Distributed mode and median across worker shards

You are given a cluster of k worker nodes, numbered 0 to k - 1. Each worker holds its own unsorted slice (shard) of a large integer dataset. A worker can read only its own shard. Workers talk only by passing messages (send(dst, payload) / recv()), and you may assume a barrier between rounds. Worker 0 can act as the coordinator.

Design and implement a distributed algorithm that computes two statistics of the whole dataset:

1. Mode: the value that occurs most often. If several values tie, return the smallest one.
2. Median: the middle value of the sorted data. If the total count is even, return the average of the two middle values.

Rules:

• No single worker may ever receive all the raw data.
• Minimize network traffic. State how many integers your algorithm sends and how many communication rounds it needs.
• Shards can be empty, values can be negative, and the dataset can contain many duplicates.

You can simulate the cluster in one process, for example a Cluster(shards: list[list[int]]) class with mode() -> int and median() -> float, as long as every piece of data that moves between workers goes through an explicit message.

Example 1

shards = [[8, 1, 8], [6, 6, 0], [4]]
mode()   -> 6      # 6 and 8 both appear twice; 6 is smaller
median() -> 6.0    # sorted: 0 1 4 6 6 8 8

Example 2

shards = [[10, 3], [], [5, 12]]
mode()   -> 3      # all values appear once; 3 is the smallest
median() -> 7.5    # sorted: 3 5 10 12 -> (5 + 10) / 2

Expect follow-ups on very many distinct values, fewer communication rounds, skewed data, and worker failures.