【Data Structure and Algorithm】Graph Exercises

topic

There are a total of seven post-school exercises related to graphs below, all of which are ideas drawing questions and not algorithm design questions, so I will list them together here~

1. Given the directed graph shown in the figure below, please answer the following questions

1. the in/out degree of each vertex;
2. adjacency matrix;
3. adjacency list;
4. inverse adjacency list;
5. strongly connected components

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answer

in/out degree of each vertex

(The following will use ①+ to indicate the in-degree of ① vertex, ①- to represent the out-degree, and other vertices by analogy)

Vertex out degree	Vertex in-degree
①+：3	①-：0
②+：2	②-：2
③+：1	③-：2
④+：1	④-：3
⑤+：2	⑤-：1
⑥+：2	⑥-：3

adjacency matrix

adjacency list

inverse adjacency list

strongly connected components

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2. The adjacency matrix of the graph represented by the known two-dimensional array is shown in the figure below, and the depth-first spanning tree and the breadth-first spanning tree obtained by traversing from vertex 1 are respectively drawn.

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answer

The depth-first spanning tree obtained by traversing from vertex 1

The breadth-first spanning tree obtained by traversing from vertex 1

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3. Answer the following two questions for the following picture

1. Write out its adjacency matrix, and find its minimum spanning tree according to Prim's algorithm
2. Write out its adjacency list, and find its minimum spanning tree according to the Kruskal algorithm

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answer

Write out its adjacency matrix, and find its minimum spanning tree according to Prim's algorithm

Write out its adjacency list, and find its minimum spanning tree according to the Kruskal algorithm

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4. Answer the following two questions for the following picture

1. List all possible topologically ordered sequences
2. Write out its adjacency list, and point out which sequence is obtained by applying the TopLogicalSort algorithm learned in class

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answer

List all possible topologically ordered sequences

• 152634
• 512634
• 156234
• 516234

Write out its adjacency list, and point out which sequence is obtained by applying the TopLogicalSort algorithm learned in class

The sequence obtained using the TopLogicalSort algorithm learned in class is: 563412

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5. For the AOE network in the figure below, calculate the relative values ​​of active arcs and vertices

$$Compute\; thee(a_i)，l(a_i),\; the\; value\; ofve(v_i)，l(v_i)$$

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answer

The numbers of the individual arcs and vertices are shown in the figure below

$Ve\left(v_i\right)$

• $Ve\left(A\right)\_=1$
• $Ve\left(B\right)=6$
• $Ve\left(C\right)\_=17(a\to G\to H\to C)$
• $Ve\left(D\right)\_=3$
• $Ve\left(E\right)=25(a\to G\to H\to C\to E)$
• $Ve(F)=4$
• $Ve\left(G\right)=3$
• $Ve(H)=13(a\to G\to H)$
• $Ve(I)=1$
• $Ve\left(J\right)\_=31(a\to G\to H\to K\to J)$
• $Ve\left(N\right)=22(a\to G\to H\to K)$
• $Ve(\infty )=43$

$I(v_i)$

• $I(A)=19$
• $I(B)=23$
• $I(C)=25$
• $I(D)=18$
• $I\left(E\right)=33$
• $I(F)=8$
• $I(G)=3$
• $I(H)=13$
• $I(I)=7$
• $I(J)=31$
• $I(K)=22$
• $I(\infty )=43$

$and(a_i)$

• $and\left(a_1\right)=0$
• $and\left(a_2\right)=0$
• $and\left(a_3\right)=0$
• $and\left(a_4\right)=0$
• $and\left(a_5\right)=0$
• $and\left(a_6\right)=0$
• $and\left(a_8\right)=6$
• $and\left(a_9\right)=3$
• $and\left(a_{10}\right)=3$
• $and\left(a_{11}\right)=3$
• $and\left(a_{12}\right)=4$
• $and(a_{13})=4$
• $and\left(a_{14}\right)=3$
• $and\left(a_{15}\right)=3$
• $and\left(a_{16}\right)=1$
• $and\left(a_{17}\right)=17$
• $and\left(a_{18}\right)=13$
• $and\left(a_{19}\right)=13$
• $and\left(a_{20}\right)=13$
• $and\left(a_{21}\right)=22$
• $and\left(a_{22}\right)=31$
• $and\left(a_{23}\right)=31$
• $and\left(a_{24}\right)=25$

$l(a_i)$

• $l\left(a_1\right)=18$
• $l(a_2)=17$
• $l\left(a_3\right)=15$
• $l\left(a_4\right)=4$
• $l\left(a_5\right)=0$
• $l\left(a_6\right)=6$
• $l\left(a_7\right)=19$
• $l\left(a_8\right)=22$
• $l\left(a_9\right)=18$
• $l\left(a_{10}\right)=25$
• $l\left(a_{11}\right)=25$
• $l\left(a_{12}\right)=27$
• $l\left(a_{13}\right)=8$
• $l\left(a_{14}\right)=22$
• $l\left(a_{15}\right)=3$
• $l\left(a_{16}\right)=7$
• $l(a_{17})=25$
• $l\left(a_{18}\right)=21$
• $l\left(a_{19}\right)=27$
• $l\left(a_{20}\right)=13$
• $l\left(a_{21}\right)=22$
• $l\left(a_{22}\right)=30$
• $l\left(a_{23}\right)=31$
• $l\left(a_{24}\right)=33$

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6. Use Dijkstra algorithm to find the shortest path from vertex a to other vertices in the graph, and write the state of each step in the process of executing the algorithm

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answer

step one

Starting with a as the initial pending path table

step two

Select the shortest edge from the pending path table, set its vertex as a new growth point, and compare each of the remaining growth points, if the path that takes the new growth point as a transit is shorter than the original path, replace it.

next steps

Repeat step 2 n-2 times (Note: Green means that there is a new path, but the path length is greater than the original path, and red means that the path length of the new path is smaller than the original path, so it is replaced)

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7. Use Floyd's algorithm to find the shortest path between each pair of vertices in the graph

answer

STEP1: Initial adjacency matrix D, and Path

STEP2: Take A as the transit point and perform the first iteration

STEP3: Take B as the transition point, and carry out the second iteration

STEP4: Take C as the transit point for the third iteration

STEP5: Take D as the turning point, and carry out the fourth iteration

STEP6: Get the final D, and Path matrix

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conclusion

  Because it is an algorithm side dish, the methods and ideas provided may not be very good, please bear with me~ If you have any questions, please leave a message to discuss. If you think this article is helpful to you, can you give me a free like? The communication between us is my biggest motivation!