Ryan's Repository of Random Reflections

Suppose we are given a vector of $n$ items $ \begin{bmatrix} x_1 & x_2 & \dots x_n \end{bmatrix}$ and we would like to perform a number of subset sums on these items.  Specifically, we would like to compute $ \sum_{i \in S} x_i $ for some collection of subsets $S \subseteq \{ 1, \dots, n \}$.   We would like to evaluate all of the summations using the minimum number of arithmetic operations (additions and subtractions). In this setting, addition and subtraction of two elements is an expensive operation (e.g., think of them as being 2D numpy arrays).  Therefore, we are willing to spend computational time and resources to find an efficient means to evaluate the subset sums.  Moreover, we may need to compute these subset sums many times, e.g., if they are computed as part of an inner loop to an iterative algorithm.  Therefore, the time spent finding the right way to evaluate the subsets sums will ultimately pay for itself. A few examples with solutions...

In this blog post, I am going to show why the work I did on  Day Three  is so important, and how using the strategy I outlined in that post can improve your odds of beating the streak by a factor of 5-10!  On day three, I analyzed the situations under which you should select a player who you think will get a hit, as opposed to not selecting that player, and instead maintaining your current streak until the next day.  In summary, I found that your decision should be guided by your current streak, the number of games left in the season, your confidence in the player (how likely is he to get a hit?), and the distribution of likelihoods across all games in the season.  After solving for the optimal strategy, I was able to approximate the probability of winning under that strategy by making some simplifying assumptions about the probability distribution of the best player getting a hit on a given day. I'm not going to get too deep into the math in this blog post, b...

I thought of this problem about two years ago and was going to send it out as the one of the weekly challenge problems for the ACM student chapter at UD, however I was unable to solve it when I first thought of it, and so I put it on the back-burner for a while.  Recently, I revisited it and made some progress on it, but was still unable to find a strategy that can solve it at all, let alone efficiently.  I am posting the problem to this blog in the hopes that you can help me figure it out by posting your ideas and code in the comments below.  Anyway, here is the problem: You are taking a quiz with $n$ true/false questions that you didn't study for, but you are allowed to retake the quiz as many times as you want. Every time you take the quiz, you are told how many questions you got correct, but not which questions were right and wrong.  Your final score is the number of answers you got correct on your last submission minus the number of times you retook the quiz. ...