Multi-hop route optimization in large-scale inhomogeneous networks is typically NP-hard, for most problem formulations, requiring the application of heuristics which, despite their relatively low processing complexity, find sub-optimal solutions. Where optimal solutions can be determined by Lagrangian based constrained optimization techniques for example, the processing complexity typically scales like $O(N^{3})$ , $N$ being the number of relays employed. Here, we propose an alternative approach to route optimization by considering the limit of infinite relay node density to develop a continuum model, which yields an optimized equivalent continuous relay path. The model is carefully constructed to maintain a constant connection density even though the node density scales without bound. This leads to a formulation for minimizing the end-to-end outage probability that can be solved using methods from the calculus of variations. With the continuum model, we show that the processing complexity scales linearly with the number of points that sample the continuous path, which can be lower than the number of relay nodes in a large scale network. We demonstrate the effectiveness of this new approach and its potential by considering a network subjected to point sources of interference.

中文翻译：

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大规模非同类多跳无线网络中路由优化的连续模型

大规模非均匀网络中的多跳路由优化通常是 NP-hard，对于大多数问题公式，需要应用启发式算法，尽管它们的处理复杂性相对较低，但可以找到次优解决方案。例如，在可以通过基于拉格朗日的约束优化技术确定最佳解决方案的情况下，处理复杂度通常按比例缩放，如 $O(N^{3})$，$N$ 是采用的继电器数量。在这里，我们提出了一种路由优化的替代方法，通过考虑无限中继节点密度的限制来开发连续模型，从而产生优化的等效连续中继路径。该模型经过精心构建，以保持恒定的连接密度，即使节点密度无限制地扩展。这导致了用于最小化端到端中断概率的公式，该公式可以使用来自变化演算的方法来解决。通过连续模型，我们表明处理复杂度与对连续路径采样的点数呈线性关系，这可能低于大规模网络中中继节点的数量。我们通过考虑受点干扰源影响的网络来证明这种新方法的有效性及其潜力。