% !TEX root = ../DES.tex \subsection{Some Online Algorihms} \subsubsection{Dating Problem} Suppose we signed up for online Dating with n people. \\ We associate every person with a Matching Number m, which tells you, how good you and this person fit together. \\ What is the best algirthm to meet the best matching person? \\ \textbf{Answer} \\ We go out with the first $\frac{n}{e}$ people. Where e corresponds to eulers number. \\ After we met these people, we remember how high the best matching was. \\ The first person we meet afterwards, who has a higher matching than everybody we have seen so far, will be the chosen one. \\ \textbf{This strategy picks the best person with 37\% probabillity} \\ \\ \textbf{Prove} \\ Let $\pi$ be the random permutation of the numbers $1,2,...,n$. \\ The process above, describes finding the maximum of the first $\pi[1],...,\pi[t]$ numbers, where $t=\frac{n}{e}$. \\ And then find finding the first j $\geq$ t + 1 such that $\pi[j] > max\{\pi[1],...,\pi[t]\}$ \\ We want to compute the probabillity that $\pi[j] = 1$, we choose the best match possible. \begin{center} P[we choose the best one] = $\displaystyle \sum_{j=t+1}^{n} P[\pi[j] = 1$ and we pick the person on the j-th Date$]$ \end{center} \begin{center} $\displaystyle \sum_{j = t + 1}^{n} P[ \pi[j] = 1$ and the max\{ $\pi[1],...,\pi[j-1] \}$ is in $\pi[1],...,\pi[t] ]$ \\ $= \sum_{j=t+1}^{n} \frac{1}{n} \cdot \frac{t}{j - 1}$ \end{center} So the probability of picking the best person is: \begin{center} $\displaystyle \frac{t}{n} \sum_{j=t+1}^{n} \frac{1}{j - 1} = \frac{t}{n} \big( \sum_{j = 1}^{n-1} \frac{1}{j} - \sum_{j=1}^{t} \frac{i}{j} \big) \simeq \frac{t}{n} \cdot (ln(n) - ln(t) = \frac{t}{n} ln( \frac{n}{t}))$ \end{center} And the last term is optimized for $t = n/e$