Distributing presents in Hawaii
Authors: Jan Hackfeld, Julie Meißner, Miriam Schlöter
Project: Design and Operation of Infrastructure Networks under Uncertainty;
DFG SPP 1736 Algorithms for Big Data“.
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Exercise
Santa Claus almost completed his plan how to deliver all his presents on
the night of December 24th just a few days before Christmas. Every year, he
starts on the Fiji Islands and then continues distributing presents in all time
zones until he reaches the last families on Hawaii. The missing bit of his tour
concerns one skyscraper, where three families whom he wants to visit live.
Santa explains to Rudolph: “This house really makes me scratch my head! I
lost the paper where I noted when the last member of each family arrives at
home. All I remember is that they all arrive at some time between midnight
and 2:30 in the morning. As they all take the bus, they will arrive at a
multiple of 10 minutes after midnight and then immediately go to bed.
Floor
Alani 1st floor
Olina 6th floor
Malu 9th floor
We have to land and take off on a balcony on the 5th floor, as this is where
the only access to the ventilation duct is. However, the duct is so narrow
that I need 5 minutes to crawl from one floor to the next. What is the best
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strategy for me to distribute the presents to every family while they’re all
asleep? This will determine when we can start our Christmas night party on
the Hawaiian beach!”
Rudolph is puzzled for a while, but then his nose begins to shine and he
suggests the following: “So, you need 5 minutes to crawl from one floor to
the next. You can also distribute the presents to the stockings of every family
so quickly that we can neglect this time. Furthermore, you can hear when
someone comes home by bus from anywhere in the duct, which can only
occur every ten minutes. Unfortunately, you don’t currently know when they
arrive because the piece of paper on which you wrote that down is lost. Thus
you need to take decisions before knowing when Alani, Olina, and Malu
arrive at home. Let’s call a strategy reacting to the times when they arrive
a strategy-without-prior-information.
To evaluate the quality of our strategy-without-prior-information, we can
compare it to the time when we could leave from the 5th floor if you hadn’t
lost the note and you knew the arrival times of the three people. In this case
we can describe an optimal strategy-with-prior-information and compute for
given arrival times the best route through the ventilation duct, such that you
finish distributing all presents as early as possible. The difference between
the departure time of a strategy-without-prior-information and a strategywith-prior-information should be as small as possible, even in the worst case
scenario of the three families’ arrival times.”
Santa Claus and Rudolph observe several things about an optimal strategywithout-prior-information. Which of their following observations is incorrect?
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Possible Answers:
1. It is an optimal strategy to land at midnight on the balcony of the 5th
floor.
2. If Alani and Olina arrive first and at the same time, then Santa shouldn’t
immediately crawl to the 6th floor.
3. Even if Olina arrives at home at least 50 minutes after Alani and Malu,
any optimal strategy-without-prior-information ends 20 minutes later
in the worst case than an optimal strategy-with-prior-information.
4. If all three arrive at 2:30 in the morning, there is an optimal strategywithout-prior-information, that finishes at the same time as any optimal strategy-with-prior-information
5. If no one has arrived by 0:30, then Santa must still be in the 5th floor
at 0:30.
6. There is a strategy-without-prior-information for Santa such that he
can take off from the 5th floor at most 20 minutes after every optimal
strategy-with-prior-information.
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7. After Santa Claus has distributed the presents on the 9th floor, he must
not wait on the 6th floor for Olina or Alani to return home.
8. No matter when Alani, Olina and Malu arrive at home exactly, Santa
can assume he will be gone from the 5th floor by 3:30.
9. If Alani arrives at least an hour after Malu, there is an optimal strategywithout-prior-information for Santa such that he can leave from the
5th floor at the same time as with an optimal strategy-with-priorinformation.
10. If Alani returns home an hour before Malu, then Santa can take off by
2:50 at the latest.
Connection to the project:
In discrete optimization, we study algorithms to find an optimal solution
among a huge, but usually finite, set of solutions. You can solve this problem
of the Christmas calendar with pen and paper. For a large number of floors
and families, however, we would only be able to solve it with suitable algorithms on a computer. A particular challenge is the fact that in this problem,
Santa needs to take decisions without knowing all information about the arrival times. The part of discrete optimization studying such problems is called
Online Optimization and what we call a strategy-without-prior-information
is referred to as an online algorithm. In Online Optimization, we search for
good algorithms solving problems where decisions have to be taken with partial information, even though these decisions influence the quality of the final
solution. In the project that inspired us to design this problem, we studied
algorithms for elevators, which could be used to operate industrial elevators
that sort products n a tall shelf.
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