**π‘ Problem Formulation:** Reversing a directed graph entails flipping the direction of all the edges. For instance, if there is an edge from node A to B in the original graph, in the reversed graph, the edge will point from B to A. Given a graph `G`

represented by an adjacency list, the desired output is a new graph `G'`

where all edge directions are reversed.

## Method 1: Using a Dictionary to Construct the Reversed Graph

This method involves creating a new adjacency list by iterating through the original graph and reversing the direction of the edges. It is suitable for sparse graphs and results in a straightforward and easily understandable code. The function takes the original graph’s adjacency list and returns a new adjacency list representing the reversed graph.

Here’s an example:

def reverse_graph(graph): reversed_graph = {node: [] for node in graph} for node, edges in graph.items(): for edge in edges: reversed_graph[edge].append(node) return reversed_graph # Example usage original_graph = {'A': ['B'], 'B': ['C', 'D'], 'C': ['D'], 'D': []} reversed_graph = reverse_graph(original_graph) print(reversed_graph)

Output:

{ 'A': [], 'B': ['A'], 'C': ['B'], 'D': ['B', 'C'] }

In the snippet above, the `reverse_graph`

function is defined to reverse the given directed graph. It first creates an empty reversed adjacency list, then iterates through the original graph, and populates the reversed graph with reversed edges.

## Method 2: Using NetworkX Library

The NetworkX library provides powerful graph manipulation capabilities within Python. It has a built-in method to reverse graphs, which is efficient and quick. The `reverse()`

function within NetworkX immediately returns a new graph with all edge directions reversed.

Here’s an example:

import networkx as nx # Create a directed graph G = nx.DiGraph([('A', 'B'), ('B', 'C'), ('B', 'D'), ('C', 'D')]) # Reverse the graph G_reversed = G.reverse() # Convert to adjacency list representation to print print(dict(G_reversed.adjacency()))

Output:

{ 'A': {}, 'B': {'A': {}}, 'C': {'B': {}}, 'D': {'B': {}, 'C': {}} }

Here, we use NetworkX’s `DiGraph`

class to represent the directed graph and its `reverse()`

method for reversing. The output is the reversed graph’s adjacency list. NetworkX handles the heavy lifting, offering a clean and concise solution.

## Method 3: Using Depth-First Search (DFS)

For a more algorithmic approach, one can use a depth-first search to traverse the graph and build the reversed version during traversal. This method is particularly educational and can handle larger graphs, although it may not be as performant as library-based solutions.

Here’s an example:

def reverse_graph_dfs(graph): def dfs(node, reversed_graph): for neighbor in graph[node]: if neighbor not in reversed_graph: reversed_graph[neighbor] = [node] dfs(neighbor, reversed_graph) else: reversed_graph[neighbor].append(node) reversed_graph = {} for node in graph: if node not in reversed_graph: reversed_graph[node] = [] dfs(node, reversed_graph) return reversed_graph # Example usage original_graph = {'A': ['B'], 'B': ['C', 'D'], 'C': ['D'], 'D': []} reversed_graph = reverse_graph_dfs(original_graph) print(reversed_graph)

Output:

{ 'A': [], 'B': ['A'], 'C': ['B'], 'D': ['C', 'B'] }

In the example above, a depth-first search is used to visit the nodes recursively and build the reversed graph. One must be cautious to handle potential issues with this approach, such as avoiding infinite loops in cyclic graphs.

## Method 4: Using Breadth-First Search (BFS)

Similar to DFS, breadth-first search can be employed to generate a reversed graph. This approach is useful for learning about graph traversal but, like DFS, may be slower than using a specialized graph library. BFS ensures all nodes at the current depth are visited before moving to the next level, which can be helpful for certain types of graph analysis.

Here’s an example:

from collections import deque def reverse_graph_bfs(graph): reversed_graph = {node: [] for node in graph} for node in graph: queue = deque([node]) while queue: current_node = queue.popleft() for neighbor in graph[current_node]: reversed_graph[neighbor].append(current_node) if neighbor not in queue: queue.append(neighbor) return reversed_graph # Example usage original_graph = {'A': ['B'], 'B': ['C', 'D'], 'C': ['D'], 'D': []} reversed_graph = reverse_graph_bfs(original_graph) print(reversed_graph)

Output:

{ 'A': [], 'B': ['A'], 'C': ['B'], 'D': ['C', 'B'] }

This code snippet uses a breadth-first search, implemented with a queue, to reverse the graph. In each step, it visits nodes level by level and reverses the edges by populating the `reversed_graph`

dictionary.

## Bonus One-Liner Method 5: Comprehension with Reversed Tuples

For those seeking a Pythonic one-liner, list comprehensions combined with tuple reversal provide a slick way to invert graph edges. This method is concise and expressive, aligning well with Python’s design philosophy, though it may trade readability for brevity in complex cases.

Here’s an example:

original_graph = {'A': ['B'], 'B': ['C', 'D'], 'C': ['D'], 'D': []} reversed_graph = {node: [] for node in original_graph} [reversed_graph[dest].append(src) for src, dests in original_graph.items() for dest in dests] print(reversed_graph)

Output:

{ 'A': [], 'B': ['A'], 'C': ['B'], 'D': ['C', 'B'] }

The one-liner comprehensively inverts the graph by iterating through each node and its destinations, appending the source to the destinations’ lists in the reversed graph. It’s an elegant use of Python’s expressive capabilities.

## Summary/Discussion

**Method 1: Using a Dictionary**. Strengths: intuitive and easy to implement; good for sparse graphs. Weaknesses: manually handling graph reversal may be error-prone and less efficient than using a library.**Method 2: Using NetworkX Library**. Strengths: extremely efficient and readable; powerful library for complex graph operations. Weaknesses: external library dependency; overkill for simple tasks.**Method 3: Using DFS**. Strengths: algorithmically interesting; suitable for educational purposes. Weaknesses: potential issues with cycle handling and stack depth.**Method 4: Using BFS**. Strengths: can handle level-based traversal logic; suitable for educational purposes. Weaknesses: potentially slower than specialized graph libraries; more complex than DFS.**Bonus Method 5: One-Liner Comprehension**. Strengths: succinct and Pythonic. Weaknesses: may sacrifice readability for brevity; not as intuitive for beginners.