利用有限资源构建图
Constructing graphs with limited resources
摘要：
我们讨论了构建一个给定的图所需的物理资源的数量，其中顶点顺序添加。我们很自然地将信息（分为指令和记忆）和随机性视为资源。毫不奇怪，我们表明，在这个框架中，阈值图是最简单的可能的图形，因为阈值图的建设需要一个单一的位的指令为每个顶点，不使用内存。没有内存的大指令不会带来任何优势。对于每个顶点，只需一位指令和一位内存，我们就可以构造出严格包含阈值图的完美图族。我们考虑的情况下，内存持续一个单一的时间步长，并表明，以及标准的阈值图，线性森林也是可生产的。我们进一步表明，所需的随机位（没有内存或指令），以构建任何图形的数量是渐近相同的所需的Erdens-Rényi随机图。我们还简要地考虑在这个方案中构建树。在所提出的框架中定义图的层次结构的问题是完全开放的。
英文摘要：
We discuss the amount of physical resources required to construct a given graph, where vertices are added sequentially. We naturally identify information -- distinct into instructions and memory -- and randomness as resources. Not surprisingly, we show that, in this framework, threshold graphs are the simplest possible graphs, since the construction of threshold graphs requires a single bit of instructions for each vertex and no use of memory. Large instructions without memory do not bring any advantage. With one bit of instructions and one bit of memory for each vertex, we can construct a family of perfect graphs that strictly includes threshold graphs. We consider the case in which memory lasts for a single time step, and show that as well as the standard threshold graphs, linear forests are also producible. We show further that the number of random bits (with no memory or instructions) needed to construct any graph is asymptotically the same as required for the Erdős-Rényi random graph. We also briefly consider constructing trees in this scheme. The problem of defining a hierarchy of graphs in the proposed framework is fully open.
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