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There are 3 popular beach resorts, A, B, and C, which reside on a line: 

                    A-----(1km)-----B-----(1km)------C.

The distances between the resorts is 1k. John owns an ice-cream truck located in beach resort A and another located in beach resort C. Two busses full of ice-cream-craving children will arrive at the beach resorts (A, B, and C) tomorrow, but John does not know for which resort each bus is headed and when each bus will arrive (the busses can arrive at different times). He plans to dispatch an ice-cream truck from its current location to a beach resort as soon as a bus arrives at that resort. Each truck can only serve a single beach resort, that is, once a truck is dispatched to a beach resort it must remain there all day. John wants to design an algorithm that minimizes the sum of distances his trucks traverse in order to reach the busses’ locations.

Show that for every deterministic algorithm ALG, there is some scenario (i.e., schedule and locations of bus arrivals) in which the total distance John’s trucks traverse under ALG is 3OPT, where OPT is the value of the optimal solution in that scenario.

DSM
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Puka Puka
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    If this is some kind of homework please tag it as such. And please ask a question ;) – Tim Dec 02 '12 at 18:06
  • @Tim if you know similar problems, that will be so helpful. thanks :-) – Puka Puka Dec 02 '12 at 18:10
  • Your question is still just a problem desctiption. So what problems do you have solving the problem? Is the description unclear? What have you tried solving the problem? Where could you start solving the problem? Just give us someting to work with here ^^ – Tim Dec 02 '12 at 18:15
  • @Tim if you give any deterministic algothim ALG that solve that problem, prove that there is a scenario for this ALG that the total distance of the John's trucks (two truks) traverse is at least 3OPT?? can you prove it. – Puka Puka Dec 02 '12 at 18:22
  • @PukaPuka Please show us what you have tried – Haile Dec 02 '12 at 18:23
  • @Halie lets say that the first bus arrives to beach B, so we have to dispatch the one of the trucks located in C or A to B (once this truck dispatched to B it can't be moved to the end of the day - let's say that the truck located at A dispatched to B), but if the second bus arrives to A then we have to dispatch the truck loacted at C to A. the sum of distance that the trucks traverse is 3km, although we could done that by leaving the truck loacted at A in A, and dispatch the truck in C to B, and the sum of the distance will be 1km in that case. – Puka Puka Dec 02 '12 at 18:51
  • but how do I prove it to all AlG's? – Puka Puka Dec 02 '12 at 18:54
  • @Tim: the homework tag has been deprecated. – Nik Bougalis Dec 03 '12 at 01:55

1 Answers1

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You already described one stategy in your comments. I will slightly modify it:

Let the strategy be that the trucks stay where they are located until children arrive at a location and then move a truck towards then. Then children arrive at B, so truck located at A dispatched to B. Now the second bus arrives to A so we have to dispatch the truck loacted at C to A. the sum of distance that the trucks traverse is 3km.

For this strategy you have already proven that there is a scenario where the distance is 3*OPT

Now think about how many other meaningful strategies can be there in this scenario (hint: there are not many). Describe such a case for all of them and you are done.

Tim
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