Compute trigger counts in a DAG

You are given a directed acyclic graph (DAG) that represents trigger dependencies between tasks.

• Each node represents a task.
• There is exactly one entry node .
• When a node is triggered once, it immediately triggers each of its outgoing neighbors once .
• A node may have multiple incoming edges; its total trigger count is the sum of triggers from all its parents .
• The entry node is triggered exactly once initially.

Your job is to compute how many times each node is triggered.

Input

• A list of directed edges edges , where each edge is a pair (u, v) meaning there is an edge u -> v .
• A designated entry node ID.
• You may assume:
  • The graph is acyclic.
  • All nodes that appear in edges are valid nodes.
  • Node IDs can be strings or integers (you can pick a representation and document it).

Example:

• Nodes: A, B, C, D, E, F
• Edges:
  • A -> B
  • B -> C
  • B -> D
  • C -> D
  • D -> E
  • D -> F
  • E -> F
• Entry node: A

Trigger propagation:

• A is triggered once (given).
• B is triggered once (from A ).
• C is triggered once (from B ).
• D is triggered twice (once from B , once from C ).
• E is triggered twice (both from D ).
• F is triggered four times (twice from D , twice from E ).

So the result is something equivalent to:

Task

Implement a function in Python with a signature such as:

The function should:

• Validate that all reachable nodes from entry are handled.
• Run efficiently for graphs with up to around 10^5 nodes and 10^5 edges.

You may assume there are no cycles in the input graph.