On the Optimality of Network Topology Discovery in Single-Hop Bounded-Interference Networks

We propose \emph{PRISM} (\textbf{Pseudorandom Residue-based Indexed Scheduling Method}), a deterministic topology-discovery framework for single-hop wireless networks with bounded interference. Each receiver has at most $L$ interfering transmitters among $K$ transmitters and identifies them through singleton transmissions. PRISM assigns finite-field labels to transmitters and schedules transmissions via modular multiplication and a second prime modulus. It achieves full discovery in $O(L(1+\delta)\log K)$ rounds in expectation with failure probability $K^{-\delta}$, and in $O(L^2\log K)$ rounds deterministically. Simulations show $\approx 0.9L\log K$ scaling, with $q/L\approx1.2$ minimizing mean completion time and $q/L\approx1.4\text{--}1.6$ improving tail performance.