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What defines a multiway search tree?
What defines a multiway search tree?
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?
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?
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?
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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?
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What is a characteristic of the leaves in a multiway search tree?
What is a characteristic of the leaves in a multiway search tree?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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 $v_i$?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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)?
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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?
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What must be preserved when inserting into a 2-3-4 tree?
What must be preserved when inserting into a 2-3-4 tree?
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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?
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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?
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Which statement correctly describes the leaves of a B-tree?
Which statement correctly describes the leaves of a B-tree?
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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?
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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?
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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?
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What is a primary reason B-trees are used in database systems?
What is a primary reason B-trees are used in database systems?
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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?
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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?
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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?
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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?
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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?
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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 $n \geq 1 + 2 + 2^2 + 2^3 + \ldots + 2^h$?
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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?
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Which statement accurately describes a key feature of 2-3-4 trees?
Which statement accurately describes a key feature of 2-3-4 trees?
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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?
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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?
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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?
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What happens when node 𝒖 is the root during an underflow condition?
What happens when node 𝒖 is the root during an underflow condition?
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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?
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What should be moved from parent 𝒖 during the transfer operation?
What should be moved from parent 𝒖 during the transfer operation?
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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?
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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?
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Which sequence correctly describes the steps of the transfer operation?
Which sequence correctly describes the steps of the transfer operation?
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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.
