Self-adjusting networks (SANs) have the ability to adapt to communication demand by dynamically adjusting the workload (or demand) embedding, i.e., the mapping of communication requests into the network topology. SANs can thus reduce routing costs for frequently communicating node pairs by paying a cost for adjusting the embedding. This is particularly beneficial when the demand has structure, which the network can adapt to. Demand can be represented in the form of a demand graph, which is defined by the set of network nodes (vertices) and the set of pairwise communication requests (edges). Thus, adapting to the demand can be interpreted by embedding the demand graph to the network topology. This can be challenging both when the demand graph is known in advance (offline) and when it revealed edge-by-edge (online). The difficulty also depends on whether we aim at constructing a static topology or a dynamic (self-adjusting) one that improves the embedding as more parts of the demand graph are revealed. Yet very little is known about these self-adjusting embeddings. In this paper, the network topology is restricted to a line and the demand graph to a ladder graph, i.e., a $2^n$ grid, including all possible subgraphs of the ladder. We present an online self-adjusting network that matches the known lower bound asymptotically and is $12$-competitive in terms of request cost. As a warm up result, we present an asymptotically optimal algorithm for the cycle demand graph. We also present an oracle-based algorithm for an arbitrary demand graph that has a constant overhead.

翻译：自调整网络能够通过动态调整工作负载（或需求）嵌入（即通信请求到网络拓扑的映射），自适应地响应通信需求。此类网络可通过调整嵌入的代价来降低频繁通信节点对的路由成本。当需求具有可被网络适应的结构性时，这种特性尤为有效。需求可采用需求图的形式表示，该图由网络节点集（顶点）和成对通信请求集（边）定义。因此，适应需求可解释为将需求图嵌入到网络拓扑中。无论需求图是预先已知（离线场景）还是逐条边暴露（在线场景），这一过程都具有挑战性。其难度还取决于目标是构建静态拓扑，还是构建能够随需求图部分暴露而逐步改进嵌入的动态（自调整）拓扑。然而，关于这类自调整嵌入的研究仍极为有限。本文将网络拓扑限制为线性结构，需求图限制为阶梯图（即$2^n$网格），并涵盖阶梯图的所有可能子图。我们提出一种在线自调整网络，它在渐近意义上匹配已知下界，且在请求代价方面实现$12$-竞争比。作为预热结果，我们针对循环需求图提出一种渐近最优算法。此外，针对任意需求图，我们提出了一种具有常数开销的基于预言机的算法。