Data Structures Chapter 4: Trees in External Memory

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Questions and Answers

What is the primary use of M-ary search trees in Secondary (External) Memory (SM)?

Representing M-ary search trees in Secondary (External) Memory (SM)

In an M-ary search tree of order N, what is the maximum number of ordered values that a node can contain?

N-1

What does the degree of a node represent in an M-ary search tree?

The current number of children

What is the relationship between the degree of a node and the number of values it contains?

The current number of values is always equal to degree - 1. Signup and view all the answers

What is the minimum degree of a node in an M-ary search tree?

2 Signup and view all the answers

All values in the subtree of Child1 in an M-ary search tree are greater than val1.

False Signup and view all the answers

All values in the subtree of Childj in an M-ary search tree are greater than valj-1 and less than valj, where j is between 2 and degree-1.

True Signup and view all the answers

All values in the subtree of Childdegree in an M-ary search tree are less than valdegree-1.

False Signup and view all the answers

What is the tree depth or height of an M-ary search tree?

The level of the farthest leaf Signup and view all the answers

What is the 'Top-Down Property' of an M-ary search tree?

All internal nodes are 100% full (optionally verified). Signup and view all the answers

How are blocks linked in a file organized as an M-ary search tree?

Using an M-ary tree structure Signup and view all the answers

Where is the root block number stored in a file organized as an M-ary search tree?

The Header Block Signup and view all the answers

What is the type of the Tval variable used as an index in SM towards a data file in M-ary search trees?

struct (key, addr) Signup and view all the answers

What is the type of the Tval variable used as a data file organized using a tree in SM in M-ary search trees?

Trecord Signup and view all the answers

How does searching for a value C in an M-ary search tree begin?

From the root node P Signup and view all the answers

In searching for a value C in an M-ary search tree, when does the search stop successfully?

If C exists in node P Signup and view all the answers

If the searched value C is not found in node P during a search in an M-ary search tree, what is the next step?

Let k be the actual position of C in P (values must remain ordered) Signup and view all the answers

If Childk in an M-ary search tree is not null, what is the next step?

P ← Childk; goto 1 Signup and view all the answers

If Childk is null in an M-ary search tree, what is the result?

The search stops with failure Signup and view all the answers

What is the variable type used for storing data related to blocks in the Search Algorithm?

TypeBlock Signup and view all the answers

In the Search Algorithm, what is the purpose of the variable numB?

The Root node number Signup and view all the answers

In the Search Algorithm, what is the initial value assigned to the variable found?

false Signup and view all the answers

In the Search Algorithm, how is the block data read?

readBlock(F, i, buf) Signup and view all the answers

In the Search Algorithm, where is the value c searched for?

buf.val[] Signup and view all the answers

In the Search Algorithm, what action is taken if the desired value 'c' is not found within the current block?

i ← buf.Child[pos] Signup and view all the answers

What is the purpose of the 'P' variable in the Search Algorithm?

A stack of < TypeBlock, integer, integer > Signup and view all the answers

In the Search Algorithm, what does the function 'CreateStack(P)' do?

Initializes a stack for P Signup and view all the answers

What is the value of 'found' at the beginning of the Search Algorithm?

false Signup and view all the answers

What value is assigned to the variable 'i' at the start of the Search Algorithm?

numB Signup and view all the answers

How is the block associated with 'i' in the Search Algorithm accessed and stored in 'buf'?

readBlock(F, i, buf) Signup and view all the answers

How is the target value 'c' searched for within the 'buf' variable in the Search Algorithm?

An internal search is performed for 'c' in buf.val[] Signup and view all the answers

What happens in the Search Algorithm if the target value 'c' is not found within the current block 'buf'?

The stack 'P' is updated with information about the current block, and the search process moves to the next relevant block Signup and view all the answers

What does 'inorder traversal' refer to in the context of M-ary search trees?

Processing the values within a tree in an ascending order Signup and view all the answers

Where is the smallest value in the tree typically located?

At the first position of the leftmost node Signup and view all the answers

How do you locate the inorder successor of a value in an M-ary search tree?

By examining the right child of the value's node and considering specific conditions based on the location within the node and whether the right child exists Signup and view all the answers

What is the rule for determining the next inorder value when the right child of a value is null and the value's position is less than degree - 1?

The next inorder value is the value immediately to the right of the current value within the same node Signup and view all the answers

What is the rule for determining the next inorder value when the right child of a value is null and the value's position is equal to degree - 1?

The next inorder value is the smallest value in the subtree rooted at the rightmost child node Signup and view all the answers

What is the rule for determining the next inorder value when the right child of a value is not null?

The next inorder value is the smallest value in the subtree rooted at the right child node Signup and view all the answers

What is an interval query in the context of M-ary search trees?

A query that seeks to retrieve all values within a specified range or interval Signup and view all the answers

What are the hints provided for implementing the algorithm for an interval query in M-ary search trees?

Use a stack to locate the start value of the interval, then use the stack to access the next elements in inorder traversal until you reach or exceed the end value of the interval Signup and view all the answers

Under what condition is a new value inserted into a node during the insertion process in M-ary search trees?

When the last accessed node through the search algorithm is not full Signup and view all the answers

What action is taken to insert a new value when the last accessed node is full during the insertion process in M-ary search trees?

A new block is allocated, the new value is placed into the block, and the block is linked into the tree Signup and view all the answers

What is the primary objective of the insertion mechanism in M-ary search trees?

To insert a new value into the tree while preserving the 'Top-Down' property and maintaining the sorted order of values within the tree Signup and view all the answers

What is one potential drawback of the insertion mechanism in M-ary search trees?

It can lead to a significant imbalance in the tree structure Signup and view all the answers

What is a possible solution or mitigation strategy for addressing imbalances in M-ary search trees caused by insertion processes?

Periodic reorganizations or re-balancing of the tree structure Signup and view all the answers

What are the two main approaches to value deletion in an M-ary search tree?

Logical deletion and physical deletion Signup and view all the answers

How is logical deletion implemented in an M-ary search tree?

By adding a logical deletion indicator for each value in the tree Signup and view all the answers

What does physical deletion involve in an M-ary search tree?

Physically removing the value from the tree structure Signup and view all the answers

When deleting a value in an M-ary search tree with physical deletion, what is the first step?

Searching for the node that contains the value Signup and view all the answers

When deleting a value in an M-ary search tree, what is the action taken if the value is found in a leaf node?

The value is removed from the node, if the node becomes empty, the node is also removed Signup and view all the answers

When deleting a value from an M-ary search tree with physical deletion, what is the first step if the value is found in an internal node?

Find the next or previous inorder value and its node Signup and view all the answers

After finding the next or previous inorder value in an M-ary search tree during the physical deletion process, what is the next step?

Replace the original value in the node with the successor/predecessor value found through inorder traversal Signup and view all the answers

After replacing the original value with the successor/predecessor value in an M-ary search tree during the physical deletion process, what is the final step?

The process repeats from step 2, re-examining the node where the value was replaced, to potentially continue the deletion cascade Signup and view all the answers

Study Notes

Chapter 4: Trees in External Memory

Organizing Files Using M-ary Search Trees:
- File blocks can be organized as a tree of blocks.
- Useful for secondary (external) memory.
- An M-ary search tree of order N has nodes containing at most N-1 ordered values and N children.
- Node degree represents the current number of children.
- Degree is at least 2 and at most N.
- Properties of M-ary search tree nodes :
  - All values in subtree of child 1 are less than val1.
  - All values in subtree of child j (j in [2, degree-1]) are greater than val(j-1) and less than val(j).
  - All values in subtree of child (degree) are greater than val(degree-1).
Organizing Files Using B-Trees:
- Definitions and Properties
- Algorithms: Insertion, Deletion, Optimized Loading
- Variants: B+-Trees, Prefix B+-Trees, B*-Trees.

Example of an M-ary Search Tree (Order = 5)

Root Node: i (32, 33, 40, 99)
Internal Nodes: Nodes containing more than one value (e.g., a, h, k, d, f).
Leaf Nodes: Nodes with no children (e.g., b, c, g, j).

Uses of M-ary Search Trees

Index:
- Used as an index in secondary memory for data files.
Data File:
- Data file can be organized as a tree in secondary memory.

Search Algorithm

Starts from the root node.
If the value is found, the search stops successfully.
If not, follows the appropriate child branch based on value order.
Search stops with failure if relevant child is not available.

Search Algorithm (Detailed)

Algorithm uses a stack to keep track of visited nodes.
Opens a file and gets the header.
Loops until the value is found or all node paths are explored.
Reads the block and searches for the value.
If not found, pushes the current block, index, and position onto the stack.
Continues until the value is found or the stack is empty.

Inorder Traversal

Process: Visits nodes in ascending order.
Result: Values are visited in ascending order.

Next Inorder

Process: Finds the next value in an inorder traversal from a given value.
Conditions: If the current child position plus one is less than the degree number and the child is not -1, the next value will be in the next child.
Algorithm: Details on finding the successor.
If a value is an internal node, finds the successor value in the branch.
If a value is a leaf node and the right child is -1, and it's the last value in the node,the next value depends on its position in the tree
- If a value is not an internal node the next value is in next node depending on its position.

Insertion Mechanism

Full Node: If the last accessed node is full, a new block is allocated and added to the tree.
It adds the value into the previously allocated block.

Deletion Mechanism

Logical Deletion: Adds a logical deletion indicator to the node containing the value to be deleted.
Physical Deletion:
- Searches for the node containing the value to be deleted.
- If node is a leaf, deletes using shifts. If it's empty free it and go to the parent.
- If node is not a leaf, find the next/previous value using an inorder traversal, replace the value to be deleted with the next/previous value, and repeat the process for the next Node recursively.

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Description

Explore the concepts of M-ary search trees and B-trees in this quiz for Chapter 4 on Trees in External Memory. Understand how these data structures organize files in secondary memory and learn about their properties, algorithms, and variants. Test your knowledge on topics such as insertion and deletion in B-trees and the structure of M-ary search trees.