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\section{Question #1}  A. \textbf{A.}  First, we will put all of the $S(u_i,u_j)$ values into an array that keeps track of the two people and their similarity level. If there is $n$ people, the number of connections will be on the order of $n^2$.From this, we can call the total number of connections, $E$.  Now, we can use counting sort to sort all of these values from the largest to smallest. TO To  use ocunting counting  sort, we have to make sure these two assumptions are satisfied first. first:  1. The values must be an integer (we know this because of how the problem is set up).  2. We must know the range of the numbers.  We can find the range of these  numbers in linear time by looking through every single value, and keeping track of the max number (at the time). So, in linear time, we can find the max value. Then, we can use counting sort which also costs \textbf{linear time}, so in total, the time for the sort is $n^2$ (to find the max)$+$ $n^2$(to do the sort) = $O(n^2)$. Although it looks like it's not, the efficiency is still linear.  Now, we take the largest value in our list and place those users into the same group. WE will use a graph data structure ot keep track of which nodes are connected to each other. This data structure will be explained in part b.