Compute Effective Letter Permissions in a DAG

Question

The question asks the candidate to compute effective letter permissions in a Directed Acyclic Graph (DAG), focusing on how permissions propagate through the graph structure.

Codemia · Accepted Answer

To compute effective letter permissions in a DAG, we will utilize a topological sorting approach. This will allow us to process each node in a way that respects the direction of the edges, ensuring that we handle permissions in the correct order. We will maintain a mapping of permissions for each node and propagate these permissions down the graph.

1. **Topological Sort**: First, perform a topological sort on the DAG to get the order of nodes.
2. **Initialize Permissions**: Create a dictionary to hold the effective permissions for each node.
3. **Process Nodes**: Iterate through the sorted nodes, updating the permissions of each child node based on the parent's permissions.
4. **Return Results**: After processing all nodes, return the effective permissions for each node.

The time complexity of this algorithm is O(V + E), where V is the number of vertices (nodes) and E is the number of edges in the DAG. This is due to the topological sorting and the subsequent processing of each edge. The space complexity is O(V) for storing the permissions.

```python
from collections import defaultdict, deque

def compute_effective_permissions(dag, permissions):
    # Step 1: Topological Sort
    in_degree = {u: 0 for u in dag}
    for u in dag:
        for v in dag[u]:
            in_degree[v] += 1
    queue = deque([u for u in dag if in_degree[u] == 0])
    sorted_nodes = []

while queue:
        u = queue.popleft()
        sorted_nodes.append(u)
        for v in dag[u]:
            in_degree[v] -= 1
            if in_degree[v] == 0:
                queue.append(v)

# Step 2: Initialize Permissions
    effective_permissions = {u: set() for u in dag}
    for node, perms in permissions.items():
        effective_permissions[node] = set(perms)

# Step 3: Process Nodes
    for u in sorted_nodes:
        for v in dag[u]:
            effective_permissions[v].update(effective_permissions[u])

return effective_permissions
```

Interviewers are looking for clarity in your thought process, how well you understand graph traversal techniques, and your ability to articulate the reasoning behind your solution. They also pay attention to how you handle edge cases and whether you can analyze the algorithm's performance effectively.

Compute Effective Letter Permissions in a DAG

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