#! /usr/bin/env python
#coding:utf-8
"""
http://www.ics.uci.edu/~eppstein/PADS/
"""

class UnionFind:
    """
    UnionFind
    Each unionFind instance X maintains a family of disjoint sets of
    hashable objects, supporting the following two methods:

    - X[item] returns a name for the set containing the given item.
      Each set is named by an arbitrarily-chosen one of its members; as
      long as the set remains unchanged it will keep the same name. If
      the item is not yet part of a set in X, a new singleton set is
      created for it.

    - X.union(item1, item2, ...) merges the sets containing each item
      into a single larger set.  If any item is not yet part of a set
      in X, it is added to X as one of the members of the merged set.
    """

    def __init__(self):
        """Create a new empty union-find structure."""
        self.weights = {}
        self.parents = {}

    def __getitem__(self, object):
        """Find and return the name of the set containing the object."""

        # check for previously unknown object
        if object not in self.parents:
            self.parents[object] = object
            self.weights[object] = 1
            return object

        # find path of objects leading to the root
        path = [object]
        root = self.parents[object]
        while root != path[-1]:
            path.append(root)
            root = self.parents[root]

        # compress the path and return
        for ancestor in path:
            self.parents[ancestor] = root
        return root
        
    def __iter__(self):
        """Iterate through all items ever found or unioned by this structure."""
        return iter(self.parents)

    def union(self, *objects):
        """Find the sets containing the objects and merge them all."""
        roots = [self[x] for x in objects]
        heaviest = max([(self.weights[r],r) for r in roots])[1]
        for r in roots:
            if r != heaviest:
                self.weights[heaviest] += self.weights[r]
                self.parents[r] = heaviest


"""

Various simple functions for graph input.

Each function's input graph G should be represented in such a way that "for v in G" loops through the vertices, and "G[v]" produces a list of the neighbors of v; for instance, G may be a dictionary mapping each vertex to its neighbor set.

D. Eppstein, April 2004.
"""

def isUndirected(G):
    """Check that G represents a simple undirected graph."""
    for v in G:
        if v in G[v]:
            return False
        for w in G[v]:
            if v not in G[w]:
                return False
    return True


def union(*graphs):
    """Return a graph having all edges from the argument graphs."""
    out = {}
    for G in graphs:
        for v in G:
            out.setdefault(v,set()).update(list(G[v]))
    return out


"""

Kruskal's algorithm for minimum spanning trees. D. Eppstein, April 2006.
"""

import unittest

def MinimumSpanningTree(G):
    """
    Return the minimum spanning tree of an undirected graph G.
    G should be represented in such a way that iter(G) lists its
    vertices, iter(G[u]) lists the neighbors of u, G[u][v] gives the
    length of edge u,v, and G[u][v] should always equal G[v][u].
    The tree is returned as a list of edges.
    """
    if not isUndirected(G):
        raise ValueError("MinimumSpanningTree: input is not undirected")
    for u in G:
        for v in G[u]:
            if G[u][v] != G[v][u]:
                raise ValueError("MinimumSpanningTree: asymmetric weights")

    # Kruskal's algorithm: sort edges by weight, and add them one at a time.
    # We use Kruskal's algorithm, first because it is very simple to
    # implement once UnionFind exists, and second, because the only slow
    # part (the sort) is sped up by being built in to Python.
    subtrees = UnionFind()
    tree = []
    for W,u,v in sorted((G[u][v],u,v) for u in G for v in G[u]):
        if subtrees[u] != subtrees[v]:
            tree.append((u,v))
            subtrees.union(u,v)
    return tree        


# If run standalone, perform unit tests

class MSTTest(unittest.TestCase):
    def testMST(self):
        """Check that MinimumSpanningTree returns the correct answer."""
        G = {0:{1:11,2:13,3:12},1:{0:11,3:14},2:{0:13,3:10},3:{0:12,1:14,2:10}}
        T = [(2,3),(0,1),(0,3)]
        for e,f in zip(MinimumSpanningTree(G),T):
            self.assertEqual(min(e),min(f))
            self.assertEqual(max(e),max(f))

if __name__ == "__main__":
    unittest.main()