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Course: COMP 108, Fall 2009
School: East Los Angeles College
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Algorithmic COMP108 Foundations Tutorial 7 7 March 2008 Note: If you've spent 10 minutes on a question without any clue of how to proceed, talk to the demonstrator. 1. Solve the following recurrence by the iterative method. T (n) = 1 3 T (n - 1) + 1 if n = 1 if n &gt; 1 (Hint: You can use the fact that 1 + x + x2 + . . . xn-1 = (xn - 1)/(x - 1) for any positive integer n and any real number x = 1.) 2....

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Algorithmic COMP108 Foundations Tutorial 7 7 March 2008 Note: If you've spent 10 minutes on a question without any clue of how to proceed, talk to the demonstrator. 1. Solve the following recurrence by the iterative method. T (n) = 1 3 T (n - 1) + 1 if n = 1 if n > 1 (Hint: You can use the fact that 1 + x + x2 + . . . xn-1 = (xn - 1)/(x - 1) for any positive integer n and any real number x = 1.) 2. Consider the following undirected graph G. c b a d g (a) List all the vertices adjacent to vertex k. (b) What is the degree of the vertex d? (c) What is the maximum degree of the vertices in G? (d) Give the adjacency matrix of the graph G. (e) State the conditions for a graph to contain an Euler circuit and an Euler path. (f) Does G contain an Euler circuit? If yes, write down the sequence of the vertices in one of these Euler circuits; if no, explain why and find the minimum number of edges needed to add to the graph so that an Euler circuit exists. (g) Does G contain an Euler path? If yes, write down the sequence of the vertices in one of these Euler paths; no, if explain why and find the minimum number of edges needed to add to the graph so that an Euler path exists. 3. Given a length-n sequence T of characters T [0..(n-1)], a length-m sequence of characters P [0..(m-1)] is called a substring of T if there exists some 0 i n - m such that T [i..(i + m-1)] = P [0..(m-1)]. For example, if T is ACGTACGGG, then ACGG is a substring appearing once in T , ACG is a substring appearing twice and ACC is not a substring at all. Design and write a pseudo code algorithm to determine if P appears as a substring of T exactly once. What is the worst case time complexity of your algorithm (in big-O ...

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