Priority-splitter balancer with near-optimal fair splitter count

Ascertain whether there exists a priority-based (2^k, 2^k)-balancer whose first-stage network using priority splitters can be implemented with only O(k·2^k) priority splitters while preserving the stated saturating-balancer behavior (i.e., pushing up to 2^{k−1} units to top rows and achieving balancing via the subsequent simple balancer), thereby reducing the total number of priority splitters from o(2^{2k}) to O(k·2^k).

Background

The authors construct a saturating (2^k, 2^k)-balancer that uses priorities to push flow into a top block and then employs a simple balancer and a final column of fair splitters. This design uses significantly fewer fair splitters—(k+1)·2^{k−2}—bringing the count close to their lower bound k·2^{k−2}.

However, the initial half-grid of priority splitters is large, leading to a total size dominated by Θ(2^{2k}). The authors note that the half-grid could be replaced by a network that ensures the top 2^{k−1} leaving arcs carry min{c(I), 2^{k−1}} total throughput, and that such a network can be defined with o(2^{2k}) priority splitters.

The unresolved question is whether one can further compress this component to O(k·2^k) priority splitters, which would dramatically improve the practicality and theoretical tightness of priority-based balancers.