Compute an Exact Distributed Median with Limited Communication

Compute the exact median of values distributed across workers. Explain partitioning, parallel work, and communication cost using send/receive-style primitives. Compare the task with computing a distributed mode. ### Constraints & Assumptions The source does not define numeric type, even-size behavior, or transport API. State them. For one concrete exact protocol, assume finite integers in stable worker partitions; use the middle value for odd N and the mean of the two middle values for even N. Define empty-input behavior separately. ### Clarifying Questions Are local partitions sorted? Is the value range bounded? Must the result be exact? How expensive are communication rounds relative to bytes? Can local data be rescanned or sorted? ### What a Strong Answer Covers Global rank selection, an exact communication protocol, local counting/search work, even-size handling, and snapshot and failure consistency. ### Follow-up Questions Why is the median of local medians generally wrong? How would you change the protocol for non-integer ordered values? When might gathering a small dataset be simpler and faster?