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Questions and Answers
What defines a multiway search tree?
What defines a multiway search tree?
- Each internal node has at least two children and stores a single key.
- Each internal node has at least two children and stores d key-object items. (correct)
- Each internal node has exactly one child and stores d key-object items.
- Each internal node can have any number of children but stores no keys.
In a multiway search tree, the keys stored in the subtrees of the children follow which order?
In a multiway search tree, the keys stored in the subtrees of the children follow which order?
- Keys in the rightmost subtree are less than the least key in the parent.
- All keys in the left child are greater than the parent key.
- Keys in all subtrees are irrelevant to the structure of their parent node.
- Keys in the subtree of each child node follow the comparison based on the parent keys. (correct)
What is the minimum number of children an internal node in a multiway search tree can have?
What is the minimum number of children an internal node in a multiway search tree can have?
- Three
- Zero
- Two (correct)
- One
What happens when reaching an external node during a search in a multiway search tree?
What happens when reaching an external node during a search in a multiway search tree?
What does the notation d represent in the context of a multiway search tree?
What does the notation d represent in the context of a multiway search tree?
What is a characteristic of the leaves in a multiway search tree?
What is a characteristic of the leaves in a multiway search tree?
Which of the following best describes the properties of a 2-3-4 tree?
Which of the following best describes the properties of a 2-3-4 tree?
What happens to the keys in a multiway search tree when nodes are added or removed?
What happens to the keys in a multiway search tree when nodes are added or removed?
If a multiway search tree has keys $k_1, k_2, ext{ and } k_{d-1}$, under what condition does the search terminate successfully?
If a multiway search tree has keys $k_1, k_2, ext{ and } k_{d-1}$, under what condition does the search terminate successfully?
In a 2-3-4 tree, how are internal nodes categorized based on the number of children?
In a 2-3-4 tree, how are internal nodes categorized based on the number of children?
What is an important feature of the structure of keys in a multiway search tree?
What is an important feature of the structure of keys in a multiway search tree?
In a multiway search tree, how are the key-object items structured in relation to the node's children?
In a multiway search tree, how are the key-object items structured in relation to the node's children?
What is the expected height of a 2-3-4 tree storing $n$ items?
What is the expected height of a 2-3-4 tree storing $n$ items?
At which state should the search in a multiway search tree continue in the child node $v_i$?
At which state should the search in a multiway search tree continue in the child node $v_i$?
What occurs during an unsuccessful search for a key not present in a multiway search tree?
What occurs during an unsuccessful search for a key not present in a multiway search tree?
If the search key is greater than $k_{d-1}$ in a multiway search tree, where does the search proceed?
If the search key is greater than $k_{d-1}$ in a multiway search tree, where does the search proceed?
What is the primary action taken when inserting a new item into a 2-3-4 tree?
What is the primary action taken when inserting a new item into a 2-3-4 tree?
In the context of a 2-3-4 tree, what does an overflow at a node signify?
In the context of a 2-3-4 tree, what does an overflow at a node signify?
What happens to the keys during a split operation in a 2-3-4 tree?
What happens to the keys during a split operation in a 2-3-4 tree?
When an overflow occurs at the leaf level in a 2-3-4 tree, which operation must be conducted?
When an overflow occurs at the leaf level in a 2-3-4 tree, which operation must be conducted?
How many keys can a node in a 2-3-4 tree hold at maximum?
How many keys can a node in a 2-3-4 tree hold at maximum?
What is the result of splitting a node with keys (𝑘𝟏, 𝑘𝟐, 𝑘𝟑, 𝑘𝟒)?
What is the result of splitting a node with keys (𝑘𝟏, 𝑘𝟐, 𝑘𝟑, 𝑘𝟒)?
What key happens to be inserted into the parent during the split operation when a node has an overflow?
What key happens to be inserted into the parent during the split operation when a node has an overflow?
What must be preserved when inserting into a 2-3-4 tree?
What must be preserved when inserting into a 2-3-4 tree?
What is the maximum number of children a non-leaf node can have in a B-tree of order 𝑑?
What is the maximum number of children a non-leaf node can have in a B-tree of order 𝑑?
What is the minimum number of children a non-root node must have in a B-tree of order 𝑑?
What is the minimum number of children a non-root node must have in a B-tree of order 𝑑?
Which statement correctly describes the leaves of a B-tree?
Which statement correctly describes the leaves of a B-tree?
If a B-tree has an order of 6, what is the maximum number of keys it can have in one node?
If a B-tree has an order of 6, what is the maximum number of keys it can have in one node?
What happens to the height of a B-tree as the order increases?
What happens to the height of a B-tree as the order increases?
Which of the following is a reason to avoid using the highest order B-tree possible?
Which of the following is a reason to avoid using the highest order B-tree possible?
What is a primary reason B-trees are used in database systems?
What is a primary reason B-trees are used in database systems?
How does the structure of a B-tree assist in managing large databases?
How does the structure of a B-tree assist in managing large databases?
What is the maximum height of a 2-3-4 tree storing n items?
What is the maximum height of a 2-3-4 tree storing n items?
What does searching for an item in a 2-3-4 tree require in terms of time complexity?
What does searching for an item in a 2-3-4 tree require in terms of time complexity?
Which of the following implementations can be used for a 2-3-4 tree?
Which of the following implementations can be used for a 2-3-4 tree?
If depth h has at least 2h items, how many items can be present at depth h+1?
If depth h has at least 2h items, how many items can be present at depth h+1?
What is the implication of the equation $n \geq 1 + 2 + 2^2 + 2^3 + \ldots + 2^h$?
What is the implication of the equation $n \geq 1 + 2 + 2^2 + 2^3 + \ldots + 2^h$?
What is the effect of inserting a new item k into a 2-3-4 tree?
What is the effect of inserting a new item k into a 2-3-4 tree?
Which statement accurately describes a key feature of 2-3-4 trees?
Which statement accurately describes a key feature of 2-3-4 trees?
What method can be used to search for a key within a node in a 2-3-4 tree?
What method can be used to search for a key within a node in a 2-3-4 tree?
What is the first step in handling an underflow at node 𝒗 when its parent 𝒖 has an adjacent sibling 𝒘 that is a 3-node or a 4-node?
What is the first step in handling an underflow at node 𝒗 when its parent 𝒖 has an adjacent sibling 𝒘 that is a 3-node or a 4-node?
In the context of a 2-3-4 tree, what occurs after the transfer operation is complete when handling an underflow?
In the context of a 2-3-4 tree, what occurs after the transfer operation is complete when handling an underflow?
What happens when node 𝒖 is the root during an underflow condition?
What happens when node 𝒖 is the root during an underflow condition?
Which of the following statements accurately describes a situation of an underflow in a 2-3-4 tree?
Which of the following statements accurately describes a situation of an underflow in a 2-3-4 tree?
What should be moved from parent 𝒖 during the transfer operation?
What should be moved from parent 𝒖 during the transfer operation?
Which scenario does NOT represent a valid action during a transfer operation when handling an underflow?
Which scenario does NOT represent a valid action during a transfer operation when handling an underflow?
What type of node must sibling 𝒘 be for the specific case of handling underflow at node 𝒗 as described?
What type of node must sibling 𝒘 be for the specific case of handling underflow at node 𝒗 as described?
Which sequence correctly describes the steps of the transfer operation?
Which sequence correctly describes the steps of the transfer operation?
Flashcards
Multiway Search Tree
Multiway Search Tree
An ordered tree where each internal node has at least two children and stores 𝑑 − 1 keys.
Internal Node
Internal Node
A node within a multiway search tree that is not a leaf.
Keys in Subtrees
Keys in Subtrees
Keys in the subtree of child 𝑣𝑖 are between 𝑘𝑖−1 and 𝑘𝑖 .
Leaf Node
Leaf Node
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𝑑
𝑑
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𝑘𝑖
𝑘𝑖
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Ordered Tree
Ordered Tree
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2-3-4 Tree
2-3-4 Tree
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Node-Size Property
Node-Size Property
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Depth Property
Depth Property
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Multiway Search Tree Search
Multiway Search Tree Search
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Height of a 2-3-4 Tree (Theorem)
Height of a 2-3-4 Tree (Theorem)
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Unsuccessful Search
Unsuccessful Search
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Successful Search
Successful Search
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2-3-4 Tree Height
2-3-4 Tree Height
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2-3-4 Tree Search Time
2-3-4 Tree Search Time
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2-3-4 Tree Structure
2-3-4 Tree Structure
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2-3-4 Tree Insertion
2-3-4 Tree Insertion
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Tree Depth Property
Tree Depth Property
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Number of Items and Depth
Number of Items and Depth
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Red-Black Trees
Red-Black Trees
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BST
BST
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Split Operation
Split Operation
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Overflow at the Bottom
Overflow at the Bottom
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Overflow at Higher Levels
Overflow at Higher Levels
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Maintaining Depth Property During Overflow
Maintaining Depth Property During Overflow
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Example: Inserting 31 & 32
Example: Inserting 31 & 32
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What is a B-tree?
What is a B-tree?
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What is the order of a B-tree?
What is the order of a B-tree?
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Why are B-trees useful for databases?
Why are B-trees useful for databases?
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What is the height of a B-tree?
What is the height of a B-tree?
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What is the impact of order on B-tree performance?
What is the impact of order on B-tree performance?
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What is the key advantage of using B-trees for large databases?
What is the key advantage of using B-trees for large databases?
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Why is it not always good to use the highest possible order for a B-tree?
Why is it not always good to use the highest possible order for a B-tree?
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What is the relationship between B-trees and 2-3-4 trees?
What is the relationship between B-trees and 2-3-4 trees?
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2-3-4 Tree Deletion (Underflow)
2-3-4 Tree Deletion (Underflow)
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Case 1: Sibling has extra keys
Case 1: Sibling has extra keys
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Case 2: Sibling is just a 3-node
Case 2: Sibling is just a 3-node
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Transfer Operation: Redistributing keys
Transfer Operation: Redistributing keys
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Underflow: Root Node
Underflow: Root Node
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Deletion Process Summary
Deletion Process Summary
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Bottom-up Propagation
Bottom-up Propagation
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Maintaining Balance
Maintaining Balance
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Study Notes
CICS 210-02 Data Structures Lecture 16
- Course: CICS 210-02 Data Structures
- Lecture: 16
- Date: 11/04/2024
- Instructor: MJ Golin
- Semester: Fall 2024
Outline
- Multiway Search Trees
- 2-3-4 Trees
- Bottom Up Insertion into 2-3-4 trees
- Deletion from a 2-3-4 Tree
- Top Down Insertion into a 2-3-4 tree
- B trees
Multiway Search Trees
- A multi-way search tree is an ordered tree.
- Each internal node has at least two children and stores d-1 key-object items (ki, oi). d = number of children.
- Keys in the subtree of v₁ are less than k₁
- Keys in the subtree of vi are between ki−1 and k₁ ( i = 2, 3, ..., d – 1)
- Keys in the subtree of va are greater than kd-1
- The leaves store no items and serve as placeholders.
- Searching in a multi-way search tree is similar to searching in a binary search tree.
- Multi-way searching starts at the root
- k = ki (i = 2, 3, ..., d - 1), the search terminates successfully
- k < k₁: continue search in child v1
- ki-1 < k < ki (i = 2, 3, ..., d - 1): continue search in child vi
- k > kd-1: continue search in child vd
- Reaching an external node terminates search unsuccessfully
2-3-4 Trees
- A 2-3-4 tree is a multi-way search tree with specific properties.
- Node-Size Property: Every internal node has at most four children. Leaves contain 1, 2, or 3 keys.
- Depth Property: All external nodes have the same depth.
- A 2-node, 3-node, or 4-node is an internal node depending on the number of children.
- Height of a 2-3-4 tree storing n items = O(log n).
- Searching for an item in a 2-3-4 tree takes O(log n) time.
- 2-3-4 tree implementation hasn't been fully specified.
- One representation is via (restricted) Binary Search Trees with a special type of edge coloring (red-black trees).
Height of a 2-3-4 Tree
- A 2-3-4 tree containing n items has a height of O(log n).
- The number of items at depth i (i = 0, 1, 2, ..., h) is at least 2^(i+1) - 1.
- So, n ≥ 2 ^(h+1) – 1, leading to h ≤ log₂(n+1) - 1.
Bottom Up Insertion into a 2-3-4 Tree
- Insert a new item (k) into the leaf that should contain it.
- Preserve the depth property.
- Maintain the node-size property.
- If an overflow occurs (node has 4 keys), split and rebalance.
Bottom Up Deletion from a 2-3-4 Tree
- Delete a key from a leaf node (if only key, there will be a few more cases).
- Remove the key and associated null pointers.
- If a node is not a leaf: replace entry with its in-order successor (or predecessor); then delete the latter entry.
- If underflow occurs: perform a fusion operation or transfer operation to maintain the node-size property and depth property.
Top Down Insertion into a 2-3-4 Tree
- Insert a key (k) top-down.
- Identify insertion path.
- Preemptively split 4-nodes in the path.
B-trees
- A B-tree of order d is a multi-way search tree with these properties:
- Every node has at most d children.
- Every non-root node has at least ⌈d/2⌉ children.
- The root has at least 2 children, unless it's a leaf.
- All leaves appear at the same level.
- A non-leaf node v with k children contains k - 1 keys.
- B-trees are generalizations of 2-3-4 trees.
- B-trees have logn height.
- Searching in B-trees takes O(log n) time.
- B-trees are used in databases and file systems to manage large amounts of data stored on external media (like hard drives).
- B+ trees are a common variant in databases to store records or objects using pointers that help organize the data.
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Description
This quiz covers the key concepts from Lecture 16 of the CICS 210-02 Data Structures course. Topics include multiway search trees, 2-3-4 trees, and B trees, focusing on their insertion and deletion methods. Test your understanding of these advanced data structures and their applications.
