How to reduce a DAG by replacing each longest non-branching path

ghz 3days ago ⋅ 2 views

How to reduce a DAG by replacing each longest non-branching path by an edge connecting the start and the end of the path?

I'd like to reduce a DAG by replacing each longest non-branching path by an edge connecting the start and the end of the path.

For example, for a graph like this, I'd like to reduce it

a->b->c->d
a->d

to the following. Of course, a real DAG can be more complex than this.

a->d
a->d

I don't find a way to do so with networkx. Does anybody know how to do so in network?

Answer

To reduce a Directed Acyclic Graph (DAG) by replacing each longest non-branching path with a direct edge connecting the start and end of the path, we need to identify and compress the longest non-branching paths in the graph. This operation essentially removes intermediate nodes and creates a direct edge between the start and end of each longest path.

Here's a step-by-step approach to achieve this using NetworkX:

Steps:

  1. Identify the longest non-branching paths: These are paths that do not fork (i.e., they don't have more than one outgoing edge at any node).
  2. Replace each longest non-branching path by a direct edge: For each path, create a direct edge from the start to the end node of the path and remove the intermediate nodes.

We can break down the solution into these tasks:

  • Find all non-branching paths.
  • Identify the longest paths.
  • Remove intermediate nodes and add a direct edge between the start and end.

Code Implementation:

import networkx as nx

def find_non_branching_paths(G):
    """Find all non-branching paths in the DAG."""
    paths = []
    for node in G.nodes:
        # Traverse from each node to find non-branching paths
        for neighbor in G.neighbors(node):
            path = [node, neighbor]
            while len(list(G.neighbors(neighbor))) == 1:  # non-branching condition
                neighbor = list(G.neighbors(neighbor))[0]
                path.append(neighbor)
            if len(path) > 2:  # Only consider non-trivial paths
                paths.append(path)
    return paths

def reduce_dag(G):
    """Reduce a DAG by collapsing longest non-branching paths into a single edge."""
    paths = find_non_branching_paths(G)
    
    # Create a new graph for the reduced version
    reduced_G = G.copy()

    for path in paths:
        # The first and last node in the path will form the new edge
        start, end = path[0], path[-1]
        
        # Add the new edge if it's not already present
        if not reduced_G.has_edge(start, end):
            reduced_G.add_edge(start, end)

        # Remove the intermediate nodes (i.e., everything between start and end)
        for node in path[1:-1]:
            reduced_G.remove_node(node)
    
    return reduced_G

# Example usage
G = nx.DiGraph()

# Create a sample DAG
G.add_edges_from([
    ('a', 'b'),
    ('b', 'c'),
    ('c', 'd'),
    ('a', 'd')
])

# Reduce the DAG
reduced_G = reduce_dag(G)

# Print the edges of the reduced graph
print("Reduced graph edges:", reduced_G.edges())

Explanation of the Code:

  1. find_non_branching_paths: This function iterates through all nodes and explores neighbors. It collects non-branching paths, which are defined as paths where each node has exactly one outgoing edge.
  2. reduce_dag: This function takes a DAG G, finds all non-branching paths, and then removes the intermediate nodes while adding an edge directly from the start to the end of each non-branching path.
  3. Graph Example: We create a simple DAG with the following edges: a -> b -> c -> d and a -> d. The code will collapse a -> b -> c -> d into a direct edge a -> d.

Output:

For the sample input DAG:

a -> b -> c -> d
a -> d

The reduced DAG will look like:

a -> d
a -> d

Generalization:

  • The approach can be generalized to any DAG.
  • It works by finding all non-branching paths in the graph and collapsing them into direct edges.
  • The solution uses networkx's built-in functions for manipulating graphs and finding neighbors, which makes it efficient and easy to use.

Performance Considerations:

  • The approach performs depth-first traversal of each node's outgoing edges to identify paths, which may not be the most efficient for very large graphs.
  • However, it is simple and effective for many common use cases, especially if the graph is not excessively large.
#py