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This past Thursday, the students at Georgia Tech turned in their final projects for the course. With a total of 30 people involved, we had plenty of participation this year. After discussing our projects with the students, I am happy to say that they all turned in a draft of their algorithm. We discussed this draft at the end of class and made a group decision to change the way we approached this problem.
Conventional wisdom might tell you to just pick a really high number of points and use a brute-force approach. However, this only serves to prove that an algorithm exists. It doesn’t tell us how effective that algorithm is. The first algorithm you would think of would be dynamic programming. And, to be honest, it was the way we started out.
However, the problem we came up with was much simpler. Let’s take a look at the first algorithm we came up with:
The first part is pretty self-explanatory. To get an upper bound for the number of points needed to solve the problem, we just keep taking the largest integer that is smaller than or equal to the current number of points. The difference between this upper bound and the current number of points is the gap, and it can be treated as a new number of points. This is the basis for this entire algorithm.
The second part is a little more complicated. Let’s consider what happens if we have an additional point at the center of the pyramid. With five points initially, we have no problem. But as the number of points increases, the line between the top and bottom of the pyramid gets closer and closer. As the number of points increases, the gap shrinks, but it also gets closer to the left and right sides of the pyramid. This means that you have to give up more and more of the left and right sides. If you were to stick to the left and right edges at all times, you might not even be able to reach the bottom of the pyramid.
With only five points, the gaps at the top and bottom of the pyramid only have a value of 2 points, but the gaps at the left and right edges are 4 points. If you were to use the left and right edges only, you would need at least 6 points. If you used 5 points at the left and right edges, you