Design distributed median and mode

You have a dataset with hundreds of billions of records spread across many machines. Design how to compute the global median and the global mode. Specify the data-partitioning strategy, local sketches or summaries, how reducers aggregate partial results, and how you minimize network I/O. Compare exact vs approximate approaches (e.g., quantile sketches, heavy-hitter sketches), discuss skew handling, fault tolerance, and incremental/streaming updates. Provide time, space, and communication complexity, and extend the design to a sliding-window median/mode over an unbounded stream.