If each directory entry maintains a byte of information for each CPU, if our CC-NUMA system has S memory blocks, and the system has T nodes, provide an equation that expresses the amount of memory that will be present in each directory.

In a CC-NUMA shared memory system, CPUs and physical memory are divided across compute nodes. Each CPU has local caches. To maintain the coherency of memory, we can add status bits into each cache block, or we can introduce dedicated memory directories. Using directories, each node provides a dedicated hardware table for managing the status of every block of memory that is "local" to that node. The size of each directory is a function of the size of the CC-NUMA shared space (an entry is provided for each block of memory local to a node). If we store coherency information in the cache, we add this information to every cache in every system (i.e., the amount of storage space is a function of the number of cache lines available in all caches). In the following problems, assume that all nodes have the same number of CPUs and the same amount memory (i.e., CPUs and memory are evenly divided between the nodes of the CC-NUMA machine).