Abstract:Circulant interconnection networks provide symmetric addressing, compact generator descriptions, and uniform local connectivity. This paper maps a degree--redundancy landscape for a fault-tolerant two-hop primitive in directed circulants: given $n$ nodes and degree budget $m$, how large can the worst-case shared-relay multiplicity $R(n,m)$ be? A node is a shared relay for an ordered terminal pair if it has outgoing links to both terminals; an $f$-relay-fault-tolerant circulant requires at least $f+1$ such relays for every pair. The underlying feasibility condition is a cyclic difference-multiplicity condition, which we use as a mathematical tool rather than claim as a new object. The contribution is the network-design framework around this tool: the parameters $R(n,m)$ and $D_f(n)$, a negative theorem for interval circulants, relay-table preprocessing and lookup algorithms, adversarial and random failure guarantees, load-balance scope, certified upper-bound interpretation of heuristic designs, exact small-$n$ calibration, a software lookup-versus-search microbenchmark, and a reproducible study of 526,539 generator sets. The results show that generator choice critically determines worst-case relay survivability: optimized threshold designs achieve $f$-relay-fault tolerance within about $1.16$--$1.63$ of the counting lower bound, while standard interval generators can fail structurally even at much larger degrees.
From: Bader Albader Dr. [view email]
[v1]
Thu, 18 Jun 2026 06:25:51 UTC (1,523 KB)
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