Please show all work so i can better understand # Topologically Sorting DAGsBelow is a Python function `naive_topsort` demonstrating how to perform a topological sort on a Directed Acyclic Graph (DAG). This implementation assumes a set-based representation of the graph nodes.```pythondef naive_topsort(G, S=None): if S is None: S = set(G) # Default: All nodes if len(S) == 1: return list(S) # Base case, single node v = S.pop() # Reduction: Remove a node seq = naive_topsort(G, S) # Recursion (assumption), n=1 min_i = 0 for i, u in enumerate(seq): if v in G[u]: min_i = i + 1 # After all dependencies seq.insert(min_i, v) return seq```- **Explanation** - **Line 2-3**: If the set `S` is not provided, initialize it with all nodes in `G`. - **Line 4-5**: Base case for recursion. If there's only one node, return it as a list. - **Line 6**: Remove a node `v` from the set `S`. - **Line 7**: Recursively call `naive_topsort` for the reduced graph. - **Lines 9-13**: Determine the position `min_i` to insert `v` after all dependencies. - **Line 14**: Insert `v` into the sequence at the appropriate position. - **Line 15**: Return the updated sequence.- **Notes** - The graph `G` is represented using adjacency sets or dictionaries. - This script poses the question, "What’s the expected runtime?," prompting a discussion on the algorithm’s efficiency.The function provides a foundational approach for educational purposes, illustrating the recursive nature of topological sorting and dependency management in DAGs.

The given algorithm recursively finds the solution of the topological sort of the graph. Let T(n) be…

Answered: Topologically Sorting DAGS def naive…

Database System Concepts

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Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

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Please show all work so i can better understand

# Topologically Sorting DAGs

Below is a Python function `naive_topsort` demonstrating how to perform a topological sort on a Directed Acyclic Graph (DAG). This implementation assumes a set-based representation of the graph nodes.

```python
def naive_topsort(G, S=None):
if S is None: S = set(G) # Default: All nodes
if len(S) == 1: return list(S) # Base case, single node
v = S.pop() # Reduction: Remove a node
seq = naive_topsort(G, S) # Recursion (assumption), n=1
min_i = 0
for i, u in enumerate(seq):
if v in G[u]: min_i = i + 1 # After all dependencies
seq.insert(min_i, v)
return seq
```

- **Explanation**
- **Line 2-3**: If the set `S` is not provided, initialize it with all nodes in `G`.
- **Line 4-5**: Base case for recursion. If there's only one node, return it as a list.
- **Line 6**: Remove a node `v` from the set `S`.
- **Line 7**: Recursively call `naive_topsort` for the reduced graph.
- **Lines 9-13**: Determine the position `min_i` to insert `v` after all dependencies.
- **Line 14**: Insert `v` into the sequence at the appropriate position.
- **Line 15**: Return the updated sequence.

- **Notes**
- The graph `G` is represented using adjacency sets or dictionaries.
- This script poses the question, "What’s the expected runtime?," prompting a discussion on the algorithm’s efficiency.

The function provides a foundational approach for educational purposes, illustrating the recursive nature of topological sorting and dependency management in DAGs.

Expert Solution

Recurrence Analysis :

The given algorithm recursively finds the solution of the topological sort of the graph.
Let T(n) be the time complexity for finding the topological sort of the graph having n number of vertices :
- T(n) = T(n - c) + O(1) where c = number of the neightbour of nth vetrex
- Here, c= O(V) i.e the maximum number of neighbors can be at a maximum of a total number of vertices.
- This recurrence equation is used for finding the solution to the time complexity of the given equation.

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