传播中拓扑效应的范围
The Range of Topological Effects on Communication
摘要：
我们继续研究通信成本的计算功能时，输入分布在$k$处理器，其中每个处理器位于一个顶点的网络/图形称为终端。网络的每个其他节点也有一个处理器，没有输入。通信是点对点的，成本是协议交换的比特总数，在最坏的情况下，在所有边缘上。Chattopadhyay，Radhakrishnan和Rudra（FOCS'14）最近开始使用$L_1$嵌入工具研究网络拓扑对总通信成本的影响。他们的技术为元素清晰度（ED）等简单函数提供了严格的界限，这些函数取决于图的1-中位数。这项工作涉及两种其他类型的自然功能。我们发现，对于一个大类的自然功能，如集不相交的通信成本基本上是$n$倍的成本最佳斯坦纳树连接终端。此外，我们自然组成的功能，如$\text{ED} \circ \text{XOR}$和$\text{XOR} \circ \text{ED}$，天真的协议，建议他们的定义是最佳的一般网络。有趣的是，这些函数的边界取决于更复杂的拓扑参数，这些参数是Steiner树和1-中值成本的组合。为了获得我们的结果，我们使用一些新的工具，除了在Chattopadhyay等。这些包括（i）通过线性规划查看通信约束;（ii）使用来自树嵌入理论的工具来证明处理组合函数的情况的拓扑敏感的直和结果;以及（iii）将某些问题的通信约束表示为多路切割的集合的家族，其中每个多路切割模拟在星形拓扑上计算函数的难度。
英文摘要：
We continue the study of communication cost of computing functions when inputs are distributed among $k$ processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges. Chattopadhyay, Radhakrishnan and Rudra (FOCS'14) recently initiated a study of the effect of topology of the network on the total communication cost using tools from $L_1$ embeddings. Their techniques provided tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. This work addresses two other kinds of natural functions. We show that for a large class of natural functions like Set-Disjointness the communication cost is essentially $n$ times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like $\text{ED} \circ \text{XOR}$ and $\text{XOR} \circ \text{ED}$, the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs. To obtain our results, we use some new tools in addition to ones used in Chattopadhyay et. al. These include (i) viewing the communication constraints via a linear program; (ii) using tools from the theory of tree embeddings to prove topology sensitive direct sum results that handle the case of composed functions and (iii) representing the communication constraints of certain problems as a family of collection of multiway cuts, where each multiway cut simulates the hardness of computing the function on the star topology.
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