🤖 AI Summary
This paper addresses the construction of edge-disjoint spanning trees (EDSTs) in star-product networks to enable efficient collective communication, enhance fault tolerance, and improve congestion control.
Method: We propose the first theoretically optimal or near-optimal EDST construction for star-product networks, leveraging star-product graph operations, combinatorial design principles, and spanning-tree algorithms, supported by rigorous structural analysis.
Contribution/Results: We formally prove that the constructed EDST set achieves the maximum or near-maximum possible size—i.e., matches or approaches the edge-connectivity upper bound—while ensuring that the depth of the primary subset of trees is asymptotically equivalent to that of EDSTs in the constituent factor graphs, significantly outperforming prior constructions. Experimental and theoretical evaluation confirms that our approach simultaneously maximizes the number of EDSTs and minimizes tree depth, thereby substantially improving communication parallelism, robustness, and scalability. The result establishes a provably efficient routing foundation for high-performance interconnection networks.
📝 Abstract
A star-product operation may be used to create large graphs from smaller factor graphs. Network topologies based on star-products demonstrate several advantages including low-diameter, high scalability, modularity and others. Many state-of-the-art diameter-2 and -3 topologies~(Slim Fly, Bundlefly, PolarStar etc.) can be represented as star products. In this paper, we explore constructions of edge-disjoint spanning trees~(EDSTs) in star-product topologies. EDSTs expose multiple parallel disjoint pathways in the network and can be leveraged to accelerate collective communication, enhance fault tolerance and network recovery, and manage congestion. Our EDSTs have provably maximum or near-maximum cardinality which amplifies their benefits. We further analyze their depths and show that for one of our constructions, all trees have order of the depth of the EDSTs of the factor graphs, and for all other constructions, a large subset of the trees have that depth.