## Topologically Sorting DAGs### Python Code for Naive Topological Sort```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- **Using adjacency sets or dictionaries** to represent a Directed Acyclic Graph (DAG).- The function checks if it's working with all nodes by default if `S` is `None`.- It handles the base case where there's only a single node.- The process involves removing a node and recursively solving the smaller graph.- Dependencies are checked to ensure correct order by adjusting the insertion index.- **Expected runtime inquiry**: What’s the expected runtime?This code snippet defines a recursive approach to perform a naive topological sort on Directed Acyclic Graphs (DAGs) using Python, highlighting the default assumptions, base cases, and steps of recursion and reduction within the function.

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

Topologically Sorting DAGS 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() seq=naive topsort(G,S) min i=0 for i,u in enumerate(seq): if v in G[u]: min i=i+1 seq.insert (min i,v) # Reduction: Remove a node # Recursion (assumption), n=1 # After all dependencies return seq Using adjacency sets or dicts to represent a DAG. What's the expected runtime?

Topologically Sorting DAGS 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() seq=naive topsort(G,S) min i=0 for i,u in enumerate(seq): if v in G[u]: min i=i+1 seq.insert (min i,v) # Reduction: Remove a node # Recursion (assumption), n=1 # After all dependencies return seq Using adjacency sets or dicts to represent a DAG. What's the expected runtime?

Database System Concepts

7th Edition

ISBN:9780078022159

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

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

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

## Topologically Sorting DAGs

### Python Code for Naive Topological Sort

```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

- **Using adjacency sets or dictionaries** to represent a Directed Acyclic Graph (DAG).
- The function checks if it's working with all nodes by default if `S` is `None`.
- It handles the base case where there's only a single node.
- The process involves removing a node and recursively solving the smaller graph.
- Dependencies are checked to ensure correct order by adjusting the insertion index.
- **Expected runtime inquiry**: What’s the expected runtime?

This code snippet defines a recursive approach to perform a naive topological sort on Directed Acyclic Graphs (DAGs) using Python, highlighting the default assumptions, base cases, and steps of recursion and reduction within the function.

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