Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system transitions to a state in one of the lattices. Then, the algorithm behaves lattice linearly while traversing to an optimal state through that lattice. We present a lattice-linear self-stabilizing algorithm for service demand based minimal dominating set (SDMDS) problem. Using this as an example, we elaborate the working of, and define, eventually lattice-linear algorithms. Then, we present eventually lattice-linear self-stabilizing algorithms for minimal vertex cover (MVC), maximal independent set (MIS), graph colouring (GC) and 2-dominating set problems (2DS). Algorithms for SDMDS, MVCc and MIS converge in 1 round plus $n$ moves (within $2n$ moves), GC in $n+4m$ moves, and 2DS in 1 round plus $2n$ moves (within $3n$ moves). These results are an improvement over the existing literature. We also present experimental results to show performance gain demonstrating the benefit of lattice-linearity.

翻译：晶格线性系统允许节点异步执行。我们引入了最终晶格线性算法，其中晶格仅在状态空间子集的各状态之间诱导产生。该算法保证系统转换到某个晶格中的状态，随后通过该晶格遍历至最优状态时表现出晶格线性行为。我们提出了一种基于服务需求的最小支配集（SDMDS）问题的晶格线性自稳定算法。以此为例，我们阐述并定义了最终晶格线性算法的工作原理。接着，我们提出了针对最小顶点覆盖（MVC）、最大独立集（MIS）、图着色（GC）和2-支配集（2DS）问题的最终晶格线性自稳定算法。SDMDS、MVC和MIS算法在1轮加n次移动内收敛（不超过2n次移动），GC算法在n+4m次移动内收敛，2DS算法在1轮加2n次移动内收敛（不超过3n次移动）。这些结果优于现有文献。我们还展示了体现性能提升的实验结果，证明了晶格线性特性的优势。