Deduplication can introduce high indexing overhead and many studies have focused on reducing the indexing overhead in deduplication. In this question, we study the indexing issues in deduplication. Suppose that we fix the chunk size as $4$KB$,$ use SHA$-256$ for chunk fingerprinting, and store the chunks in $64-$bit address space. Note that the data units are assumed to be in power of $2.$

D$)$Sparse indexing further reduces the memory usage of indexing by leveraging chunk locality. Suppose that the segment size is $10$MB and the sampling rate is $1/64.$ What is the expected memory size for the sparse index to support the physical storage space of $10$TB$?$ Derive your answer.