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Each problem is worth 25 points. For each problem you must specify your algorithm clearly and prove it correct.
Reminder: For each problem you will receive 6 of the 25 points if you write nothing. If you feel that you have a partial answer worth more than 6 points, e.g., you have an algorithm but can’t prove it correct, or your algorithm can be proven correct but is too slow, you can still write this information down, as long as you begin your answer by assessing the state of your work. If we find your work as assessed, you will receive at least 6 points.
1 Star Wars
In a galaxy far, far away, the Empire has built n Death Satellites to attack n planets controlled by the rebels (each satellite will attack exactly one planet). The Rebels have installed a defensive shield in each planet. The i-th satellite has power fi, while the shield on planet j can tolerate up to sj power. A successful attack on the i-th planet brings benefit bi to the Empire. That is, if planet i is attacked by the j -th satellite and fj > si, then the Empire will receive benefit bi; otherwise, it will receive benefit 0. Match satellites to planets to maximize the Empire’s benefit in
O(n log n) time, using a greedy algorithm. Hint: Make one or more heaps.
2 Super Slam
You are a wrestling promoter putting together a SuperSlam and you need to pair up your fighters by weight, in order to have the most balanced fights. Specifically, there are two pools of fighters, A = fa1; a2; : : : ; ang and
B = fb1; b2; : : : ; bng and each fighter from A must be paired with exactly one fighter from B. Your job is to minimize the sum of jw(ai) w(bj )j over all matched pairs, where w(f) denotes the weight of fighter f. Being an entrepreneurial person, you are naturally inclined to design a greedy algorithm for this task. Do it.
Bonus (10 points): What if the notion of imbalance changed from jw(ai) w(bj )j to [w(ai) w(bj )]2 ? Would your algorithm still work? If not, could you modify it so that it does?
3 You Can’t Pause Toast
Leon needs to make breakfast toast for the n members of his family. His toaster can accommodate one person’s toast at a time, and each family member, j, wants their toast to spend tj seconds in the toaster. In Leon’s household, certain family members must have their toast before others (the precedence relationships form a directed acyclic graph). Violating precedence constraints is not an option, as it will make everything topsy turvy. Also, the amount of complaining that will be voiced by family member, j, if they receive their toast at time t is cj (t), where cj is a member-specific, non-negative, strictly increasing function.
One thing Leon has learned is that you cannot pause toast. Pausing toast causes it to lose its essence. So, once Leon starts toasting someone’s toast, he cannot stop until it is done. Now, truth be told, Leon does not want to bring the ruckus to this breakfast. He asks you to develop a greedy algorithm that finds the ordering (schedule) that minimizes
the maximum complaining among all valid toast schedules. Hint: start at the end
4 Dock of the Bay
You are in charge of the port of Oakland, where n ships are present. Ship i holds ci containers and must depart the port by time ti. Unloading a ship takes one hour, independently of how many containers are on it. Only a single ship may be unloaded at any given moment. The unloading of any ship can start as early as time 0. Develop a greedy algorithm to select which ships to unload so as to maximize the total number of containers unloaded.
Hint: make one safe decision after another