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 (\tds). Algorithms for SDMDS, \mvc and \mis converge in 1 round plus $n$ moves (within $2n$ moves), \gc in $n+4m$ moves, and \tds 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：格点线性系统允许节点异步执行。我们引入了最终格点线性算法，其中格点仅诱导于状态空间子集内的状态之间。该算法保证系统转换到其中一个格点的状态。随后，算法在该格点中遍历至最优状态时表现出格点线性行为。我们针对基于服务需求的最小支配集（SDMDS）问题提出了一种格点线性自稳定算法。以此为例，我们详细阐述了最终格点线性算法的工作机制并给出了其定义。接着，我们针对最小顶点覆盖（\mvc）、最大独立集（\mis）、图着色（\gc）和二支配集问题（\tds）提出了最终格点线性自稳定算法。针对SDMDS、\mvc和\mis的算法在1轮加n步移动内收敛（2n步内），\gc在n+4m步内收敛，\tds在1轮加2n步内收敛（3n步内）。这些结果优于现有文献。我们还通过实验结果展示了性能提升，证明了格点线性化的优势。