D-way Heap. D-way heaps (aka d-ary heaps or d-heaps) are a simple but effective extension of standard binary heaps, but nonetheless the allow to drastically cut down the running time over the most common operation on this data structure. They are not as advanced as binomial or Fibonacci's heap: the latter, in particular, allows to improve the ...I implemented a D-ary max heap backed by a vector for resizing. I would like to know any possible improvements in performance, design, and in the code in general. #pragma once #include <vector...May 9, 2017 · When the tree in question is the infinite d-ary tree, this algorithm becomes (naively) initialize a queue Q = [1] nextID = 2 forever (Q is always nonempty) pop the head of Q into v repeat d times let w = nextID (w is a child of v) increment nextChildID push w into Q Aug 10, 2019 · A d-ary heap is just like a regular heap but instead of two childrens to each element, there are d childrens! d is given when building a heap, either by giving an argument or by passing it while calling init. Here is my Implementation: import math class DHeap: ''' creates d-heap ''' ''' heap: A python's list ''' def __init__ (self, heap: list ... D-way Heap. D-way heaps (aka d-ary heaps or d-heaps) are a simple but effective extension of standard binary heaps, but nonetheless the allow to drastically cut down the running time over the most common operation on this data structure. They are not as advanced as binomial or Fibonacci's heap: the latter, in particular, allows to improve the ...c. Give an efficient implementation of Extract-Max in a d-ary max-heap. (Hint: How would you modify the existing code?) Analyze the running time of your implementation in terms of n and d. (Note that d must be part of your Θexpression even if it occurs in a constant term.) d. Give an efficient implementation of Insert in a d-ary max-heapIf so, I tend to think it is indeed tight. For a hint, this paper: The Analysis of Heapsort mentions that (in Abstract) The number of keys moved during 2 2 -ary heap-sort when sorting a random file of n n distinct elements is n lg n + O(n) n lg n + O ( n) in the worst case. It even further proves that (Notice that it is for the best case)boost.heap is an implementation of priority queues. Priority queues are queue data structures, that order their elements by a priority. The STL provides a single template class std::priority_queue , which only provides a limited functionality. To overcome these limitations, boost.heap implements data structures with more functionality and ...Based on my understanding, different questions where HEAP is common data structure to use can be categorized in following 4 categories: Top K Pattern. Merge K Sorted Pattern. Two Heaps Pattern. Minimum Number Pattern. All questions under one patterns has some similarities in terms of using HEAP as a data structure.Description. This class implements an immutable priority queue. Internally, the d-ary heap is represented as dynamically sized array (std::vector), that directly stores the values. The template parameter T is the type to be managed by the container. The user can specify additional options and if no options are provided default options are used.Apr 7, 2016 · By using a $ d $-ary heap with $ d = m/n $, the total times for these two types of operations may be balanced against each other, leading to a total time of $ O(m \log_{m/n} n) $ for the algorithm, an improvement over the $ O(m \log n) $ running time of binary heap versions of these algorithms whenever the number of edges is significantly ... Computer Science. Computer Science questions and answers. c++ part 1 answer questions 1) List 5 uses of heaps 2) Define a d-ary heap 3) Define a complete binary heap 4) Why do most implementations of heaps use arrays or vectors 5) What is a heap called a Parent Child sort order heap ?the heap property, a single node's two children can be freely interchanged unless doing so violates the shape property (compare with treap).The binary heap is a special case of the d-ary heap in which d = 2. Heap operations Both the insert and remove operations modify the heap to conform to the shape property first, by adding or •Can think of heap as a completebinary tree that maintains the heap property: –Heap Property: Every parent is better-than[less-than if min-heap, or greater-than if max-heap] bothchildren, but no ordering property between children •Minimum/Maximum value is always the top element Min-Heap 7 18 9 19 35 14 10 2839 3643 1625 Always a complete tree D-way Heap. D-way heaps (aka d-ary heaps or d-heaps) are a simple but effective extension of standard binary heaps, but nonetheless the allow to drastically cut down the running time over the most common operation on this data structure. They are not as advanced as binomial or Fibonacci's heap: the latter, in particular, allows to improve the ... May 9, 2017 · When the tree in question is the infinite d-ary tree, this algorithm becomes (naively) initialize a queue Q = [1] nextID = 2 forever (Q is always nonempty) pop the head of Q into v repeat d times let w = nextID (w is a child of v) increment nextChildID push w into Q The d-ary heap or d-heap is a priority queue data structure, a generalization of the binary heap in which the nodes have d children instead of 2 This data structure allows decrease priority operations to be performed more quickly than binary heaps, at the expense of slower delete minimum operations.This C++ Program demonstrates the implementation of D-ary Heap. Here is source code of the C++ Program to demonstrate the implementation of D-ary Heap. The C++ program is successfully compiled and run on a Linux system. The program output is also shown below. /* * C++ Program to Implement D-ary-Heap */#include <iostream>#include <cstring>#include <cstdlib>using namespace std;/* * D-ary ...A Heap is a special Tree-based data structure in which the tree is a complete binary tree. More on Heap Data Structure. Question 1. What is the time complexity of Build Heap operation. Build Heap is used to build a max (or min) binary heap from a given array. Build Heap is used in Heap Sort as a first step for sorting.When creating a d-ary heap from a set of n items, most of the items are in positions that will eventually hold leaves of the d-ary tree, and no downward swapping is performed for those items. At most n / d + 1 items are non-leaves, and may be swapped downwards at least once, at a cost of O( d ) time to find the child to swap them with.5. (CLRS 6-2) Analysis of d-ary heaps A d-ary heap is like a binary heap, but instead of 2 children, nodes have d children. a. How would you represent a d-ary heap in a array? b. What is the height of a d-ary heap of n elements in terms of n and d? c. Give an e cient implementation of Extract-Max. Analyze its running time in terms of d and n. d. Dijkstra using k-ary heap Timeform decrease-priorityoperations: O m log n log k Timeforn find-and-remove-minoperations:O nk log n log k Tominimizetotaltime,choosek tobalancethesetwobounds k = max(2,⌈m/n⌉) Totaltime= O m log n log m/n ThisbecomesO(m) wheneverm = Ω(n1+ε) foranyconstantε > 0boost.heap is an implementation of priority queues. Priority queues are queue data structures, that order their elements by a priority. The STL provides a single template class std::priority_queue , which only provides a limited functionality. To overcome these limitations, boost.heap implements data structures with more functionality and ...Question. A d-ary heap is like a binary heap, but (with one possible exception) non-leaf nodes have d children instead of 2 children. a. How would you represent a d-ary heap in an array? b. What is the height of a d-ary heap of n elements in terms of n and d? c. Give an efficient implementation of EXTRACT-MAX in a d-ary max-heap. Analyze its ...Answer: A d-ary heap can be represented in a 1-dimensional array by keeping the root of the heap in A[1], its d children in order in A[2] through A[d+1], their children in order in A[d+2] through A[d2 +d+1], and so on. The two procedures that map a node with index i to its parent and to its jth child (for 1 ≤j ≤d) are D-PARENT(i) 1 return d ... Therefore, the total amount of time to create a heap in this way is. The exact value of the above (the worst-case number of comparisons during the construction of d-ary heap) is known to be equal to:, where s d (n) is the sum of all digits of the standard base-d representation of n and e d (n) is the exponent of d in the factorization of n ... The problem is that d d can exceed n n, and if d d keeps increasing while n n is fixed, then logd n log d n will approach 0 0. Also, one can show that the height is at least logd(n(d − 1) + 1) − 1 ≥ logd n − 1 log d ( n ( d − 1) + 1) − 1 ≥ log d n − 1 for d d sufficiently large. Why is this in Ω(logd n) Ω ( log d n)?Expert Answer. (a) In d-ary heaps, every non-leaf nodes have d childern. So, In array representation of d-ary heap, root is present in A [1], the d children of root are present in the cells having index from 2 to d+1 and their children are in cells having index from …. A d-ary heap is like a binary heap, but (with one possible exception) non ... Since the number of nodes in each layer of a d-ary heap grows exponentially by a factor of d at each step, the height of a d-ary heap is O (log d n) = O (log n / log d). This means that if you increase the value of d, the height of the d-ary heap will decrease, so decrease-keys and insertions will take less time.This C++ Program demonstrates the implementation of D-ary Heap. Here is source code of the C++ Program to demonstrate the implementation of D-ary Heap. The C++ program is successfully compiled and run on a Linux system. The program output is also shown below. /* * C++ Program to Implement D-ary-Heap */#include <iostream>#include <cstring>#include <cstdlib>using namespace std;/* * D-ary ... Explanation: Although pairing heap is an efficient algorithm, it is worse than the Fibonacci heap. Also, pairing heap is faster than d-ary heap and binary heap. 13.Now I have this d-ary heap data structure. Note that for d = 2 this is a binary heap. The client programmer specifies the value of d when constructing the heap. See what I have: heap.h: #ifndef H...According to some experiments, d-ary heap (d>2, typically d=4) generally performs better than binary heap. GitHub - hanmertens/dary_heap: A d-ary heap in Rust GitHub - skarupke/heap: Looking into the performance of heaps, starting with the Min-Max Heap They have the same compact memory layout as binary heap. I don't see any drawback compared to binary heap. Plus, Rust has already chosen b-tree ...Construction of a binary (or d-ary) heap out of a given array of elements may be performed in linear time using the classic Floyd algorithm, with the worst-case number of comparisons equal to 2N − 2s 2 (N) − e 2 (N) (for a binary heap), where s 2 (N) is the sum of all digits of the binary representation of N and e 2 (N) is the exponent of 2 ... boost::heap::priority_queue. The priority_queue class is a wrapper to the stl heap functions. It implements a heap as container adaptor ontop of a std::vector and is immutable. boost::heap::d_ary_heap. D-ary heaps are a generalization of binary heap with each non-leaf node having N children. For a low arity, the height of the heap is larger ...Internally, the d-ary heap is represented as dynamically sized array (std::vector), that directly stores the values. The template parameter T is the type to be managed by the container. The user can specify additional options and if no options are provided default options are used.D-ary heap. D-ary heap is a complete d-ary tree filled in left to right manner, in which holds, that every parent node has a higher (or equal value) than all of its descendands. Heap respecting this ordering is called max-heap, because the node with the maximal value is on the top of the tree. Analogously min-heap is a heap, in which every ...The binary heap is a special case of the d-ary heap in which d = 2. Summary of running times. Here are time complexities of various heap data structures. Function names assume a min-heap. For the meaning of "O(f)" and "Θ(f)" see Big O notation.D-ary heap. D-ary heap is a complete d-ary tree filled in left to right manner, in which holds, that every parent node has a higher (or equal value) than all of its descendands. Heap respecting this ordering is called max-heap, because the node with the maximal value is on the top of the tree. Analogously min-heap is a heap, in which every ...Binomial Heaps - Princeton Universitythe heap property, a single node's two children can be freely interchanged unless doing so violates the shape property (compare with treap).The binary heap is a special case of the d-ary heap in which d = 2. Heap operations Both the insert and remove operations modify the heap to conform to the shape property first, by adding orเป็นการคิดค้นโดย Johnson (ปี 1975) D- Heap , D-ary Heap , m-ary Heap หรือ k-ary Heap คือ Heap ที่มี children node ไม่เกิน d node ซึ่งลำดับความสำคัญของแต่ละโหนดสูงกว่าลำดับความสำคัญของ children node5. (CLRS 6-2) Analysis of d-ary heaps A d-ary heap is like a binary heap, but instead of 2 children, nodes have d children. a. How would you represent a d-ary heap in a array? b. What is the height of a d-ary heap of n elements in terms of n and d? c. Give an e cient implementation of Extract-Max. Analyze its running time in terms of d and n. d. Apr 7, 2016 · By using a $ d $-ary heap with $ d = m/n $, the total times for these two types of operations may be balanced against each other, leading to a total time of $ O(m \log_{m/n} n) $ for the algorithm, an improvement over the $ O(m \log n) $ running time of binary heap versions of these algorithms whenever the number of edges is significantly ... A d-ary heap is like a binary heap, but (with one possible exception) non-leaf nodes have d children instead of 2 children. . a. How would you represent a d-ary heap in an array? . b. What is the height of a d-ary heap of n elements in terms of n and d? . c. Give an efficient implementation of EXTRACT-MAX in a d-ary max-heap. (b) Write an e cient implementation of Heapify and Heap-Insert for a d-ary heap. The Heapify algorithm is somewhat di erent from the binary-heap version, whereas Heap-Insert is identical to the corresponding algorithm for binary heaps. The running time of Heapify is O(dlogd n), and the running time of Heap-Insert is O(logd n). Heapify(A;i;n;d ...1. In a d-ary heap, up-heaps (e.g., insert, decrease-key if you track heap nodes as they move around) take time O (log_d n) and down-heaps (e.g., delete-min) take time O (d log_d n), where n is the number of nodes. The reason that down-heaps are more expensive is that we have to find the minimum child to promote, whereas up-heaps just compare ...d-ARY-MAX-HEAPIFY (A, i) largest = i for k = 1 to d if d-ARY-CHILD (k, i) ≤ A. heap-size and A [d-ARY-CHILD (k, i)] > A [i] if A [d-ARY-CHILD (k, i)] > largest largest = A [d-ARY-CHILD (k, i)] if largest!= i exchange A [i] with A [largest] d-ARY-MAX-HEAPIFY (A, largest) Therefore, the total amount of time to create a heap in this way is. The exact value of the above (the worst-case number of comparisons during the construction of d-ary heap) is known to be equal to:, where s d (n) is the sum of all digits of the standard base-d representation of n and e d (n) is the exponent of d in the factorization of n ... Computer Science. Computer Science questions and answers. c++ part 1 answer questions 1) List 5 uses of heaps 2) Define a d-ary heap 3) Define a complete binary heap 4) Why do most implementations of heaps use arrays or vectors 5) What is a heap called a Parent Child sort order heap ?Sep 9, 2016 · 1 Answer. In a ternary heap, each node has up to three children. The heap is represented in the array in breadth-first order, with the root node at 0, and the children of node x at locations (x*3)+1, (x*3)+2, and (x*3)+3. The node at location x is at (x-1)/3. So, your array, [90,82,79,76,46,1,49,44,61,62], looks like this when displayed the ... A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior.1 Answer. Since you declared your heap as mutable, the push operation is supposed to return the handle_t you typedefed as the handle_type: mpl::if_c< is_mutable, handle_type, void >::type push (value_type const & v); In the respect of obtaining the handle, your code is fine. To simplify a bit to make it clearer:Give an efficient implementation of INSERT in a d-ary max-heap. Analyze its running time in terms of d and n. Give an efficient implementation of INCREASE-KEY(A, i, k), which flags an error if k < A[i] = k and then updates the d-ary matrix heap structure appropriately. Jun 30, 2023 · Implementation (Max Heap) We will store the n-ary heap in the form of an array where: The maximum value node will be at the 0th index. The parent of a node at the ith index will be at (i-1)/k. The children of a node at the ith index will be at indices: (k*i)+1, (k*i)+2 … (k*i)+k. getMax (): It returns the maximum element in the heap. เป็นการคิดค้นโดย Johnson (ปี 1975) D- Heap , D-ary Heap , m-ary Heap หรือ k-ary Heap คือ Heap ที่มี children node ไม่เกิน d node ซึ่งลำดับความสำคัญของแต่ละโหนดสูงกว่าลำดับความสำคัญของ children node"""Implementation of a d-ary heap. The branching factor for the heap can be passed as an argument. It's 2 by default, which is also the minimum possible value. The branching factor is the maximum number of children that each internal node can have. For regular heaps, a node an have at most 2 children, so the branching factor is 2.The d-ary heap or d-heap is a priority queue data structure, a generalization of the binary heap in which the nodes have d children instead of 2 This data structure allows decrease priority operations to be performed more quickly than binary heaps, at the expense of slower delete minimum operations.I implemented a D-ary max heap backed by a vector for resizing. I would like to know any possible improvements in performance, design, and in the code in general. #pragma once #include <vector...It seems like if you got unlucky with your heap structure this could easily be causing your infinite loop. Similarly, in this loop you're never reassigning tempChild, so on each iteration tempChild will pick up where it left off on the previous iteration. If on one of those iterations tempChild was equal to size, then the inner loop will never ...Now I have this d-ary heap data structure. Note that for d = 2 this is a binary heap. The client programmer specifies the value of d when constructing the heap. See what I have: heap.h: #ifndef H...5. (CLRS 6-2) Analysis of d-ary heaps A d-ary heap is like a binary heap, but instead of 2 children, nodes have d children. a. How would you represent a d-ary heap in a array? b. What is the height of a d-ary heap of n elements in terms of n and d? c. Give an e cient implementation of Extract-Max. Analyze its running time in terms of d and n. d.Answer: A d-ary heap can be represented in a 1-dimensional array by keeping the root of the heap in A[1], its d children in order in A[2] through A[d+1], their children in order in A[d+2] through A[d2 +d+1], and so on. The two procedures that map a node with index i to its parent and to its jth child (for 1 ≤j ≤d) are D-PARENT(i) 1 return d ... 2 The number of items in a full d-heap of n levels is (1-d n. A little algebra tells us that the number of levels required to hold n items in a d-heap is log d (n*(d - 1) + 1). So a 4-heap with 21 items takes log 4 (20*(4 - 1)+1), or 2.96 levels. We can’t have a partial level, so we round up to 3. See my blog post, The d-ary heap, for more ...boost.heap is an implementation of priority queues. Priority queues are queue data structures, that order their elements by a priority. The STL provides a single template class std::priority_queue , which only provides a limited functionality. To overcome these limitations, boost.heap implements data structures with more functionality and ...c. Give an efficient implementation of Extract-Max in a d-ary max-heap. (Hint: How would you modify the existing code?) Analyze the running time of your implementation in terms of n and d. (Note that d must be part of your Θ expression even if it occurs in a constant term.) d. Give an efficient implementation of Insert in a d-ary max-heap.Expert Answer. Question 7 (Analysis of d-ary heaps). As mentioned in Lecture L0301 Slide 23, we define a d-ary heap (for d > 2) analogously like a binary heap, but instead, in the d-ary tree visualization of a d-ary heap, we allow every node to have at most d children. In this question, you will extend several binary heap operations to the case ... Nov 14, 2022 · Suppose the Heap is a Max-Heap as: 10 / \ 5 3 / \ 2 4 The element to be deleted is root, i.e. 10. Process : The last element is 4. Step 1: Replace the last element with root, and delete it. 4 / \ 5 3 / 2 Step 2: Heapify root. Final Heap: 5 / \ 4 3 / 2. Time complexity: O (logn) where n is no of elements in the heap. The d-ary heap or d-heap is a priority queue data structure, a generalization of the binary heap in which the nodes have d children instead of 2 This data structure allows decrease priority operations to be performed more quickly than binary heaps, at the expense of slower delete minimum operations.When the tree in question is the infinite d-ary tree, this algorithm becomes (naively) initialize a queue Q = [1] nextID = 2 forever (Q is always nonempty) pop the head of Q into v repeat d times let w = nextID (w is a child of v) increment nextChildID push w into QDec 1, 2010 · A d-ary heap is like a binary heap, but (with one possible exception) non-leaf nodes have d children instead of 2 children.. How would you represent a d-ary heap in an array?A d-ary heap can be implemented using a dimensional array as follows.The root is kept in A[1], its d children are kept in order in A[2] through A[d+1] and so on. The d_ary_heap_indirect is designed to only allow priorities to increase. If in the update () and push_or_update () functions you change: preserve_heap_property_up (index); to. preserve_heap_property_up (index); preserve_heap_property_down (); it seems to allow increasing or decreasing the priorities while keeping the queue sorted.May 6, 2015 · 1. In a d-ary heap, up-heaps (e.g., insert, decrease-key if you track heap nodes as they move around) take time O (log_d n) and down-heaps (e.g., delete-min) take time O (d log_d n), where n is the number of nodes. The reason that down-heaps are more expensive is that we have to find the minimum child to promote, whereas up-heaps just compare ... A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior.Sep 3, 2012 · The d_ary_heap_indirect is designed to only allow priorities to increase. If in the update () and push_or_update () functions you change: preserve_heap_property_up (index); to. preserve_heap_property_up (index); preserve_heap_property_down (); it seems to allow increasing or decreasing the priorities while keeping the queue sorted. Jun 23, 2012 · 2 Answers. Sorted by: 4. This uses the common identity to convert between logarithmic bases: logx(z) = logm(z) / logm(x) By multiplying both sides by log m (x), you get: logm(z) = logx(z) * logm(x) Which is equivalent to the answer in the question you site. More information is available here. c. Give an efficient implementation of Extract-Max in a d-ary max-heap. (Hint: How would you modify the existing code?) Analyze the running time of your implementation in terms of n and d. (Note that d must be part of your Θexpression even if it occurs in a constant term.) d. Give an efficient implementation of Insert in a d-ary max-heapc. Give an efficient implementation of Extract-Max in a d-ary max-heap. (Hint: How would you modify the existing code?) Analyze the running time of your implementation in terms of n and d. (Note that d must be part of your Θ expression even if it occurs in a constant term.) d. Give an efficient implementation of Insert in a d-ary max-heap.
Jun 23, 2012 · 2 Answers. Sorted by: 4. This uses the common identity to convert between logarithmic bases: logx(z) = logm(z) / logm(x) By multiplying both sides by log m (x), you get: logm(z) = logx(z) * logm(x) Which is equivalent to the answer in the question you site. More information is available here. . Epoxy paint for bathtub lowe
Explanation: Although pairing heap is an efficient algorithm, it is worse than the Fibonacci heap. Also, pairing heap is faster than d-ary heap and binary heap. 13.c. Give an efficient implementation of Extract-Max in a d-ary max-heap. (Hint: How would you modify the existing code?) Analyze the running time of your implementation in terms of n and d. (Note that d must be part of your Θ expression even if it occurs in a constant term.) d. Give an efficient implementation of Insert in a d-ary max-heap.Feb 25, 2022 · Contact Datils (You can follow me at)Instagram: https://www.instagram.com/ahmadshoebkhan/LinkedIn: https://www.linkedin.com/in/ahmad-shoeb-957b6364/Faceboo... A d -ary heap is like a binary heap, but (with one possible exception) non-leaf nodes have d children instead of 2 children. How would you represent a d -ary heap in an array? What is the height of a d -ary heap of n elements in terms of n and d? Give an efficient implementation of EXTRACT-MAX in a d -ary max-heap.May 6, 2015 · 1. In a d-ary heap, up-heaps (e.g., insert, decrease-key if you track heap nodes as they move around) take time O (log_d n) and down-heaps (e.g., delete-min) take time O (d log_d n), where n is the number of nodes. The reason that down-heaps are more expensive is that we have to find the minimum child to promote, whereas up-heaps just compare ... Explanation: Although pairing heap is an efficient algorithm, it is worse than the Fibonacci heap. Also, pairing heap is faster than d-ary heap and binary heap. 13.Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.Computer Science. Computer Science questions and answers. c++ part 1 answer questions 1) List 5 uses of heaps 2) Define a d-ary heap 3) Define a complete binary heap 4) Why do most implementations of heaps use arrays or vectors 5) What is a heap called a Parent Child sort order heap ?The d_ary_heap_indirect is designed to only allow priorities to increase. If in the update () and push_or_update () functions you change: preserve_heap_property_up (index); to. preserve_heap_property_up (index); preserve_heap_property_down (); it seems to allow increasing or decreasing the priorities while keeping the queue sorted.•Can think of heap as a completebinary tree that maintains the heap property: –Heap Property: Every parent is better-than[less-than if min-heap, or greater-than if max-heap] bothchildren, but no ordering property between children •Minimum/Maximum value is always the top element Min-Heap 7 18 9 19 35 14 10 2839 3643 1625 Always a complete tree Jun 29, 2022 · K-ary heap. K-ary heaps are similar to the binary heap (where K = 2) just having one difference that instead of 2 child nodes, there can be k child nodes for every node in the heap. It is nearly like a complete binary tree, i.e. all the levels are having maximum number of nodes except the last level, which is filled from left to right. A Heap is a special Tree-based data structure in which the tree is a complete binary tree. More on Heap Data Structure. Question 1. What is the time complexity of Build Heap operation. Build Heap is used to build a max (or min) binary heap from a given array. Build Heap is used in Heap Sort as a first step for sorting.1. Which of the following is true? a) Prim’s algorithm initialises with a vertex. b) Prim’s algorithm initialises with a edge. c) Prim’s algorithm initialises with a vertex which has smallest edge. d) Prim’s algorithm initialises with a forest. View Answer. 2. Consider the given graph. .