随机加权路径的总捕获数
The Total Acquisition Number of the Randomly Weighted Path
摘要：
存在一个显着的身体上的工作，以确定收购数$a_t(G)$的各种图形时，这些图形的顶点最初被分配一个单位的权重。我们确定的路径，星，完整的，完整的二分图，周期和车轮图的变化，这个初始的加权方案的收购数量的属性，与我们的大部分工作集中在预期的收购数量的随机加权图。特别地，当整数权重的$n$可区分“单元”或码片随机分布在其顶点之间的$0.242n$和$0.375n$之间时，我们约束$n$路径的预期采集数$E(a_t(P_n))$。在计算机的支持下，我们通过显示$E(a_t(P_n))$位于$0.29523n$和$0.29576n$之间来改进它。然后，我们使用次可加性表明，存在的限制比$\lim E(a_t(P_n))/n$，和模拟揭示更确切的极限值等于。利用Hoeffding-Azuma不等式证明了收购数紧紧集中在其期望值附近。此外，在不同的上下文中，我们提供了一个非最佳的采集协议算法的随机加权路径，并准确地计算所得到的残差集的预期大小。
英文摘要：
There exists a significant body of work on determining the acquisition number $a_t(G)$ of various graphs when the vertices of those graphs are each initially assigned a unit weight. We determine properties of the acquisition number of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the expected acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number $E(a_t(P_n))$ of the $n$-path when $n$ distinguishable "units" of integral weight, or chips, are randomly distributed across its vertices between $0.242n$ and $0.375n$. With computer support, we improve it by showing that $E(a_t(P_n))$ lies between $0.29523n$ and $0.29576n$. We then use subadditivity to show that the limiting ratio $\lim E(a_t(P_n))/n$ exists, and simulations reveal more exactly what the limiting value equals. The Hoeffding-Azuma inequality is used to prove that the acquisition number is tightly concentrated around its expected value. Additionally, in a different context, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and exactly compute the expected size of the resultant residual set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
