Link PTR to NEXT, making NEXT the parent of the node pointed by PTR. PREV PTR NEXT Head [H] →(12 4 8 18 19 10 14 14 (36 27 12 29 18 21 12 18 21 (31) 36 (24 19 (39 Figure 12.19(e)
Link PTR to NEXT, making NEXT the parent of the node pointed by PTR. PREV PTR NEXT Head [H] →(12 4 8 18 19 10 14 14 (36 27 12 29 18 21 12 18 21 (31) 36 (24 19 (39 Figure 12.19(e)
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![### Explanation of Binomial Heap Linking Process
In these diagrams, we demonstrate the process of linking nodes within a binomial heap. A binomial heap is a data structure that acts as a collection of binomial trees.
#### Figure 12.19(e)
- **Description**: This figure illustrates the configuration where `PTR` and `NEXT` are two consecutive nodes in the heap's linked list.
- **Action**: The operation demonstrates linking `PTR` to `NEXT`, effectively making `NEXT` the parent of the node pointed to by `PTR`.
- **Details**:
- The linked list begins with a `Head [H]` that points to `PREV` node with the value `12`.
- `PREV` is followed by node `PTR` with the value `4`.
- `PTR` connects to `NEXT` with the value `9`, which further connects to node `8`.
- `NEXT`'s children are `17`, `10`, `14`, and `36`, while `PTR`'s subtree remains unchanged with nodes like `19`, `10`, `12`, `29`, etc.
#### Figure 12.19(f)
- **Description**: This figure is a continuation of the linking process shown previously.
- **Action**: Here, `NEXT` is linked to `PTR`, making `PTR` the new parent of the node initially pointed to by `NEXT`.
- **Details**:
- The structure starts with the same `Head [H]` pointing to `PREV` which remains `12`.
- In this step, `PTR` (node `4`) assumes the parent role for what was `NEXT` (node `9` previously under `8`).
- `PTR` now has an expanded subtree that includes the former `NEXT` children `17`, `10`, `14`, `36`, and so forth, integrating with its own previous subtree of nodes `9`, `19`, `7`, `27`, etc.
These figures illustrate different linking strategies used to maintain the properties of a binomial heap, ensuring the heap remains properly organized as operations are performed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac965a5e-260b-4b5a-b5dd-dbb137feedb3%2F889390c2-834f-46a3-9ec1-e89c136e2c72%2Fve26s6_processed.png&w=3840&q=75)
Transcribed Image Text:### Explanation of Binomial Heap Linking Process
In these diagrams, we demonstrate the process of linking nodes within a binomial heap. A binomial heap is a data structure that acts as a collection of binomial trees.
#### Figure 12.19(e)
- **Description**: This figure illustrates the configuration where `PTR` and `NEXT` are two consecutive nodes in the heap's linked list.
- **Action**: The operation demonstrates linking `PTR` to `NEXT`, effectively making `NEXT` the parent of the node pointed to by `PTR`.
- **Details**:
- The linked list begins with a `Head [H]` that points to `PREV` node with the value `12`.
- `PREV` is followed by node `PTR` with the value `4`.
- `PTR` connects to `NEXT` with the value `9`, which further connects to node `8`.
- `NEXT`'s children are `17`, `10`, `14`, and `36`, while `PTR`'s subtree remains unchanged with nodes like `19`, `10`, `12`, `29`, etc.
#### Figure 12.19(f)
- **Description**: This figure is a continuation of the linking process shown previously.
- **Action**: Here, `NEXT` is linked to `PTR`, making `PTR` the new parent of the node initially pointed to by `NEXT`.
- **Details**:
- The structure starts with the same `Head [H]` pointing to `PREV` which remains `12`.
- In this step, `PTR` (node `4`) assumes the parent role for what was `NEXT` (node `9` previously under `8`).
- `PTR` now has an expanded subtree that includes the former `NEXT` children `17`, `10`, `14`, `36`, and so forth, integrating with its own previous subtree of nodes `9`, `19`, `7`, `27`, etc.
These figures illustrate different linking strategies used to maintain the properties of a binomial heap, ensuring the heap remains properly organized as operations are performed.
![# How to Unite Two Binomial Heaps in C++
## Example 12.5 - Unite the Binomial Heaps
This example demonstrates how to merge two binomial heaps into one. You should implement a function in C++ that displays the unified binomial heap, using the following process as a guide.
### Initial Binomial Heaps
- **Head [H1]:** 12 → 7 → 9, with subtrees rooted at 27 and 10. Node 12 further connects to node 10, followed by node 14, and 12.
- **Head [H2]:** 18 → 4, with subtrees rooted at 19, 29, and 8. Node 4 connects to nodes 7, 10, 14, and 36. The root node 8 is connected directly to 36.
**Figure 12.19(a)** shows the initial state of the two heap structures.
### Step-by-Step Integration of Heaps
#### Step 1: Merging Heaps
After invoking `Merge_Binomial-Heap()`, the heaps are combined to form a new structure:
- **Head [H]:** 12 → 18 → 7 → 4 → 9 → 8
- The merged tree follows with nodes: 27, 19, 10, 12, 29, 18, 21, 12, 8 and ends with nodes 31, 36, 24, 19, and 39.
**Figure 12.19(b)** shows the heap structure after initial merging.
#### Step 2: Linking Nodes
- Connect the `NEXT` pointer to `PTR`, where `PTR` becomes the parent of the node indicated by `NEXT`.
**Updated Structure:**
- **Head [H]:** 12 → 7 → 4 → 9 → 8
- Internal subtrees: 18, 27, 19, 10, 14, 12, and a secondary chain of 29, 18, 21, 12, with terminal nodes 31, 36, 24, 19, and 39.
**Figure 12.19(c)** shows the tree after this linking step.
#### Step 3: Adjusting Pointers
- Now `PTR` represents the first of three roots of the same degree. Adjust the pointers:
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac965a5e-260b-4b5a-b5dd-dbb137feedb3%2F889390c2-834f-46a3-9ec1-e89c136e2c72%2Frovw2r_processed.png&w=3840&q=75)
Transcribed Image Text:# How to Unite Two Binomial Heaps in C++
## Example 12.5 - Unite the Binomial Heaps
This example demonstrates how to merge two binomial heaps into one. You should implement a function in C++ that displays the unified binomial heap, using the following process as a guide.
### Initial Binomial Heaps
- **Head [H1]:** 12 → 7 → 9, with subtrees rooted at 27 and 10. Node 12 further connects to node 10, followed by node 14, and 12.
- **Head [H2]:** 18 → 4, with subtrees rooted at 19, 29, and 8. Node 4 connects to nodes 7, 10, 14, and 36. The root node 8 is connected directly to 36.
**Figure 12.19(a)** shows the initial state of the two heap structures.
### Step-by-Step Integration of Heaps
#### Step 1: Merging Heaps
After invoking `Merge_Binomial-Heap()`, the heaps are combined to form a new structure:
- **Head [H]:** 12 → 18 → 7 → 4 → 9 → 8
- The merged tree follows with nodes: 27, 19, 10, 12, 29, 18, 21, 12, 8 and ends with nodes 31, 36, 24, 19, and 39.
**Figure 12.19(b)** shows the heap structure after initial merging.
#### Step 2: Linking Nodes
- Connect the `NEXT` pointer to `PTR`, where `PTR` becomes the parent of the node indicated by `NEXT`.
**Updated Structure:**
- **Head [H]:** 12 → 7 → 4 → 9 → 8
- Internal subtrees: 18, 27, 19, 10, 14, 12, and a secondary chain of 29, 18, 21, 12, with terminal nodes 31, 36, 24, 19, and 39.
**Figure 12.19(c)** shows the tree after this linking step.
#### Step 3: Adjusting Pointers
- Now `PTR` represents the first of three roots of the same degree. Adjust the pointers:
-
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