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One example of a greedy algorithm is the Dijkstra algorithm for finding the lowest cost path through a weighted graph. The diagram below shows two weighted graphs that a student wants to investigate using Dijkstra's algorithm. In each case the task it to find the lowest cost of reaching every node from v₁. Each graph has a single negative weight in it. V₁ 10 12 V₂ V3 10 -9 V4 V₁ 15 8 V₂ V3 30 -7 V₁ Graph (a) Graph (b) One of the graphs will yield a correct analysis of the lowest cost for all vertices, and the other will produce an incorrect analysis. Which of the two graphs will produce the incorrect analysis, and explain why the greedy nature of Dijkstra's algorithm is responsible for the incorrect analysis. Your answer should include the key concept of an invariant.

One example of a greedy algorithm is the Dijkstra algorithm for finding the lowest cost path through a weighted graph. The diagram below shows two weighted graphs that a student wants to investigate using Dijkstra's algorithm. In each case the task it to find the lowest cost of reaching every node from v₁. Each graph has a single negative weight in it. V₁ 10 12 V₂ V3 10 -9 V4 V₁ 15 8 V₂ V3 30 -7 V₁ Graph (a) Graph (b) One of the graphs will yield a correct analysis of the lowest cost for all vertices, and the other will produce an incorrect analysis. Which of the two graphs will produce the incorrect analysis, and explain why the greedy nature of Dijkstra's algorithm is responsible for the incorrect analysis. Your answer should include the key concept of an invariant.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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(b) One example of a greedy algorithm is the Dijkstra algorithm for finding the lowest cost path through a weighted graph. The diagram below shows two weighted graphs that a student wants to investigate using Dijkstra's algorithm. In each case the task it to find the lowest cost of reaching every node from v₁. Each graph has a single negative weight in it. ● V₁ ● 10 ● 12 V₂ V3 10 V4 V₁ 15 Graph (a) Graph (b) One of the graphs will yield a correct analysis of the lowest cost for all vertices, and the other will produce an incorrect analysis. Which of the two graphs will produce the incorrect analysis, and explain why the greedy nature of Dijkstra's algorithm is responsible for the incorrect analysis. Your answer should include the key concept of an invariant. V₂ (c) The priority queue is a widely used data structure. Priority queues may be implemented using binary heaps and simple linear arrays. For the basic priority queue operations of: 30 Building an initial queue Taking the highest priority item off the queue Adding a new item to the queue V4 Compare and contrast the running time complexities (best and worst cases) associated with implementations using binary heaps and simple linear arrays. You may find it helpful to use diagrams to support your answer.
Transcribed Image Text:(b) One example of a greedy algorithm is the Dijkstra algorithm for finding the lowest cost path through a weighted graph. The diagram below shows two weighted graphs that a student wants to investigate using Dijkstra's algorithm. In each case the task it to find the lowest cost of reaching every node from v₁. Each graph has a single negative weight in it. ● V₁ ● 10 ● 12 V₂ V3 10 V4 V₁ 15 Graph (a) Graph (b) One of the graphs will yield a correct analysis of the lowest cost for all vertices, and the other will produce an incorrect analysis. Which of the two graphs will produce the incorrect analysis, and explain why the greedy nature of Dijkstra's algorithm is responsible for the incorrect analysis. Your answer should include the key concept of an invariant. V₂ (c) The priority queue is a widely used data structure. Priority queues may be implemented using binary heaps and simple linear arrays. For the basic priority queue operations of: 30 Building an initial queue Taking the highest priority item off the queue Adding a new item to the queue V4 Compare and contrast the running time complexities (best and worst cases) associated with implementations using binary heaps and simple linear arrays. You may find it helpful to use diagrams to support your answer.
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