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Convergence of the non-uniform Physarum dynamics
Theoretical Computer Science ( IF 0.9 ) Pub Date : 2020-02-28 , DOI: 10.1016/j.tcs.2020.02.032
Andreas Karrenbauer , Pavel Kolev , Kurt Mehlhorn

The Physarum computing model is an analog computing model motivated by the network dynamics of the slime mold Physarum Polycephalum. In previous works, it was shown that it can solve a class of linear programs. We extend these results to a more general dynamics motivated by situations where the slime mold operates in a non-uniform environment.

Let cZ>0m, AZn×m, and bZn. We show under fairly general conditions that the non-uniform Physarum dynamicsx˙e=ae(x,t)(|qe|xe) converges to the optimum solution x of the weighted basis pursuit problem minimize cTx subject to Af=b and |f|x. Here, f and x are m-dimensional vectors of real variables, q minimizes the energy e(ce/xe)qe2 subject to the constraints Aq=b and supp(q)supp(x), and ae(x,t)>0 is the reactivity of edge e to the difference |qe|xe at time t and in state x. Previously convergence was only shown for the uniform case ae(x,t)=1 for all e, x, and t.

We also show convergence for the dynamicsx˙e=xe(ge(|qe|xe)1), where each ge is an increasing differentiable function with ge(1)=1 (satisfying some mild conditions). Previously, convergence was only shown for the special case of the shortest path problem on a graph consisting of two nodes connected by parallel edges.



中文翻译:

非均匀动力的收敛

Physarum计算模型是一种模拟计算模型,该模型是由粘液霉菌Physarum Polycephalum的网络动力学驱动的。在以前的工作中,它可以解决一类线性程序。我们将这些结果扩展到更广泛的动态范围,这些动态范围是由粘液模在非均匀环境中运行的情况引起的。

Cž>0一种žñ×bžñ。我们在相当普遍的条件下证明了非均匀的Phys骨动力学X˙Ë=一种ËXŤ|qË|-XË 收敛到最佳解决方案 X 加权基准追踪问题的最小化 CŤX 服从 一种F=b|F|X。在这里,fx是实变量的m维向量,q使能量最小ËCË/XËqË2 受约束 一种q=b支持q支持X一种ËXŤ>0是边e对差异的反应性|qË|-XË在时间t和状态x。以前仅在统一情况下显示收敛一种ËXŤ=1个对于所有ext

我们还展示了动力学的收敛性X˙Ë=XËGË|qË|XË-1个 每个在哪里 GË 是一个越来越微分的功能 GË1个=1个(满足一些温和的条件)。以前,仅在图上最短路径问题的特殊情况下才显示收敛性,该图由两个由平行边连接的节点组成。

更新日期:2020-02-28
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