PolarStar: Expanding the Scalability Horizon of Diameter-3 Networks

TL;DR

PolarStar targets scalable, low-diameter interconnects by using a star-product framework that combines a large diameter-2 structure graph with a diameter-3 supernode to create diameter-3 networks with unprecedented scale. By selecting ER_q as the structure graph (property $R$) and Inductive-Quad (IQ) as the supernode (property $R^*$), PolarStar achieves near-Moore-bound efficiency and outperforms existing diameter-3 topologies across radixes in the range $[8,128]$, including 1.3x over Bundlefly, 1.9x over Dragonfly, and 6.7x over 3-D HyperX on geometric means. The design supports a modular, bundlable layout and a rich design space with multiple configurations per radix, along with a routing approach that exploits the underlying graph properties to minimize state and latency. Through synthetic and real-world motif simulations, PolarStar demonstrates competitive or superior performance while delivering higher scalability and resilience to link failures, making it a practical candidate for future exascale and co-packaged HPC interconnects.

Abstract

We present PolarStar, a novel family of diameter-3 network topologies derived from the star product of low-diameter factor graphs. PolarStar gives the largest known diameter-3 network topologies for almost all radixes, thus providing the best known scalable diameter-$3$ network. Compared to current state-of-the-art diameter-$3$ networks, PolarStar achieves $1.3\times$ geometric mean increase in scale over Bundlefly, $1.9\times$ over Dragonfly, and $6.7\times$ over {3-D} HyperX. PolarStar has many other desirable properties, including a modular layout, large bisection, high resilience to link failures and a large number of feasible configurations for every radix. We give a detailed evaluation with simulations of synthetic and real-world traffic patterns and show that PolarStar exhibits comparable or better performance than current diameter-3 networks.

Figures (14)

• Figure 1: Scalability of direct diameter-3 topologies with respect to the Moore bound. Data labels show the largest number of nodes and corresponding radix in each topology for radix $\leq 64$. StarMax denotes an upper bound on the largest graphs theoretically achievable with $P-$ and $R-$star products -- the mathematical constructs used in state-of-the-art Bundlefly and PolarStar networks. For Spectralfly, which is not a fixed diameter topology, we only compare design points with diameter $\leq 3$ and largest scale for a given radix (if it exists). For Kautz networks, we consider each link as bidirectional.
• Figure 2: A comparison of the Cartesian product $L_3 \times C_4$ with an example star product $L_3 * C_4$. The structure graph $L_3$ is the path graph on three vertices, and the supernode $C_4$ is the cycle graph on four vertices.
• Figure 3: This figure illustrates Theorem \ref{['thm_diam-3']} on a diameter-2 structure graph with property $R$, showing how paths are constructed from $2$- and $3$-hop alternating paths. The star product path may include a $1$-hop supernode detour, shown in \ref{['fig:diameter_p2']} and \ref{['fig:diameter_p3']}, which is permitted by Property \ref{['prop_R_star']} on $G'$.
• Figure 4: Moore-bound comparison for some known families of diameter-$2$ graphs: the $ER$ graph, the McKay-Miller-Širáň graphs mckay98, the best Cayley graphs abas_2017, and the Paley graph. It can be seen that any larger structure graph would only marginally increase the size of the star product.
• Figure 5: Construction of the star product $ER_3*Paley($5$)$.

Theorems & Definitions (9)

• Definition 1
• Proposition 1
• Definition 2
• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 2
• Corollary 1