On the Continuous CNN Problem

TL;DR

This paper introduces an online algorithm for the continuous CNN problem, achieving a competitive ratio of approximately 6.464, improving significantly over the previous ratio of 9, and tight analysis confirms optimality.

Contribution

The paper presents the first online algorithm with a competitive ratio of about 6.464 for the continuous CNN problem, surpassing previous bounds and extending the discrete case.

Findings

Achieved a competitive ratio of 3+2√3 (~6.464) for the continuous CNN problem.

Proved the lower bound on competitive ratio is 3.

Provided a tight analysis confirming the optimality of the algorithm.

Abstract

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).