  In the (discrete) CNN problem, online requests appear as points in
$\mathbb{R}^2$. Each request must be served before the next one is revealed. We
have a server that can serve a request simply by aligning either its $x$ or $y$
coordinate with the request. The goal of the online algorithm is to minimize
the total $L_1$ distance traveled by the server to serve all the requests. The
best known competitive ratio for the discrete version is 879 (due to Sitters
and Stougie).
  We study the continuous version, in which, the request can move continuously
in $\mathbb{R}^2$ and the server must continuously serve the request. A simple
adversarial argument shows that the lower bound on the competitive ratio of any
online algorithm for the continuous CNN problem is 3. Our main contribution is
an online algorithm with competitive ratio $3+2 \sqrt{3} \approx 6.464$. Our
analysis is tight. The continuous version generalizes the discrete orthogonal
CNN problem, in which every request must be $x$ or $y$ aligned with the
previous request. Therefore, Our result improves upon the previous best
competitive ratio of 9 (due to Iwama and Yonezawa).
