B-trees for databases in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

What is the primary purpose of a B-tree in databases?

Choose the best description of why B-trees are used in database systems.

A To maintain sorted data and allow efficient insertion, deletion, and search operations.

B To compress data to save storage space in databases.

C To store data in a way that allows fast sequential access only.

D To encrypt data for security purposes during storage.

💡 Hint

Think about how databases need to quickly find and update records.

B-trees keep data sorted and balanced, enabling fast search, insert, and delete operations, which is essential for database indexing.

📋 Factual

intermediate

What is the minimum number of keys a non-root node in a B-tree of order 4 must have?

In a B-tree of order 4 (maximum 4 children per node), how many keys must a non-root node have at minimum?

A 2

B 1

C 3

D 4

💡 Hint

Recall that the minimum number of keys is about half the maximum children minus one.

For a B-tree of order 4, each node can have up to 4 children and thus up to 3 keys. The minimum number of keys in a non-root node is ceil(4/2)-1 = 1.

🔍 Analysis

advanced

What happens during a node split in a B-tree insertion?

When inserting a key causes a node to overflow in a B-tree, what is the correct sequence of actions during the node split?

A 1,3,2,4

B 1,2,4,3

C 2,1,3,4

D 1,2,3,4

💡 Hint

Think about the order of splitting keys and updating the parent.

First, find the median key to split the node. Then create two nodes from keys less and greater than median. Next, move the median key up to the parent. Finally, update the parent's children pointers.

❓ Comparison

advanced

How does a B-tree differ from a binary search tree in database indexing?

Choose the statement that best explains the difference between B-trees and binary search trees (BST) in the context of databases.

A B-trees do not maintain sorted order of keys, unlike BSTs.

B B-trees are always binary, but BSTs can have multiple children per node.

C B-trees store multiple keys per node and keep the tree balanced, while BSTs store one key per node and can become unbalanced.

D BSTs are optimized for disk storage, while B-trees are only used in memory.

💡 Hint

Consider how nodes and balance affect search speed in large datasets.

B-trees store multiple keys per node and keep the tree balanced to minimize disk reads. BSTs store one key per node and can become unbalanced, leading to slower searches.

❓ Reasoning

expert

Why do B-trees reduce disk I/O operations in databases?

Explain why B-trees are designed to reduce the number of disk input/output (I/O) operations when searching or updating data in databases.

A Because B-trees store many keys in each node, they reduce the tree height, minimizing the number of disk reads needed to find a key.

B Because B-trees compress data, they reduce the amount of data read from disk.

C Because B-trees use hashing internally, they directly access data without searching.

D Because B-trees store data in random order, they avoid sequential disk reads.

💡 Hint

Think about how storing multiple keys per node affects tree height and disk access.

B-trees store multiple keys per node, which reduces the height of the tree. A shorter tree means fewer nodes to read from disk during search or update, thus reducing disk I/O operations.

Practice

1. What is the main purpose of a B-tree in databases?

easy

A. To compress data to save disk space

B. To keep data sorted and balanced for fast searching and updating

C. To encrypt data for security

D. To store data in a linear list for quick access

Solution

1. Step 1: Understand B-tree structure

   B-trees organize data in a sorted and balanced tree structure.

2. Step 2: Identify the purpose in databases

   This structure allows fast searching, insertion, and deletion by minimizing tree height and disk reads.

3. Final Answer:

   To keep data sorted and balanced for fast searching and updating -> Option B

4. Quick Check:

   B-tree purpose = fast, balanced data access [OK]

Hint: B-trees balance data for speed, not encryption or compression [OK]

Common Mistakes:

• Confusing B-trees with simple lists
• Thinking B-trees encrypt data
• Assuming B-trees compress data

2. Which of the following correctly describes a property of B-tree nodes?

easy

A. Each node can contain multiple keys and multiple children

B. Nodes contain only keys but no children

C. Each node contains exactly one key and two children

D. Nodes are always leaf nodes without children

Solution

1. Step 1: Recall B-tree node structure

   B-tree nodes hold multiple keys to reduce tree height.

2. Step 2: Understand children count

   Each node has one more child than the number of keys it holds.

3. Final Answer:

   Each node can contain multiple keys and multiple children -> Option A

4. Quick Check:

   Multiple keys and children per node = C [OK]

Hint: B-tree nodes hold many keys and children, not just one [OK]

Common Mistakes:

• Thinking nodes have only one key
• Assuming nodes have no children
• Confusing B-trees with binary trees

3. Consider a B-tree of order 3 (each node can have at most 2 keys). If a node currently has keys [10, 20] and a new key 15 is inserted, what will happen?

medium

A. The key 15 will be discarded as duplicates are not allowed

B. The node will hold keys [10, 15, 20] without splitting

C. The node will split because it exceeds the max keys, promoting a key up

D. The tree will become unbalanced and require rebalancing later

Solution

1. Step 1: Check node capacity for order 3 B-tree

   Max keys per node = 2. Current keys are [10, 20]. Inserting 15 adds a third key.

2. Step 2: Understand insertion rules

   When a node exceeds max keys, it splits and promotes the middle key to the parent.

3. Final Answer:

   The node will split because it exceeds the max keys, promoting a key up -> Option C

4. Quick Check:

   Node over capacity causes split and promotion [OK]

Hint: If keys exceed max, node splits and middle key moves up [OK]

Common Mistakes:

• Thinking node can hold 3 keys without splitting
• Assuming duplicates are discarded here
• Believing tree becomes unbalanced without immediate fix

4. A B-tree node is supposed to split when it exceeds its maximum keys. Which of the following is a common mistake that can cause the tree to become unbalanced after insertion?

medium

A. Not promoting the middle key to the parent node after splitting

B. Always inserting keys in sorted order

C. Using nodes with multiple keys and children

D. Searching for keys before insertion

Solution

1. Step 1: Understand node splitting in B-trees

   When a node splits, the middle key must be promoted to keep the tree balanced.

2. Step 2: Identify the error impact

   If the middle key is not promoted, the tree structure breaks and becomes unbalanced.

3. Final Answer:

   Not promoting the middle key to the parent node after splitting -> Option A

4. Quick Check:

   Missing promotion causes imbalance [OK]

Hint: Always promote middle key on split to keep balance [OK]

Common Mistakes:

• Skipping promotion step after split
• Thinking sorted insertion causes imbalance
• Confusing node structure rules

5. You have a B-tree of order 4 (max 3 keys per node). After several insertions, a leaf node has keys [5, 10, 15] and you want to insert 12. Describe the sequence of steps the B-tree will perform to maintain balance.

hard

A. Insert 12 and increase the node capacity temporarily

B. Discard 12 because the node is full

C. Insert 12 and rebalance by merging with sibling nodes without splitting

D. Insert 12 into the leaf node, then split the node and promote the middle key to the parent

Solution

1. Step 1: Attempt to insert 12 into leaf node

   The leaf node has max 3 keys [5, 10, 15]. Inserting 12 adds a 4th key, exceeding capacity.

2. Step 2: Split the node and promote middle key

   The node splits into two nodes, and the middle key (10) is promoted to the parent to maintain balance.

3. Final Answer:

   Insert 12 into the leaf node, then split the node and promote the middle key to the parent -> Option D

4. Quick Check:

   Insert, split, promote middle key = balanced B-tree [OK]

Hint: Insert, then split and promote middle key if node is full [OK]

Common Mistakes:

• Discarding keys when node is full
• Merging instead of splitting on insertion
• Temporarily increasing node capacity