Detect cycle in directed dependency graph

You are given a directed dependency graph representing services in a system.

• There are n services, labeled from 0 to n - 1 .
• You are given a list of directed edges dependencies , where each element is a pair [a, b] meaning service a depends on service b (i.e., there is a directed edge a -> b ).

Assume:

• 1 <= n <= 10^5
• 0 <= a, b < n
• There may be zero or more edges.

Task:

1. Implement a function that determines whether there is at least one cycle in this directed graph.
   • Return true if there is a cycle.
   • Return false otherwise.
2. Follow-up: Now you are given multiple dependency arrays instead of just one. Each array represents an additional list of edges on the same set of services. For example:
   • dependencies1 , dependencies2 , ..., dependenciesk
   Each dependenciesi is a list of [a, b] pairs as defined above. Treat the union of all these edges as a single directed graph and determine whether this combined graph contains any cycle.

Describe the approach, data structures, and algorithm you would use, and analyze the time and space complexity.