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When we create a Hamiltonian circuit in a graph, it is a closed loop. We can start at any vertex, follow the path, and arrive back at the starting vertex. For instance, if we use the greedy algorithm to create a Hamiltonian circuit starting at vertex A, we are actually creating a circuit that could start at any vertex in the circuit. If we use the greedy algorithm and start from different vertices, will we always get the same result? Try it on Figure lat the left. If we start at vertex A, we get the circuit A-C-E-B-D-A, with a total weight of 26 (see Figure 2). However, if we start at vertex B, we get B-E-D-C-A-B, with a total weight of 18 (see Figure 3). Even though we found the second circuit by starting at B, we could traverse the same circuit beginning at A, namely A-B-E-D-C-A, or, equivalently, A-C-D-E-B-A. This circuit has a smaller total weight than the first one we found. D. gure 1

When we create a Hamiltonian circuit in a graph, it is a closed loop. We can start at any vertex, follow the path, and arrive back at the starting vertex. For instance, if we use the greedy algorithm to create a Hamiltonian circuit starting at vertex A, we are actually creating a circuit that could start at any vertex in the circuit. If we use the greedy algorithm and start from different vertices, will we always get the same result? Try it on Figure lat the left. If we start at vertex A, we get the circuit A-C-E-B-D-A, with a total weight of 26 (see Figure 2). However, if we start at vertex B, we get B-E-D-C-A-B, with a total weight of 18 (see Figure 3). Even though we found the second circuit by starting at B, we could traverse the same circuit beginning at A, namely A-B-E-D-C-A, or, equivalently, A-C-D-E-B-A. This circuit has a smaller total weight than the first one we found. D. gure 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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EXPLORATION: When we create a Hamiltonian circuit in a graph, it is a closed loop. We can start at any vertex, follow the path, and arrive back at the starting vertex. For instance, if we use the greedy algorithm to create a Hamiltonian circuit starting at vertex A, we are actually creating a circuit that could start at any vertex in the circuit. 10 If we use the greedy algorithm and start from different vertices, will we always get the same result? Try it on Figure lat the left. If we start at vertex A, we get the circuit A-C-E-B-D-A, with a total weight of 26 (see Figure 2). However, if we start at vertex B, we get B-E-D-C-A-B, with a total weight of 18 (see Figure 3). Even though we found the second circuit by starting at B, we could traverse the same circuit beginning at A, namely A-B-E-D-C-A, or, equivalently, A-C-D-E-B-A. This circuit has a smaller total weight than the first one we found. Figure 1 B. 10 C. Figure 2 Figure 3 In other words, it may pay to try the greedy algorithm starting from different vertices, even when we know that we actually want to begin the circuit at a particular vertex. To be thorough, we can extend the greedy algorithm by using it at each and every vertex, generating several different circuits. We then choose from among these the circuit with the smallest total weight and begin the circuit at the vertex we want. Continue investigating Hamiltonian circuits in Figure lby using the greedy algorithm starting at vertices C, D, and E. Then compare the various Hamiltonian circuits to identify the one with the smallest total weight.
Transcribed Image Text:EXPLORATION: When we create a Hamiltonian circuit in a graph, it is a closed loop. We can start at any vertex, follow the path, and arrive back at the starting vertex. For instance, if we use the greedy algorithm to create a Hamiltonian circuit starting at vertex A, we are actually creating a circuit that could start at any vertex in the circuit. 10 If we use the greedy algorithm and start from different vertices, will we always get the same result? Try it on Figure lat the left. If we start at vertex A, we get the circuit A-C-E-B-D-A, with a total weight of 26 (see Figure 2). However, if we start at vertex B, we get B-E-D-C-A-B, with a total weight of 18 (see Figure 3). Even though we found the second circuit by starting at B, we could traverse the same circuit beginning at A, namely A-B-E-D-C-A, or, equivalently, A-C-D-E-B-A. This circuit has a smaller total weight than the first one we found. Figure 1 B. 10 C. Figure 2 Figure 3 In other words, it may pay to try the greedy algorithm starting from different vertices, even when we know that we actually want to begin the circuit at a particular vertex. To be thorough, we can extend the greedy algorithm by using it at each and every vertex, generating several different circuits. We then choose from among these the circuit with the smallest total weight and begin the circuit at the vertex we want. Continue investigating Hamiltonian circuits in Figure lby using the greedy algorithm starting at vertices C, D, and E. Then compare the various Hamiltonian circuits to identify the one with the smallest total weight.
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