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Chapter 14: Problem 6

What is the order \(\mathrm{p}\) of a \(\mathrm{B}^{+}\) -tree? Describe the structure of both internal and leaf nodes of a \(\mathrm{B}^{+}\) -tree.

Short Answer

Expert verified
The order p of a B+ tree is the maximum number of children that a node can have. The internal nodes of a B+ tree store upto p-1 keys and p pointers with each key separating the range of keys and the pointers pointing to child nodes. A node with k keys will have k+1 children. The leaf nodes store a maximum of p-1 keys and same number of pointers, with the pointers pointing to the actual data records. All leaf nodes are linked at the same level for ordered access to the records.

Step by step solution

01

Understanding the order of a B+ tree.

The order (p) of a B+ tree is defined as the maximum number of children that a node can have. The order of a B+ tree explains how many elements can be stored in each node of the tree.
02

Structure of internal nodes

In a B+ tree, an internal node can store up to p-1 keys and p pointers. Each key acts as a separation value which separates the range of keys and each pointer points to child nodes. A node with k keys will have k+1 children.
03

Structure of leaf nodes

Leaf nodes in a B+ tree contain a maximum of p-1 keys and the same number of pointers. Unlike internal nodes, pointers in leaf nodes actually point to the storage location (records) of the specific values or data. Furthermore, all leaf nodes are linked together at the same level to provide ordered access to the records.

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Most popular questions from this chapter

What are the differences among primary, secondary, and clustering indexes? How do these differences affect the ways in which these indexes are implemented? Which of the indexes are dense, and which are not?

Consider a disk with block size B "" 512 bytes. A block pointer is P "" 6 bytes long, and a record pointer is PR "" 7 bytes long. A file has r "" 30,000 EMPLOYEE records of fixed length. Each record has the following fields: NAME (30 bytes), SSN (9 bytes), DEPARTMENTCODE (9 bytes), ADDRESS (40 bytes), PHONE (9 bytes), BIRTHDATE (8 bytes), SEX (l byte), JOBCODE (4 bytes), SALARY (4 bytes, real number). An additional byte is used as a deletion marker. a. Calculate the record size R in bytes. b. Calculate the blocking factor bfr and the number of file blocks b, assuming an unspanned organization. c. Suppose that the file is ordered by the key field SSN and we want to construct a primary index on SSN. Calculate (i) the index blocking factor bfri (which is also the index fan-out fa); (ii) the number of first-level index entries and the number of first-level index blocks; (iii) the number of levels needed if we make it into a multilevel index; (iv) the total number of blocks required by the multilevel index; and (v) the number of block accesses needed to search for and retrieve a record from the file-given its SSN value-using the primary index. d. Suppose that the file is not ordered by the key field SSN and we want to construct a secondary index on SSN. Repeat the previous exercise (part c) for the secondary index and compare with the primary index. e. Suppose that the file is not ordered by the nonkey field DEPARTMENTCODE and we want to construct a secondary index on DEPARTMENTCODE, using option 3 of Section 14.1.3, with an extra level of indirection that stores record pointers. Assume there are 1000 distinct values of DEPARTMENTCODE and that the EMPLOYEE records are evenly distributed among these values. Calculate (0 the index blocking factor bfr, (which is also the index fan-out fa); (ii) the number of blocks needed by the level of indirection that stores record pointers; (iii) the number of firstlevel index entries and the number of first-level index blocks; (iv) the number of levels needed if we make it into a multilevel index; (v) the total number of blocks required by the multilevel index and the blocks used in the extra level of indirection; and (vi) the approximate number of block accesses needed to search for and retrieve all records in the file that have a specific DEPARTMENTCODE value, using the index. f. Suppose that the file is ordered by the nonkey field DEPARTMENTCODE and we want to construct a clustering index on DEPARTMENTCODE that uses block anchors (every new value of DEPARTMENTCODE starts at the beginning of a new block). Assume there are 1000 distinct values of DEPARTMENTCODE and that the EMPLOYEE records are evenly distributed among these values. Calculate (i) the index blocking factor bfr, (which is also the index fan-out fa); (ii) the number of first-level index entries and the number of first-level index blocks; (iii) the number of levels needed if we make it into a multilevel index; (iv) the total number of blocks required by the multilevel index; and (v) the number of block accesses needed to search for and retrieve all records in the file that have a specific DEPARTMENT CODE value, using the clustering index (assume that multiple blocks in a cluster are contiguous). g. Suppose that the file is not ordered by y the key field SSN and we want to construct a B+-tree access structure (index) on SSN. Calculate (i) the orders p and Pleof of the W -tree: (ii) the number of leaf-level blocks needed if blocks are approximately 69 percent full (rounded up for convenience); (iii) the number of levels needed if internal nodes are also 69 percent full (rounded up for convenience); (iv) the total number of blocks required by the W-tree; and (v) the number of block accesses needed to search for and retrieve a record from the file-given its SSN value-using the B+-tree. h. Repeat part g, but for a B-tree rather than for a B+-tree. Compare your results for the B-tree and for the W-tree.

How does a B-tree differ from a \(\mathrm{B}^{+}\) -tree? Why is a \(\mathrm{B}^{+}\) -tree usually preferred as an access structure to a data file?

What is the order \(p\) of a B-tree? Describe the structure of B-tree nodes.

Why can we have at most one primary or clustering index on a file, but several secondary indexes?
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