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Problem E: For each of the following parts, state True or False. If true, give a short proof. If ...
Problem A: Consider the following graph.
Problem A: Consider the following graph. (a). Find a minimum spanning tree of the graph using Kruskal's algorithm. List the edges in the order they are put into the tree. (b). Apply Prim's algorithm to the same graph starting with node A. List the edges, in order added to the MST. (c). Suppose the cost of every edge touching node A is increased by a constant. Are we guaranteed that the MST remains the MST? Explain.
In the correctness proof of Kruskal's algorithm (taught in lecture), an important step is to show...
In the correctness proof of Kruskal's algorithm (taught in lecture), an important step is to show the greedy choice of the algorithm leads to an optimal solution. In order to show this, an induction is performed. The algorithm progressively adds more edges to the final solution. Assume by certain point, the algorithm has selected and added a few edges to T, T CT*, where T* is an assumed minimum spanning tree. Now the algorithm selects an edge e=(x,y) based its...
Let G=(V, E) be a connected graph with a weight w(e) associated with each edge e....
Let G=(V, E) be a connected graph with a weight w(e) associated with each edge e. Suppose G has n vertices and m edges. Let E’ be a given subset of the edges of E such that the edges of E’ do not form a cycle. (E’ is given as part of input.) Design an O(mlogn) time algorithm for finding a minimum spanning tree of G induced by E’. Prove that your algorithm indeed runs in O(mlogn) time. A minimum...
3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answ...
3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answered (or blank) independently of the other ( subject to the 4 total blank for partial credit rule). s MST algorithm on the graph below and left, starting with vertex all work done so far: al (40 points) You are runing Prim' a. You are about to take vertex g out of the min-ehave not done so yet. Show the order that vertices wer...
Problem 3's picture are given below. 5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n...
Problem 3's picture are given below.
5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n 1, let cycles in G. Modify {e1, e2,.. . ,en} be a subset of edges (from E) that includes no Kruskal's algorithm in order to obtain a spanning tree of G that is minimal among all the spanning trees of G that include the edges e1, e2, . . . , Cn. (b) Apply your algorithm in (a)...
C. 7. True/False Questions. (2 points each) a. Applying Horner's Rule, an n-degree polynomial can be...
C. 7. True/False Questions. (2 points each) a. Applying Horner's Rule, an n-degree polynomial can be evaluated at a given point using only n multiplications and n additions. b. Quick Sort and Merge Sort are comparison-based sorting algorithms. Heap Sort and Distribution Counting Sort are not comparison-based sorting algorithms. An AVL tree applies four types of rotations: RIGHT, LEFT, RIGHT-LEFT, and LEFT-RIGHT. d. When an AVL tree's left sub-tree is left-heavy, a LEFT rotation is needed. e. When an AVL...
Problem 1 (20 points). For each of the following statements, either give a (short) proof to show ...
Problem 1 (20 points). For each of the following statements, either give a (short) proof to show that it 1. Let G- (V,E) be a directed graph. Let s E V. During a BFS run on G starting from s, vertex vis 2. Let G-(V,E) be a directed graph. Let e (u,v) E E. During a DFS nun on G, edge e is a cross 3. Let G (V,E) be a directed graph without negative cycles. Let e e E...
State if each of the statements is true or false. Assume that n is a positive...
State if each of the statements is true or false. Assume that n is a positive integer. Dijkstra’s single-source-shortest-paths algorithm relaxes edges without considering path costs. IoT technology is useful for remotely monitoring elderly people who are living alone. The average of the n positive integers ranging from 1 to n can be found in O(1) time. The average of the greatest and the least numbers among the given n numbers can be found in O(n) time. Optional toolboxes augment...
Question 8 (Chapters 1-8) [1 x 14 14 marks For the statements bellow, say if they are true or false. If true, give a short mathematical proof, if false, give a counterexample. (h) If f : Rn → R i...
Question 8 (Chapters 1-8) [1 x 14 14 marks For the statements bellow, say if they are true or false. If true, give a short mathematical proof, if false, give a counterexample. (h) If f : Rn → R is convex and h : R → Rnxn s strictly convex and nondecreasing, then ho f is strictly convex (i) If f is strictly convex, then it is coercive. ) If f : Rn → R is such that the level...
4. True or False. Label each of the following statements as true or false. If true,...
4. True or False. Label each of the following statements as true or false. If true, give a proof, if false, give a counterexample. (a) Every nontrivial subgroup of Q* contains some positive and some negative numbers (b) Let G be a finite group. Let a E G. If o(a) 5, then o(a1) 5. (c) Let G be a group and H a normal subgroup of G. If G is cyclic, then G/H is also cyclic. (d) Le t R...
In the correctness proof of Kruskal's algorithm (taught in lecture), an important step is to show the greedy choice of the algorithm leads to an optimal solution. In order to show this, an induction is performed. The algorithm progressively adds more edges to the final solution. Assume by certain point, the algorithm has selected and added a few edges to T, T CT*, where T* is an assumed minimum spanning tree. Now the algorithm selects an edge e=(x,y) based its...
3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answered (or blank) independently of the other ( subject to the 4 total blank for partial credit rule). s MST algorithm on the graph below and left, starting with vertex all work done so far: al (40 points) You are runing Prim' a. You are about to take vertex g out of the min-ehave not done so yet. Show the order that vertices wer...
Problem 3's picture are given below.
5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n 1, let cycles in G. Modify {e1, e2,.. . ,en} be a subset of edges (from E) that includes no Kruskal's algorithm in order to obtain a spanning tree of G that is minimal among all the spanning trees of G that include the edges e1, e2, . . . , Cn. (b) Apply your algorithm in (a)...
C. 7. True/False Questions. (2 points each) a. Applying Horner's Rule, an n-degree polynomial can be evaluated at a given point using only n multiplications and n additions. b. Quick Sort and Merge Sort are comparison-based sorting algorithms. Heap Sort and Distribution Counting Sort are not comparison-based sorting algorithms. An AVL tree applies four types of rotations: RIGHT, LEFT, RIGHT-LEFT, and LEFT-RIGHT. d. When an AVL tree's left sub-tree is left-heavy, a LEFT rotation is needed. e. When an AVL...
Problem 1 (20 points). For each of the following statements, either give a (short) proof to show that it 1. Let G- (V,E) be a directed graph. Let s E V. During a BFS run on G starting from s, vertex vis 2. Let G-(V,E) be a directed graph. Let e (u,v) E E. During a DFS nun on G, edge e is a cross 3. Let G (V,E) be a directed graph without negative cycles. Let e e E...
Question 8 (Chapters 1-8) [1 x 14 14 marks For the statements bellow, say if they are true or false. If true, give a short mathematical proof, if false, give a counterexample. (h) If f : Rn → R is convex and h : R → Rnxn s strictly convex and nondecreasing, then ho f is strictly convex (i) If f is strictly convex, then it is coercive. ) If f : Rn → R is such that the level...
4. True or False. Label each of the following statements as true or false. If true, give a proof, if false, give a counterexample. (a) Every nontrivial subgroup of Q* contains some positive and some negative numbers (b) Let G be a finite group. Let a E G. If o(a) 5, then o(a1) 5. (c) Let G be a group and H a normal subgroup of G. If G is cyclic, then G/H is also cyclic. (d) Le t R...
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