Sorted Edges 0566Ef

1. **State the problem:** We have a weighted complete graph with vertices $C, B, A, D, E$ and edges with weights: $C-D=2$, $B-E=9$, $A-E=1$, $B-A=8$, $C-B=15$, $D-E=11$, $C-A=13$, $D-A=3$, $C-E=4$, $D-B=6$. We need to apply the sorted edges algorithm (also known as the greedy algorithm for the Traveling Salesman Problem) to find a circuit that visits all vertices exactly once and returns to the start, and find the total cost of this circuit. 2. **Explain the sorted edges algorithm:** - Sort all edges in ascending order by weight. - Add edges one by one from the sorted list to the circuit, skipping any edge that would create a vertex with degree more than 2 or create a smaller cycle before all vertices are included. - Continue until all vertices have degree 2, forming a Hamiltonian circuit. 3. **Sort edges by weight:** $$ A-E=1,\ C-D=2,\ D-A=3,\ C-E=4,\ D-B=6,\ B-A=8,\ B-E=9,\ D-E=11,\ C-A=13,\ C-B=15 $$ 4. **Add edges step-by-step:** - Add $A-E=1$ (degrees: $A=1$, $E=1$) - Add $C-D=2$ (degrees: $C=1$, $D=1$) - Add $D-A=3$ (degrees: $D=2$, $A=2$) - Add $C-E=4$ (degrees: $C=2$, $E=2$) At this point, vertices $A, C, D, E$ all have degree 2. - Next edge is $D-B=6$. Adding $D-B$ would increase $D$'s degree to 3, which is not allowed. - Next edge is $B-A=8$. Adding $B-A$ would increase $A$'s degree to 3, not allowed. - Next edge is $B-E=9$. Adding $B-E$ would increase $E$'s degree to 3, not allowed. We have not connected $B$ yet. We need to connect $B$ without violating the rules. - Next edge is $D-E=11$. Adding $D-E$ would increase $D$ and $E$ degrees beyond 2, not allowed. - Next edge is $C-A=13$. Adding $C-A$ would increase $C$ and $A$ degrees beyond 2, not allowed. - Next edge is $C-B=15$. Adding $C-B$ would increase $C$'s degree beyond 2, not allowed. Since $B$ is isolated, we must reconsider the edges chosen. 5. **Reconsider edges to include $B$:** We must avoid closing the cycle too early. The edge $C-E=4$ closes a cycle among $C, D, A, E$. Try skipping $C-E=4$ and instead add edges involving $B$: - Add $B-D=6$ (degrees: $B=1$, $D=2$) - Add $B-A=8$ (degrees: $B=2$, $A=2$) Now degrees: - $A=2$ (edges $A-E=1$, $B-A=8$) - $B=2$ (edges $B-D=6$, $B-A=8$) - $D=2$ (edges $C-D=2$, $B-D=6$) - $C=1$ (edge $C-D=2$) - $E=1$ (edge $A-E=1$) Add $C-E=4$ to connect $C$ and $E$ (degrees $C=2$, $E=2$). 6. **Final edges in circuit:** $$ A-E=1,\ C-D=2,\ D-A=3,\ B-D=6,\ B-A=8 $$ 7. **Check for Hamiltonian circuit:** Vertices $A, B, C, D, E$ all have degree 2. 8. **Calculate total cost:** $$ 1 + 2 + 3 + 6 + 8 = 20 $$ **Final answer:** The total cost of the resulting circuit using the sorted edges algorithm is $20$.