4.(5 points) Five people occupy five seats. If five seats are arranged in a circle, bow many different ways can the five people select their seats?

(d) Given the pre-order and level-order traversal sequences, we can construct a unique binary tree.

(a) Please sequentially insert the following keys into the given AVL tree: 17, 19, 18. Please sbow the final result of the AVL tree after all the keys are inserted. Only the final result is needed, no step-by-step illustration is required.

(b) Continue with the previous sub-problem. After the keys in sub-problem (a) are inserted, please sequentially delete keys 25 and 17 (when deleting a non-leaf node from the AVL tree, please replace it by the node with the largest key in its left subtree). Please show the final AVL tree only (no step-by-step illustration is required).

(a) (3 points) A center of a graph is a vertex that incurs the minimum eccentricity. That is, a center c is defined as:. Is it possible for a graph to have more than one center? If yes, please provide an example; If no, please provide a proof.

(b) (2 x 3 points) In contrast to the center, which must be a vertex, an absolute center is a point that can be on an edge or on a vertex, such that its maximum sbortest path distance to all vertices is minimum. Take the following graph as an example, the point x is an absolute center, because the shortest path distances from x to vertices a, b, c, d are 6.5, 2.5, 6.5, 2.5, respectively. That is, the maximum sbortest path distance from x to all the other vertices is 6.5, which is minimum among all possible cases.  Given the definition of the absolute center, we know that there may be multiple absolute centers in a graph. So, please answer the following questions. (b-i) If there are multiple absolute centers in a graph, can all of them be on vertices, i.e, no absolute center is on an edge? If yes, please provide an example; If no, please provide a proof. (b-ii) If there are multiple absolute centers in a graph, can some of them be on vertices, and some of them be on edges at the same time? If yes, please provide an example; If no, please provide a proof.

1.(10 points) Consider L= f{anbn } and the statement S=integer m. Write the statement of 7S (the negation of S). Hint:

(5 points) (a) What is a spanning tree?

(5 points) (b) Given an undirected, weighted graph in Figure 1, what is the minimum spanning tree (MST)?

(5 points) (c) Describe the sequence of adding edges to form the MST of the graph in Figure 1 using the greedy Kruskal's algorithm.

3.(8 points) Use the Euclidcan algorithm to find the greatest common divisor of 167,076 and 1,928,737.

110年 - 110 國立清華大學_碩士班招生考試_資訊工程學系:基礎計算機科學#105771