Triple-loop networks with arbitrarily many minimum distance diagrams

Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form “L-shape’’. In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.

💡 Research Summary

The paper investigates the combinatorial and algebraic structure of minimum‑distance diagrams (MDDs) in multi‑loop interconnection networks, focusing on the case of triple‑loop networks. An MDD is a geometric encoding of the network’s diameter and the set of shortest‑path routings; in the well‑studied double‑loop (two‑parameter) case it is known that every network admits at most two MDDs, each of which has a characteristic “L‑shape”. This strong restriction follows from the fact that the lattice ideal associated with a double‑loop network has only two distinct monomial initial ideals, each corresponding to one of the two possible term orders.

The authors extend this framework to triple‑loop networks, which are defined by three step sizes ((p,q,r)) modulo a common modulus (N). A node pair ((u,v)) is connected by a shortest path described by a non‑negative integer triple ((a,b,c)) satisfying (ap+bq+cr\equiv 0\pmod N). The set of all such triples forms a three‑dimensional lattice, and an MDD can be visualised as a polyhedral region in (\mathbb{Z}^3) whose lattice points correspond to feasible routing vectors.

The central technical contribution is a construction that shows the number of distinct MDDs for a triple‑loop network can be made arbitrarily large. The authors first establish a bijection between MDDs and monomial initial ideals of the lattice ideal
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