Edge-Disjoint Spanning Trees on Star-Product Networks

TL;DR

This work studies edge-disjoint spanning trees (EDSTs) in star-product network topologies, addressing fault tolerance and parallelism for large-scale communications. It develops a universal construction that yields $t_s+t_n-2$ EDSTs on any star product by combining EDSTs from the factor graphs, and introduces additional maximum constructions that can reach $t_s+t_n$ (or $t_s+t_n-1$) under specific edge-availability conditions; it also analyzes the depth of these EDSTs, showing the universal solution has depth closely tied to the factor-graph EDST depths, while maximum constructions may incur higher (often quadratic) depth. The results apply to contemporary star-product networks such as Slim Fly, BundleFly, and PolarStar, providing tight upper bounds and practical tradeoffs between EDST cardinality and latency. Overall, the paper advances topology-aware EDST design for scalable in-network collective communication and fault-tolerant computing, offering constructive tools to tailor EDST sets to application needs in HPC and datacenter environments.

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.

Figures (10)

• Figure 1: Comparing Cartesian and star products on structure graph $C_4$ and supernode $C_3$. Figure \ref{['fig:cart_star_comp_2_cart']} is the unique Cartesian product on $C_4$ and $C_3$, joining equivalent vertices of the same color. Figures \ref{['fig:cart_star_comp_20_star']} and \ref{['fig:cart_star_comp_21_star']} are two non-Cartesian star products.
• Figure 2: The $f$-transforms of Cartesian spanning trees to a star product may not be spanning trees. Figures \ref{['fig:const_counterex_cart1']} and \ref{['fig:const_counterex_cart2']} show the two EDSTs from Product_STs_2003 for the Cartesian product in Figure \ref{['fig:cart_star_comp_2_cart']}. The bijection $f$ from Figure \ref{['fig:cart_star_comp_21_star']} joins blue and green vertices between supernodes and transforms these spanning trees to get Figures \ref{['fig:const_counterex_star1']} and \ref{['fig:const_counterex_star2']}. These edge sets are not spanning trees: they have dotted-line cycles and exclude white vertices.
• Figure 3: The constructions producing $t_s+t_n-2$ of the trees in the Maximum solution when $r_s=t_s$ and $r_n=t_n$. The legend depicts the choices of vertices and edges used to construct these EDSTs coming from the edges in the EDSTS $\mathbf{X} = \{X_1, X_2, \ldots, X_{t_s}\}$ of $G_s$ and $\mathbf{Y} = \{Y_1, Y_2, \ldots, Y_{t_n}\}$ of $G_n$. The constructions \ref{['construction:t1_trees']} and \ref{['construction:t2_trees']} producing the Universal solution of $t_s+t_n-2$ EDSTs can be seen as generalizations of Constructions \ref{['construction:t1_trees_maximum']} and \ref{['construction:t2_trees_maximum']}, where we allow $u_i$ to be an arbitrary vertex in $V_s$ that is unique for each $i$ (corresponds to arbitrary supernode in product graph).
• Figure 4: To construct low-depth EDSTs, we modify the Universal Construction \ref{['construction:t2_trees']} which is a generalization of Construction \ref{['construction:t2_trees_maximum']} shown in Figure \ref{['fig:max_solutions_universal']}. Instead of using edges from different copies of $\bar{X}_1$, we trace out a single copy of $\bar{X}_1$. Inter-supernode traversal is done along this copy and thus, avoids hopping within intermediate supernodes.
• Figure 5: The constructions producing the additional two trees in the Maximum solution. The legend depicts the choices of vertices and edges used to construct these EDSTs. As in Figure \ref{['fig:max_solutions_universal']}, the gray boxes represent supernodes. Supernode $o$ is distinct from the supernodes where $Y_1$ is instantiated in Construction \ref{['construction:t1_trees_maximum']} (Figure \ref{['fig:max_solutions_universal']}).

Theorems & Definitions (51)

• Definition 2.2.1
• Definition 2.2.2
• Example 2.4.1
• Example 2.4.2
• Example 2.4.3
• Example 2.4.4
• Example 2.4.5
• Example 2.4.6
• Proposition 4.0.0
• proof