We continue the study of the communication complexity of gap cycle counting problems. These problems have been introduced by Verbin and Yu [SODA 2011] and have found numerous applications in proving streaming lower bounds. In the noisy gap cycle counting problem (NGC), there is a small integer $k \geq 1$ and an $n$-vertex graph consisted of vertex-disjoint union of either $k$-cycles or $2k$-cycles, plus $O(n/k)$ disjoint paths of length $k-1$ in both cases (``noise''). The edges of this graph are partitioned between Alice and Bob whose goal is to decide which case the graph belongs to with minimal communication from Alice to Bob. We study the robust communication complexity -- `a la Chakrabarti, Cormode, and McGregor [STOC 2008] -- of NGC, namely, when edges are partitioned randomly between the players. This is in contrast to all prior work on gap cycle counting problems in adversarial partitions. While NGC can be solved trivially with zero communication when $k < \log{n}$, we prove that when $k$ is a constant factor larger than $\log{n}$, the robust (one-way) communication complexity of NGC is $\Omega(n)$ bits. As a corollary of this result, we can prove several new graph streaming lower bounds for random order streams. In particular, we show that any streaming algorithm that for every $\varepsilon > 0$ estimates the number of connected components of a graph presented in a random order stream to within an $\varepsilon \cdot n$ additive factor requires $2^{\Omega(1/\varepsilon)}$ space, settling a conjecture of Peng and Sohler [SODA 2018]. We further discuss new implications of our lower bounds to other problems such as estimating size of maximum matchings and independent sets on planar graphs, random walks, as well as to stochastic streams.

翻译：我们继续研究间隙环计数问题的通信复杂度。这些问题由Verbin和Yu [SODA 2011]提出，并在证明流算法下界方面具有广泛应用。在噪声间隙环计数问题（NGC）中，存在一个小整数$k \geq 1$和一个由顶点不相交的$k$-环或$2k$-环组成的$n$顶点图，两种情况下均包含$O(n/k)$条长度为$k-1$的不相交路径（即“噪声”）。该图的边被分配到Alice和Bob之间，目标是通过最小化Alice到Bob的通信量，判断图属于哪种情况。我们研究NGC的鲁棒通信复杂度——遵循Chakrabarti、Cormode和McGregor [STOC 2008]的定义，即边在玩家之间随机分配。这与以往所有在对抗性划分下研究间隙环计数问题的工作形成对比。虽然当$k < \log{n}$时，NGC可以在零通信下简单解决，但我们证明当$k$比$\log{n}$大一个常数因子时，NGC的鲁棒（单向）通信复杂度为$\Omega(n)$比特。作为该结果的推论，我们可以证明随机顺序流中多个新的图流算法下界。特别地，我们证明：任何对于每个$\varepsilon > 0$都能在随机顺序流中估计图的连通分量个数，且误差在$\varepsilon \cdot n$加法因子内的流算法，需要$2^{\Omega(1/\varepsilon)}$空间，从而解决了Peng和Sohler [SODA 2018]的一个猜想。我们进一步讨论这些下界对其他问题的新影响，包括平面图上的最大匹配和独立集大小估计、随机游走以及随机流问题。