LP-Star: Embedding Longest Paths into Star Networks With Large-Scale Missing Edges Under an Emerging Assessment Model

Star networks play an essential role in designing parallel and distributed systems. With the massive growth of faulty edges and the widespread applications of the longest paths and cycles, it is crucial to embed the longest fault-free paths and cycles in edge-faulty networks. However, the traditional fault model allows a concentrated distribution of faulty edges and thus can only tolerate faults that depend on the minimum degree of the network vertices. This article introduces an improved fault model called the partitioned fault model, which is an emerging assessment model for fault tolerance. Based on this model, we first explore the longest fault-free paths and cycles by proving the edge fault-tolerant Hamiltonian laceability, edge fault-tolerant strongly Hamiltonian laceability, and edge fault-tolerant Hamiltonicity in the $n$-dimensional star network $S_{n}$. Furthermore, based on the theoretical proof, we give an $O(nN)$ algorithm to construct the longest fault-free paths in star networks based on the partitioned fault model, where $N$ is the number of vertices in $S_{n}$. We also make comparisons to show that our result of edge fault tolerance has exponentially improved other known results.