Consider an inverted index containing, for each term, the posting list (i.e. the list of documents and occurrences within documents) for that term. The posting lists are accessed through a B+ tree with the terms serving as search keys. Each leaf of the B+ tree holds a sublist of alphabetically consecutive terms, and, with each term, a pointer to the posting list for that term. 

Part a. An artificially small example of a B+ tree is shown here (pdf). (Note only part of the tree is shown in detail.) What nodes of the example B+ tree are visited to find the posting list for "dune"?

Part b. Suppose there are 2 million terms for a collection of 32 million documents of total size 200 gigabytes. We would like each internal node of the B+ tree and each leaf of the B+ tree to fit in one 8-kilobyte page of the file system. Recall that a B+ tree has a parameter m called the order of the tree, and each internal node of a B+ tree has between m+1 and 2m+1 children (except the root, which has between 2 and 2m+1). Assume that each term is represented using 16 bytes, and each pointer to a child in the tree or to a posting list is represented using 8 bytes. Find a value for the order m of the B+ tree so that one 8 kilobyte page can be assigned to each internal node and leaf, and so that an internal node will fill, but not overflow, its page when it has 2m+1 children. If you need to make additional assumptions, state what assumptions you are making.

Part c. For your m of Part b, estimate the height of the B+ tree. (Giving a range of heights is fine.) Also estimate the amount of memory needed to store the tree, including leaves but not including the posting lists themselves.

Part d. Estimate the aggregate size of the posting lists.