merge_set(set_u, set_v)
            mst.append(edge)
            if len(mst) == n-1: 
                # If we have selected n-1 edges, all the other 
                # edges will be discarted, so, we can stop here
                break

    return sum([edge.weight for edge in mst])




if __name__ == "__main__":
    # Test. How input works:
    # Input consists of different weighted, connected, undirected graphs. 
    # line 1: 
    #   integers n, m
    # lines 2..m+2:
    #   edge with the format -> node index u, node index v, integer weight
    #
    # Samples of input:
    #
    # 5 6
    # 1 2 3
    # 1 3 8
    # 2 4 5
    # 3 4 2
    # 3 5 4
    # 4 5 6
    # 
    # 3 3
    # 2 1 20
    # 3 1 20
    # 2 3 100
    #
    # Sum of weights of the optimal paths:
    # 14, 40
    import sys
    for n_m in sys.stdin:
        n, m = map(int, n_m.split())
        ds = DisjointSet(m)
        edges = [None] * m # Create list of size <m>

        # Read <m> edges from input
        for i in range(m):
            u, v, weight = map(int, input().split())
            u -= 1 # Convert from 1-indexed to 0-indexed
            v -= 1 # Convert from 1-indexed to 0-indexed
            edges[i] = Edge(u, v, weight)

        # After finish input and graph creation, use Kruskal algorithm for MST:
        print("MST weights sum:", kruskal(n, edges, ds))