BT, BST, AVL. Web Resources page on textbook website:

Transcription

1 Excercises

2 BT, BST, AVL Web Resources page on textbook website:

3 Follow the execution. Consider the following Java code involving a Sequence: for (int i=0; i<10 ; i++) { int rank = (i/2)+1 ; S.insertAtRank(rank,i) ; } Note here that / represents integer division. i.e., only the integral part of the quotient is considered. 1. Show the sequence (just list its elements in order) after each iteration of the for loop. Assume that S was initialized to an empty sequence before the for loop. 2. Show the sequence after n iterations of the for loop.

4 The time delay of a long distance call can be determined by multiplying a small fixed constant to the number of communication links on the telephone network between the caller and callee. Suppose the telephone network of RT&T is a tree. The engineers of RT&T want to compute the maximum possible time delay that may be experienced in a long-distance call. Given a tree T with N nodes, the diameter of T is the length of a longest path between two nodes of T. Give an efficient algorithm for computing the diameter of T.

5 Write a function (Java or pseudocode) which for a given binary tree computes the average depth of the external nodes

6 A team of biologists keeps information about DNA structures in an AVL tree using as key the specific weight (an integer) of the structure. The biologists routinely ask questions of the type Are there any structures in the tree with specific weight between a and b (inclusive)? and they hope to get an answer as soon as possible. Design an efficient algorithm that given integers a and b, returns true if there exists a key x in the tree such that a <= x <= b, and false if no such key exists in the tree. Describe your algorithm in pseudocode or Java. What is the time complexity of your algorithm?

7 RT&T has a network of N switching stations connected by M high-speed communication links, represented by a graph. Each customer's phone is directly connected to one station in his/her area. The engineers of RT&T have developed a prototype video-phone system that allows two customers to see each other during the phone call. In order to have acceptable image quality, the number of links used to transmit video signals between the two parties cannot exceed 4. Design an efficient algorithm that prints, for a given station, the set of stations it can reach using no more than 4 links You can not use any additional data structures.

8 Suppose that graph G has the following adjacency list: 1 (2; 3; 4) 2 (1; 3; 4) 3 (1; 2; 4) 4 (1; 2; 3; 6) 5 (6; 7; 8) 6 (4; 5; 7) 7 (5; 6; 8) 8 (5; 7) 1. Draw G. 2. Give the sequence of vertices visited using depthfirst search starting at vertex Give the sequence of vertices visited using breadthfirst search starting at vertex 1.

9 Reword the following statement as a theorem about undirected graphs and then prove it. Assume that if A is a friend of B then, B is a friend of A and that for all A, A is not a friend of A. In any group of n >= 2 people, there are two people with the same number of friends in the group.

10 Prove that every graph has an even number of odd degree vertices.

11 Symbol table Operations Insert Retrieve Delete Options Array Linked list Balanced binary tree Hash table Investigate the options by creating a table that gives, for each option, the expected time of each operation Discuss advantages and disadvantages

12 Strongly connected components A strongly connected component is a subgraph in which every vertex is reachable from every other vertex. The number of strongly connected components of a graph G is k. By how much can this number change if we add a new edge?