Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 6.5, Problem 5E
Program Plan Intro
To describe the correctness of HEAP-INCREASE-KEY for the loop invariants.
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Develop a topological sort implementation thatmaintains a vertex-indexed array that keeps track of the indegree of each vertex. Initialize the array and a queue of sources in a single pass through all the edges Then, perform the following operations until the source queue is empty:■ Remove a source from the queue and label it.■ Decrement the entries in the indegree array corresponding to the destinationvertex of each of the removed vertex’s edges.
Develop a topological sort implementation thatmaintains a vertex-indexed array that keeps track of the indegree of each vertex. Initialize the array and a queue of sources in a single pass through all the edges. Then, perform the following operations until the source queue is empty:■ Remove a source from the queue and label it.■ Decrement the entries in the indegree array corresponding to the destination vertex of each of the removed vertex’s edges
If decrementing any entry causes it to become 0, insert the corresponding vertex onto the source queue.
30. In the implementation for a breadth-first search we studied, a queue was used. The code below replaces
the queue with a stack. List the pre-order enumeration that the vertices in the graph below are visited
using this modified method, starting from vertex 0.
}
3
/** stack-based search */
static void modSearch (Graph G, int start) {
Stack S new AStack (G.n());
S.push(start);
}
6
G.setMark (start, VISITED);
while (S.length) > 0) {
}
modSearch:
intv S.pop();
PreVisit (G, v);
for (int w = G.first (v); w < G.n(); w = G.next(v, w))
if (G.getMark (w) == UNVISITED) {
G.setMark (w, VISITED);
S.push(w);
Chapter 6 Solutions
Introduction to Algorithms
Ch. 6.1 - Prob. 1ECh. 6.1 - Prob. 2ECh. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.2 - Prob. 1ECh. 6.2 - Prob. 2ECh. 6.2 - Prob. 3E
Ch. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6 - Prob. 1PCh. 6 - Prob. 2PCh. 6 - Prob. 3P
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- 3. The following is a recursive algorithm for a preorder binary tree traversal. Determine the worst-case runtime of this algorithm where the proper binary tree t has n internal nodes (non null nodes). Do this by specify the recurrence equation for the runtime T(n) and then deriving the closed form. Count the comparison v != null as 1 primary operation and the call processNode(t,v) as c primary operations, where c is some constant. Algorithm preorder(Tree t, TreeNode v) Input: Binary tree t of size n and node v of t Output: None if v!= null then end end processNode(t.v) preorder(t,v.getLeft()) preorder(t,v.getRight())arrow_forwardFun problem. Given a linked list, check in O(n) if it is a palin-drome. For example, [1 →2 →3 →2] is not a palindrome, while[1 →2 →2 →1] and [3 →5 →3] are.1This class has all necessary methods of the stack data structure, e.g., push, pop, peek, isEmpty, etc.arrow_forwardcreate the remove method for a threaded binary search tree. here is a description for a binary search tree: Since a binary search tree with N nodes has N + 1 NULL pointers, half the space allocated in a binary search tree for pointer information is wasted. Suppose that if a node has a NULL left child, we make its left child pointer link to its inorder predecessor, and if a node has a NULL right child, we make its right child pointer link to its inorder successor. This is known as a threaded tree and the extra links are called threads. here is my .h file: #ifndef THREADEDBST_H #define THREADEDBST_H #include "Node.cpp" class ThreadedBST { private: Node* root; public: /** * Constructs an empty ThreadedBST object. * Pre-condition: None. * Post-condition: An empty ThreadedBST object is created with a null root. */ ThreadedBST(); /** * Destroys the ThreadedBST object and frees the associated memory. * Pre-condition: None. * Post-condition: The ThreadedBST object is destroyed, and all the…arrow_forward
- For the operation below, provide the worst case running time in terms of n (Big O notation). Briefly justify your answer >>> Using binary search in a sorted singly linked list (as you completed in LAB 4). The implementation for __len__ traverses the list to count each nodearrow_forwardConstruct a priority queue using a heapordered binary tree, but instead of an array, use a triply linked structure. Each node will require three links: two to go down the tree and one to traverse up the tree. Even if no maximum priority-queue size is specified ahead of time, your solution should ensure logarithmic running time per operation.arrow_forwardWrite Algorithm to illustrates a pseudo-code procedure for insertion in a singly linkedlist that is non-empty.arrow_forward
- 1.For a full binary tree, the number of leaf-nodes is more than non-leaf nodes. True False 2.For a Max-Heap, the functions Max and Extract-Max have same runtime complexity. True False 3.Heap-increase-Key and Heap-Decrease-Key both have same runtime complexity because both call Heapify function. True False 4.A sorted linked-list has fast insertion but slow extraction. True False 5.Don’t use Max-Heap in case you often perform search operation. Use sorted linked-list inserted. True Falsearrow_forwardIt is required to implement the TDA graph (variant 1, Shiflet) using adjacency represented by simply chained unordered lists. The following will be implemented operators: InitGraf, GrafVid, InserNod, InserArc, DeleteNode, DeleteArc. the performance of the operators implemented in terms of the O function.arrow_forwardQ. No. 1. Write a program that implement a generic binary search tree of the nodes 19, 2, 70, 5, 3, 20, 3, 3,40, 50, 82, 1, 2, 15. Create a function for searching 90 in the developed binary search tree. and traverse the tree following in order traversal.arrow_forward
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