Optimal Online Ring Routing Jessen T. Havill Department of Mathematics and Computer Science Denison University Granville, OH 43023 havill@denison.edu Kevin R. Hutson Department of Mathematics Furman University Greenville, SC 29613 kevin.hutson@furman.edu Abstract We study how to route online splittable flows in bidirectional ring networks to minimize maximum load. We show that the competitive ratio of any deterministic online algorithm for this problem is at least 2 − 2/n, where n is the size of the ring, and that the online algorithm that splits each flow inversely proportionally to the length of the flow’s shortest path achieves this competitive ratio for all integers n ≥ 2. 1 Introduction In recent years, there has been renewed interest in bidirectional ring networks, especially for Metropolitan Area Networks (MANs) [1, 14]. The ring topology offers two primary benefits. First, the connectivity of the cycle protects against failure of either a node or a link and increases the survivability of the network. In addition, if there is high demand congesting a link, the ring offers an alternative route for communication. Second, the ring topology allows the network resources to be shared in a cost-effective way. There are currently two prevalent modern MAN ring technologies: the Synchronous Optical Network (SONET) ring [14], which is virtual circuit based, and the newer Resilient Packet Ring (RPR) [1], which is packet based but allows for reserved bandwidth transmis- sions. Both technologies allow for bidirectional traffic in either the clockwise or counterclockwise direction. The ring capacity is equal in both directions and fixed in the design phase. Therefore, it is important to route requested demand efficiently to avoid congestion (load) on any particular link in the system. Here we consider the problem of routing online splittable flows in ring networks to minimize max- imum load. We define a problem instance (n, F ) as follows. Let R =(V,E) be a directed ring network with node set V = {0, 1,...,n − 1} and arc set E = {(i, (i + 1) mod n), ((i + 1) mod n, i): i ∈ V }. Let F = f 1 ,f 2 ,...,f k be a sequence of flow requests on R. Each flow f j is a triple (s j ,t j ,l j ) where s j ∈ V is the source of the flow, t j ∈ V is the destination of the flow, and l j is the bandwidth (or demand) required for the flow. Flows may also have arrival and departure times assigned to them. For each flow f j , an algorithm must decide what fraction of l j to route from s j to t j in the clockwise direction (involving arcs of type e i =(i, (i + 1) mod n)) and what fraction to route in the counter-clockwise direction (involving arcs of type e i = ((i + 1) mod n, i)). In this online problem, an algorithm must irrevocably make the decision for each flow f j independent of knowledge about future flows f i where i>j (but may use knowledge of previous flows f i where i<j ). For an online algorithm A and flow sequence F , we define the load on arc e at time t, denoted A e (F ,t), to be the sum of the fractional flows that an algorithm A has assigned to e at time t. The 1