Multi-Orientation Edge-Minimum Repair for Non-Redundant Fault-Tolerant Broadcasting in Dense Eisenstein--Jacobi Networks

Computer Science > Distributed, Parallel, and Cluster Computing [Submitted on 18 Jun 2026] Title:Multi-Orientation Edge-Minimum Repair for Non-Redundant Fault-Tolerant Broadcasting in Dense Eisenstein--Jacobi Networks View PDF HTML (experimental)Abstract:Dense Eisenstein--Jacobi (EJ) networks are degree-six algebraic interconnection networks whose finite quotient geometry is naturally represented by a hexagonal axial-coordinate ball. This paper studies non-redundant one-to-all broadcast repair in the dense EJ network generated by $\alpha=(t+1)+t\omega$, where $t$ is the network diameter. We propose EJ-MOEM, a multi-orientation edge-minimum repair method that evaluates a constant-size family of hexagonal broadcast-tree orientations, selects a fault-aware candidate, contracts the fault-pruned tree into healthy components, and reconnects these components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph: every healthy node receives the message exactly once, no faulty node is used, and the original healthy tree components are preserved. We prove that, for a chosen orientation whose fault-pruned component graph is connected, exactly $c-1$ external repair edges are necessary and sufficient, where $c$ is the number of healthy components. We also prove a depth-certificate theorem for EJ coordinate-reduction trees: every one-fault placement admits a repair of depth at most $t+1$, and every two-fault placement admits a repair of depth at most $t+2$. The proof uses the three-strip representation of EJ hexagons, a sector-suffix attachment lemma, a non-adjacent-sector separation lemma, and a six-direction shielding classification for paired cuts. Extended validation includes exhaustive one- and two-fault enumeration for $t=2,\ldots,12,14,16,18$ (up to $N=1027$ and 525,825 two-fault placements at $t=18$), structured theorem-critical tests through $t=30$, and large random tests through $t=200$, all with 100\% success and no violation of the theorem. Current browse context: cs.DC References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.