Problem

Meta network operations automation often pulls device state from multiple REST endpoints where some requests depend on others. Given a set of API calls, their dependencies, and a per-time-slot rate limit, compute the earliest execution schedule or report that the workflow is impossible.

Formal Specification

Implement a function schedule_api_calls(calls, dependencies, rate_limit).

• calls: a list of unique strings, where each string is a REST operation ID such as "get_fabric_health".
• dependencies: a list of pairs [a, b] meaning call a must complete before call b can start.
• rate_limit: a positive integer representing the maximum number of calls that can be executed in one time slot.

Return a list of lists, where each inner list contains the call IDs executed in that time slot. If the dependency graph contains a cycle, return [].

When multiple calls are available in the same slot, execute them in lexicographically sorted order for deterministic output.

Examples

Example 1 Input: calls = ["auth", "devices", "ports", "alerts"], dependencies = [["auth", "devices"], ["auth", "alerts"], ["devices", "ports"]], rate_limit = 2 Output: [["auth"], ["alerts", "devices"], ["ports"]] Explanation: auth must run first. Then both alerts and devices are ready and fit in the same slot.

Example 2 Input: calls = ["a", "b", "c"], dependencies = [["a", "b"], ["b", "c"], ["c", "a"]], rate_limit = 2 Output: [] Explanation: The dependencies form a cycle, so no valid schedule exists.

Constraints

• 1 <= len(calls) <= 10^5
• 0 <= len(dependencies) <= 2 * 10^5
• 1 <= rate_limit <= len(calls)
• All call IDs are unique strings of length 1 to 50
• Every dependency references valid call IDs