Minimal Deadlock-Free Routing for Degree-Six Triangular-Lattice Meshes and Tori with Two Forbidden Turns

RESUMO

Degree-six triangular-lattice interconnection networks offer substantial minimal-path diversity, but their additional directions complicate deadlock-free routing under wormhole flow control. We study a finite hexagon-shaped mesh and its periodic torus quotient in a common six-direction coordinate system. For the finite mesh, we construct a minimal partially adaptive routing relation that uses one virtual channel and forbids only two directed turns. For the torus, we prove that every source-destination pair has a unique closest lattice lift, but that the same two-turn physical routing relation still has a cyclic one-VC resource CDG for every n >= 3. We eliminate this residual periodic dependency by combining two virtual channels with Hamiltonian coordinates and group-specific datelines. Each same-group segment crosses its dateline at most once, which permits a global rank on VC-labelled channel resources. We prove minimal all-pairs connectivity for both physical routing relations and acyclicity of the complete resource CDG for the proposed one-VC mesh and two-VC torus constructions. For a single static bidirectional link failure known before a routing epoch, we further rotate the turn rule toward the failed orientation and replace a failed hop by a same-group two-hop triangle bypass. This restricted extension preserves all-pairs connectivity and the original VC counts, with at most one additional hop relative to the healthy shortest-path distance.