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.

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