I am working with an undirected, weighted graph. It contains two types of nodes, called "A-type" and "B-type". I am starting with a set of tuples of the following form that describe the graph edges:

(string: Name of First Node, string: Name of Second Node, float: Weight)

Example graph:

Original Graph

My goal is to reduce the graph by removing any nodes of B-type that have exactly two neighbors, and their associated edges, and replacing them with a single edge that has a weight equal to the sum of the weights of removed edges. The reduced version of the example graph would look like this:

Reduced Graph

And printing the set of edges after reduction would produce this:

{('a1', 'a2', 1.1),
('a1', 'a4', 1.0),
('a1', 'a5', 1.1),
('a2', 'a3', 0.4),
('a3', 'a4', 2.3),
('a3', 'b5', 1.6),
('a3', 'b6', 3.0),
('a4', 'a5', 1.2),
('a4', 'b5', 1.7),
('a5', 'b5', 1.8)}

Note that...

  • Nodes B6 and B5 were not removed, even though they are B-type nodes, because they do not have exactly two neighbors.
  • A2 was not removed because it is not a B-type node
  • The weights of the new edges are the sum of the weights of the removed edges.

import re
import pprint
import operator
import collections
from itertools import chain

def series_reduction(edge_set):
    Reduces a set of graph edges by eliminating B-type nodes with two neighbors

    Nodes of type B are extracted from the nodes in the set of edges, and become 
    candidates for removal if they have only two neighbors. The set of removal
    candidates is iterated over,  and each node it contains becomes the target
    node. The target node's edges are removed and replaced with a new edge that
    connects the target node's neighbors to each other. The weight of the new
    edge is equal to the sum of the removed edges' weights.

    @param edge_set: graph edges to perform series reduction on
    @type edge_set: set

    @rtype: set

    neighbor_counter = collections.defaultdict(int)

    # identify nodes of B type and count their neighbors
    for node in chain.from_iterable(
            (node1, node2) 
            for node1, node2, w in edge_set):

        if re.search('\Ab', node):
            neighbor_counter[node] += 1

    # identify nodes that should be removed because they only have two neighbors
    reduce_nodes = set(
            for node, occurences in neighbor_counter.items() 
            if occurences == 2)

    # create final edge set - initialize with edges that have no nodes to be
    # reduced
    final_edge_set = set(
            (node1, node2, weight)
            for node1, node2, weight in edge_set
            if node1 not in reduce_nodes and node2 not in reduce_nodes)

    # create the set of edges that need to be reduced
    reduce_edge_set = edge_set - final_edge_set

    # iterate over nodes that will be removed
    for target_node in reduce_nodes:

        # find the edges with the target node
        target_edges = set(
                (node1, node2, weight) 
                for node1, node2, weight in reduce_edge_set 
                if node1 == target_node or node2 == target_node)

        # find the nodes from those edges that are NOT the node we're removing
        first_nodes, second_nodes, weights = zip(*target_edges)
        end_node1, end_node2 = [
                for node in first_nodes+second_nodes 
                if node != target_node]

        # removed the old edges from the set of edges to be reduced
        for edge in target_edges:

        # add the new edge to the set of remaining edges - it may contain nodes
        # that need to be reduced further, so we can't put it in the final set
        # yet
        reduce_edge_set.add((end_node1, end_node2, sum(weights)))

    # were done removing edges, so update the final set with the reduced edges

    return final_edge_set

I am looking for suggestions regarding:

  • Any optimizations - memory usage, runtime, length of code required, etc
  • Pythonic code, idioms, PEP8 guidelines
  • Maintainability
  • Any other suggestions


This code is going to be part of a company-internal project, so I had to remove descriptions of what the actual node types are, and have described them as "A-type" and "B-type". The only differences between this code and what will be used are the regular expression for determining node type, and the node type references in the docstring/comments.


1 Answer 1


1. Data structure

If you look at the Wikipedia article on graph representations, you'll see that it describes three common data structures that are used to represent graphs: adjacency list, adjacency matrix, and incidence matrix. Notice that the representation you've chosen (set of edges) does not appear. This is because:

  1. The set-of-edges representation can't represent all graphs! If there's an isolated node (a node with no edges) there's no way to represent it.

  2. The set-of-edges representation doesn't support efficient implementations of the operations that are needed by graph algorithms. For example, it is a hard problem just to iterate over the nodes. You use:

    for node in chain.from_iterable(
            (node1, node2) 
            for node1, node2, w in edge_set):

    but nodes appear in this iteration multiple times (once for each edge incident to the node), so you have to de-duplicate the nodes by storing them in a set.

So the first priority is to transform your graph representation into a more convenient data structure. We could use a package like python-graph, but it's not hard to write your own. Here's a graph implementation using the adjacency list representation:

from collections import defaultdict

class Graph:
    """An undirected weighted graph."""
    def __init__(self):
        # Map from node to set of its adjacent nodes.
        self._graph = defaultdict(set)
        # Map from sorted tuple of two nodes to the weight of the edge
        # between them.
        self._weight = dict()

    def has_node(self, n):
        """Return True iff the graph contains the node n."""
        return n in self._graph

    def nodes(self):
        """Return an iterator over the nodes of the graph."""
        return iter(self._graph)

    def node_count(self):
        """Return the number of nodes of the graph."""
        return len(self._graph)

    def neighbours(self, n):
        """Return an iterator over the neighbours of the node n."""
        return iter(self._graph[n])

    def neighbour_count(self, n):
        """Return the number of neighbours of the node n."""
        return len(self._graph[n])

    def add_node(self, n):
        """Add the node n."""

    def remove_node(self, n):
        """Remove the node n and all incident edges."""
        for n1 in self.neighbours(n):
            if n1 != n:
            del self._weight[self._edge(n, n1)]
        del self._graph[n]

    def _edge(self, n1, n2):
        """Return the representation of the edge between nodes n1, n2."""
        return (n1, n2) if n1 <= n2 else (n2, n1)

    def has_edge(self, n1, n2):
        """Return True iff the graph contains an edge between nodes n1, n2."""
        return self._edge(n1, n2) in self._weight

    def edges(self):
        """Return an iterator over the edges of the graph."""
        for n, w in self._weight.items():
            yield n + (w,)

    def edge_count(self):
        """Return the number of edges of the graph."""
        return len(self._weight)

    def edge_weight(self, n1, n2):
        """Return the weight of the edge between n1 and n2."""
        return self._weight[self._edge(n1, n2)]

    def add_edge(self, n1, n2, w):
        """Add an edge between nodes n1 and n2 with weight w."""
        self._weight[self._edge(n1, n2)] = w

    def remove_edge(self, n1, n2):
        """Remove the edge between nodes n1 and n2."""
        if n1 != n2:
        del self._weight[self._edge(n1, n2)]

Using this data structure it is easy to implement the algorithm:

def series_reduction(edge_set):
    # Build graph from edge set.
    g = Graph()
    for e in edge_set:

    # Remove B-type nodes with exactly two neighbours.
    btype_nodes = [n for n in g.nodes() if n.startswith('b')]
    for n in btype_nodes:
        if g.neighbour_count(n) == 2:
            n1, n2 = g.neighbours(n)
            if g.has_edge(n1, n2):
                pass # see §2.2 below
                g.add_edge(n1, n2, g.weight(n, n1) + g.weight(n, n2))

    # Return modified edge set.
    return set(g.edges())

The code would be even simpler if you were able to use the adjacency list representation throughout your program: then you'd be able to drop the conversion steps.

2. Other review comments

  1. Instead of:

    re.search('\Ab', node):


  2. series_reduction goes wrong if, when removing a B-type node between two nodes \$a\$ and \$c\$, there was already an edge between \$a\$ and \$c\$:

    >>> series_reduction({('a', 'b', 1), ('b', 'c', 1), ('a', 'c', 1)})
    {('a', 'c', 1), ('a', 'c', 2)}

    The result is not a valid graph: it has two edges between the same pair of nodes.

    You'll see that my revised code above tests for this case and does nothing if so. But maybe it should raise an exception? Maybe the old edge should be replaced with the new edge? Or maybe only if the new edge has a smaller weight? Because you've concealed the details of your problem, I can't tell what the right thing is to do here.

  3. series_reduction goes wrong if a graph contains an edge from a B-type node to itself:

    >>> series_reduction({('b', 'b', 2)})
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
      File "cr81880.py", line 158, in series_reduction
        for node in first_nodes+second_nodes 
    ValueError: need more than 0 values to unpack
  • \$\begingroup\$ Thank you for this insightful answer. I appreciate the amount of work you put into explaining everything. Some of your feedback actually doesn't apply to my use case, but I obviously just didn't explain myself well enough. (I really do need to use regex to identify types - the real strings I parse to identify types are complex. Also, I don't have to worry about incoming graphs with isolated nodes.) Your feedback is spot-on for the question/code as presented, so thank you. I really like how simple your implementation is. \$\endgroup\$
    – skrrgwasme
    Jun 10, 2015 at 2:22

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