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Scheduling Activities

Your friend is working as a camp counselor at a camp. He needs to organize activities for the kids. One of his plans is the following marathon: each contestant must swim 20 laps of a pool, then bike 10 miles, and then run 3 miles. The plan is to send the contestants out in a staggered fashion via the following rule: the contestants must use the pool one at a time. In other words, first one contestant swims the 20 laps, gets out, and starts biking. As soon as this first person is out of the pool a second contestant begins swimming the 20 laps; as soon as he is out of the pool and starts biking, a third contestant begins swimming and so on.

Each contestant has projected swimming time (the expected time it will take him or her to complete 20 laps), a projected biking time (the expected time it will take him or her to complete the 10 miles of bicycling), and a projected running time (the time it will take him or her to complete the 3 miles of running). Your friend wants to decide on a schedule for the triathlon: an order in which to sequence the starts of the contestants. Let's say that the completion time of a schedule is the earliest time at which all contestants will be finished with all three legs of the triathlon, assuming they each spend exactly their projected swimming, biking and running times on the 3 parts. (Again, note that participants can bike and run simultaneously but at most one person can be in the pool at any time.) What is the best order for sending people out if one wants the whole competition to be over as early as possible? More precisely give an efficient algorithm that produces a schedule whose completion time is as small as possible.

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Since only one participant at a time can swim, sum of the swimming times is constant. However, since ...