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DAT600: Algorithm Theory - Greedy Algorithms - Dynamic Programming - Engineering Assignment Help
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Task: 1 Finding the majority Let E = {e1, e2, ..., en} be a sequence of integers. We say that an integer ei forms a majority in E if it appears more than ceil(n/2) times in E. For instance, the integer 3 is a majority in sequence E = {2, 3, 3, 2, 3, 3}, whereas the sequence E = {6, 3, 2, 7, 3, 1} has no majority. It can be proved that if ei ? ej and E has value em as majority, then sequence E – {ei , ej} also has em as majority. a) Present a greedy algorithm to decide whether E has a majority. b) Explain why we can or can’t use Dynamic programming to solve this problem. Task: 2 Maria is taking her algorithms exam. On the exam paper, the professor has clearly assigned points to each problem; the professor assigns points to the problems depending on the difficulty of the problem. Being a clever student (just like all the students who take Reggie’s famous course on Algorithms), Maria’s opinion of the difficulty of each problem differs from the professor’s. However, as an expert on greedy algorithms, Maria decides to apply a greedy algorithm to assign her own points to the problems; for each problem, she estimates how much time it will take her to solve the problem. Maria’s goal is to maximize the points she will receive given that:- {p1, p2, ..., pn} are the number of points assigned by the professor to each of the n problems, where ?pi =100%.
- {t1, t2, ..., tn} are Maria’s estimates of the time required to do each problem, where ?ti need not be equal to T, T is the total time available for taking the exam, and
- It is possible to get partial points, if a problem is solved partially.
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