For high-speed data processing in a Java-based Hadoop application, the combination of Apache Spark with Apache Kafka is most effective. Spark provides fast in-memory data processing, and Kafka ensures high-throughput, fault-tolerant data streaming.

The MapReduce Shuffle phase contributes to data processing efficiency by performing data sorting. During this phase, the output of the Map tasks is sorted and partitioned based on keys, ensuring that the data is grouped appropriately before reaching the Reduce tasks. Sorting facilitates faster data processing during the subsequent Reduce phase.

When tuning a Hadoop cluster, optimizing the Input Split Size is crucial for MapReduce job performance. It determines the amount of data each mapper processes, and an appropriate split size helps in achieving better parallelism and efficiency in job execution.

A Hadoop administrator should prioritize data in different HDFS directories based on the replication factor. Critical data should have a higher replication factor to ensure availability and fault tolerance in the event of node failures.

Parquet is a highly efficient file format in Hadoop designed for fast data serialization and deserialization. It is columnar-oriented, supports schema evolution, and is optimized for both compression and performance.

The ResourceManager in YARN architecture is responsible for allocating system resources to different applications running on the Hadoop cluster. It keeps track of available resources and schedules tasks based on the requirements of the applications.

The format of input data is crucial in Hadoop application development as it directly impacts job performance. Choosing the right input format, such as Hadoop's preferred formats like SequenceFile or Avro, can enhance data processing efficiency.

Compaction is the process in HBase that involves combining smaller files into larger ones for efficiency. It helps in reducing the number of files and improving read and write performance in HBase.

Apache Storm integrates with the Hadoop ecosystem by running on Hadoop YARN. YARN (Yet Another Resource Negotiator) allows Storm to utilize Hadoop's resource management capabilities, making it easier to deploy and manage real-time processing applications alongside batch processing in a Hadoop cluster.

Sqoop is commonly used in Hadoop for importing data from relational databases into HDFS. It provides a command-line interface and supports the transfer of data between Hadoop and relational databases like MySQL, Oracle, and others.