Optimal Oblivious Load-Balancing for Sparse Traffic in Large-Scale Satellite Networks

TL;DR

This work tackles oblivious load-balancing on large-scale torus networks under sparse traffic, motivated by LEO satellite constellations. It derives a universal lower bound on the worst-case edge load for oblivious routing and shows that Valiant Load Balancing is suboptimal when traffic is sparse, motivating a specialized scheme. The authors introduce Local Load-Balancing (LLB), a symmetry-aware, three-phase routing policy that achieves the lower bound (up to constants) and is proven optimal in key cases (e.g., when $2k$ is a perfect square), with extensions to general $N\times M$ tori and unequal link capacities. Numerical results corroborate the theory, showing LLB matching the best oblivious performance and outperforming VLB in hotspot scenarios, highlighting the practical impact for distributed satellite networks and torus-based interconnects.

Abstract

Oblivious load-balancing in networks involves routing traffic from sources to destinations using predetermined routes independent of the traffic, so that the maximum load on any link in the network is minimized. We investigate oblivious load-balancing schemes for a $N\times N$ torus network under sparse traffic where there are at most $k$ active source-destination pairs. We are motivated by the problem of load-balancing in large-scale LEO satellite networks, which can be modelled as a torus, where the traffic is known to be sparse and localized to certain hotspot areas. We formulate the problem as a linear program and show that no oblivious routing scheme can achieve a worst-case load lower than approximately $\frac{\sqrt{2k}}{4}$ when $1<k \leq N^2/2$ and $\frac{N}{4}$ when $N^2/2\leq k\leq N^2$. Moreover, we demonstrate that the celebrated Valiant Load Balancing scheme is suboptimal under sparse traffic and construct an optimal oblivious load-balancing scheme that achieves the lower bound. Further, we discover a $\sqrt{2}$ multiplicative gap between the worst-case load of a non-oblivious routing and the worst-case load of any oblivious routing. The results can also be extended to general $N\times M$ tori with unequal link capacities along the vertical and horizontal directions.

Figures (10)

• Figure 1: $N\times N$ torus with $N=4$.
• Figure 2: World traffic heatmap at particular time of day, overlaid with a 10 orbit constellation with 100 satellites. Dark regions indicate high traffic. Satellites that would experience high traffic demand are boxed. Data provided by our sponsor.
• Figure 3: The minimum cut set for $k = 4$. The set of vertices $S$ is highlighted. The edges to be cut are marked with a cross.
• Figure 4: Set of source nodes $S$ is shaded with red vertical lines. Set of sink nodes $T$ is shaded with blue horizontal lines. The dark edges in the figure lie in the cut-set of both $S$ and $T$. These edges would see a load of at least $\frac{2\sqrt{k}}{4}(1-k/N^2)$ under the VLB scheme.
• Figure 5: Symmetries of Routing Policies

Theorems & Definitions (14)

• Lemma 1
• Definition 1: Automorphism chitavisutthivong_optimal_2023
• Lemma 2: Adopted from chitavisutthivong_optimal_2023
• Lemma 3
• Theorem 1
• Lemma 4
• Theorem 2
• proof : Proof of Lemma \ref{['lemma:k_sparse_k_limited_equivalence']}
• proof : Proof of Lemma \ref{['lemma:symmetry_sufficiency_condition']}
• Lemma 5