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.

翻译：网络中的无感知负载均衡涉及使用独立于流量的预定路由将流量从源节点路由至目的节点，以最小化网络中任意链路上的最大负载。我们研究了$N\times N$环面网络在稀疏流量下的无感知负载均衡方案，其中最多存在$k$个活跃的源-目的对。该研究受到大规模低地球轨道卫星网络负载均衡问题的启发，此类网络可建模为环面结构，且已知其流量具有稀疏性并集中于特定热点区域。我们将该问题表述为线性规划，并证明当$1<k \leq N^2/2$时，任何无感知路由方案的最坏情况负载下界约为$\frac{\sqrt{2k}}{4}$；当$N^2/2\leq k\leq N^2$时，下界为$\frac{N}{4}$。此外，我们论证了著名的Valiant负载均衡方案在稀疏流量下并非最优，并构建了能够达到该下界的最优无感知负载均衡方案。进一步地，我们发现非无感知路由的最坏情况负载与任意无感知路由的最坏情况负载之间存在$\sqrt{2}$倍的乘性差距。该结果亦可推广至垂直与水平方向链路容量不等的广义$N\times M$环面网络。