DAG-Inducing Problems and Algorithms
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In this paper, we show that in a parallel processing system, if a directed acyclic graph (DAG) can be induced in the state space and execution is enforced along that DAG, then synchronization cost can be eliminated. Specifically, we show that in such systems, correctness is preserved even if the nodes execute asynchronously and rely on old/inconsistent information of other nodes. We present two variations for inducing DAGs – DAG-inducing problems, where the problem definition itself induces a DAG, and 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) and 2-approximation vertex cover (VC) are not DAG-inducing problems. However, DAG-inducing algorithms can be developed for them. Among these, the algorithm for MM allows self-stabilization and the 2-approx. algorithm for VC does not. Our algorithm for MM converges in 2n moves and does not require a synchronous environment, which is an improvement over the existing algorithms in the literature. Algorithms for DC, SP and 2-approx. VC converge in 2m, 2m and n moves respectively. 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).
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