CMPT 405/705 — Design and Analysis of Algorithms Exercises on Graphs and Greedy Algorithms. Due: Thursday, September 24th (at the beginning of the class) Reminder: the work you submit must be your own. Any collaboration and consulting outside resourses must be explicitely mentioned on your submission. 1. Consider the problem of making change for n cents using the fewest number of coins. Assume each coin’s value is an integer. (a) Describe a greedy algorithm to make change consisting of quaters, dimes, nickels, and pennies. Prove that your algorithm is optimal. (b) Give a set of coins for which the greedy algorithm does not yield an optimal solution. Your set should include a penny so that there is a solution for every value of n . 2. Give an algorithm to detect whether a given undirected graph contains a cycle. If the graph contains a cycle, then your algorithm should output one (not all). The running time of your algorithm should be O ( m + n )foragraphw ith n vertices and m edges.
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