Consider a database of fixed-length records, stored as an ordered file. The database has 25,000 records, with each record being 100 bytes, of which the primary key occupies 15 bytes. The data file is block-aligned in that each data record is fully contained within a block. The database is indexed by a primary index file, which is also stored as a block-aligned ordered file. The figure below depicts this indexing scheme. Suppose the block size of the file system is 1024 bytes, and a pointer to a block occupies 5 bytes. The system uses binary search on the index file to search for a record with a given key. You may assume that a binary search on an index file of $b$ blocks takes $\lceil \log_2 b \rceil$ block accesses in the worst case. Given a key, the number of block accesses required to identify the block in the data file that may contain a record with the key, in the worst case, is ________.

To find the number of block accesses required to identify the target block in the data file, we determine the number of blocks $b$ in the primary index file:

1. Data file blocks:

   • Record size, $R = 100\text{ bytes}$

   • Block size, $B = 1024\text{ bytes}$

   • Blocking factor for data file (records per block), $Bfr = \lfloor 1024 / 100 \rfloor = 10\text{ records/block}$

   • Total data records, $N = 25000$

   • Number of data blocks, $B_{\text{data}} = \lceil 25000 / 10 \rceil = 2500\text{ blocks}$

2. Primary index file blocks:

   • A primary index has one entry for each data block (sparse index).

   • Total index entries, $N_{\text{index}} = B_{\text{data}} = 2500$

   • Index entry size, $R_i = \text{Primary key size} + \text{Block pointer size} = 15 + 5 = 20\text{ bytes}$

   • Index blocking factor (entries per index block), $Bfr_i = \lfloor 1024 / 20 \rfloor = 51\text{ entries/block}$

   • Total index blocks, $b = \lceil 2500 / 51 \rceil = \lceil 49.0196 \rceil = 50\text{ blocks}$

3. Worst-case block accesses to identify the data block:

   • Searching the index file using binary search takes $\lceil \log_2 b \rceil$ block accesses.

   • Accesses $= \lceil \log_2 50 \rceil = 6$

Thus, the number of block accesses required to identify the block in the data file is 6.