Lattice-linearity was introduced as modelling problems using predicates that induce a lattice among the global states (Garg, SPAA 2020). Such modelling enables permitting asynchronous execution in multiprocessor systems. A key property of \textit{the predicate} representing such problems is that it induces \textit{one} lattice in the state space. Such representation guarantees the execution to be correct even if nodes execute asynchronously. However, many interesting problems do not exhibit lattice-linearity. This issue was alleviated with the introduction of eventually lattice-linear algorithms (Gupta and Kulkarni, SSS 2021). They induce \textit{single or multiple} lattices in a subset of the state space even when the problem cannot be defined by a predicate under which the states form a lattice. In this paper, we focus on analyzing and differentiating between lattice-linear problems and algorithms. We introduce a new class of algorithms called \textit{fully lattice-linear algorithms}. These algorithms partition the \textit{entire} reachable state space into \textit{one or more lattices}. For illustration, we present lattice-linear self-stabilizing algorithms for minimal dominating set (MDS) and graph colouring (GC) problems, and a parallel processing lattice-linear 2-approximation algorithm for vertex cover (VC). The algorithms for MDS and GC converge in {\boldmath $n$} moves and {\boldmath $n+2m$} moves respectively. These algorithms preserve this time complexity while allowing the nodes to execute asynchronously, where these nodes may execute based on old or inconsistent information about their neighbours. The algorithm for VC is the first lattice-linear approximation algorithm for an NP-Hard problem; it converges in {\boldmath $n$} moves.

翻译：格线性是将问题建模为使用谓词在全局状态之间诱导格结构的一种方法（Garg, SPAA 2020）。这种建模方式允许多处理器系统中的异步执行。表征此类问题的谓词的关键性质是它在状态空间中诱导出**一个**格结构。这种表示保证了即使节点异步执行，计算也正确无误。然而，许多有趣的问题并不具备格线性。这一缺陷通过后来引入的最终格线性算法得到了缓解（Gupta and Kulkarni, SSS 2021）。这类算法在状态空间的子集中诱导出**单个或多个**格结构，即使该问题无法用状态形成格的谓词来定义。在本文中，我们重点分析和区分格线性问题与算法。我们引入了一类新的算法，称为**完全格线性算法**。这些算法将**整个**可达状态空间划分为**一个或多个格**。作为说明，我们提出了最小支配集（MDS）和图着色（GC）问题的格线性自稳定算法，以及顶点覆盖（VC）问题的并行处理格线性2-近似算法。MDS和GC的算法分别收敛于{\boldmath $n$}步和{\boldmath $n+2m$}步。这些算法在允许节点异步执行的同时保持了这一时间复杂度，其中这些节点可能基于关于邻居的过时或不一致信息执行。VC算法是首个针对NP难问题的格线性近似算法；它收敛于{\boldmath $n$}步。