In this paper, we show that in a parallel processing system, if a partial order is induced among the local states visited by a node, then synchronization cost can be eliminated. As a result of this partial order, a DAG is induced among the global states. Specifically, we show that in such systems, correctness is preserved even if the nodes execute asynchronously and read old information of other nodes. We present two variations for inducing DAGs -- \textit{DAG-inducing problems}, where the problem definition itself induces a DAG, and \textit{DAG-inducing algorithms}, where a DAG is induced by the algorithm. We demonstrate that the dominant clique (DC) problem and shortest path (SP) problem are DAG-inducing problems. Among these, DC allows self-stabilization, whereas the algorithm that we present for SP does not. We demonstrate that maximal matching (MM) is not a DAG-inducing problem. However, a DAG-inducing algorithm can be developed for it. The algorithm for MM allows self-stabilization. This algorithm converges in $2n$ moves and does not require a synchronous environment, which is an improvement over the existing algorithms in the literature. The algorithm for DC converges in $2m$ moves, and the algorithm for SP converges in $\mathcal{D}$ rounds. ($n$ is the number of nodes and $m$ is the number of edges in the input graph, and $\mathcal{D}$ is its diameter.) We also note that DAG-inducing problems are more general than, and encapsulate, lattice linear problems (Garg, SPAA 2020). Similarly, DAG-inducing algorithms encapsulate lattice linear algorithms (Gupta and Kulkarni, SSS 2022). We also show that a partial order induced among the local states visited by a node, as discussed above, is a necessary and sufficient condition to allow asynchrony.

翻译：本文证明：在并行处理系统中，若节点所访问的局部状态间存在偏序关系，则可消除同步开销。这种偏序关系将在全局状态间诱导出有向无环图（DAG）。具体而言，我们证明在此类系统中，即使节点异步执行并读取其他节点的陈旧信息，系统正确性仍得以保持。我们提出两种DAG诱导方式——**DAG诱导问题**（问题定义本身诱导DAG）与**DAG诱导算法**（算法的执行过程诱导DAG）。研究表明，主团（DC）问题和最短路径（SP）问题均为DAG诱导问题：前者允许自稳定，而本文提出的SP算法不具备此性质。最大匹配（MM）问题虽非DAG诱导问题，但可为其设计DAG诱导算法。该MM算法具有自稳定性，仅需$2n$步收敛且无需同步环境，优于现有文献中的算法。DC算法需$2m$步收敛，SP算法需$\mathcal{D}$轮收敛（$n$为输入图节点数，$m$为边数，$\mathcal{D}$为图直径）。此外，我们指出DAG诱导问题比格线性问题（Garg，SPAA 2020）更具一般性且涵盖后者；类似地，DAG诱导算法亦涵盖格线性算法（Gupta 和 Kulkarni，SSS 2022）。最后证明了前述节点局部状态间偏序关系的存在性，是系统支持异步执行的充要条件。