Near-Optimal Four-Cycle Counting in Graph Streams

Abstract

We study four-cycle counting in arbitrary order graph streams. We present a 3-pass algorithm for $(1+\varepsilon)$-approximating the number of four-cycles using $\widetilde{O}(m/\sqrt{T})$ space, where $m$ is the number of edges and $T$ the number of four-cycles in the graph. This improves upon a 3-pass algorithm by Vorotnikova using space $\widetilde{O}(m/T^{1/3})$ and matches a multi-pass lower bound of $Ω(m/\sqrt{T})$ by McGregor and Vorotnikova.

Figures (4)

• Figure 1: Illustrations of the three hard instances we consider in \ref{['sec:challenges-node-sampling']}.
• Figure 2: Illustration of our assigned configurations $(A,\kappa,x,y)$. If $x$ and $y$ are opposite nodes in a four-cycle $A$, we will only consider $A$ if $x,y\in S_{1,\kappa}$ and its other two nodes are in $S_{2,\kappa}$and it does not contain an edge $e$ of heaviness $t(e)>\sqrt{T}$. Otherwise, if $x$ and $y$ are adjacent, we only consider $A$ if $x\in R_{1a,\kappa}$ and $y\in R_{1b,\kappa}$and it has exactly one edge $e$ of heaviness $t(e)\geq \sqrt{T}$. The dashed line represents the heavy edge $e$ and we note that our algorithm collects it in the second pass.
• Figure 3: Visualization of the probabilities $p_{(z,\kappa,\ell)}$ and the heaviness thresholds $\theta_{(z,\kappa,\ell)}$ for given labeled substructures $(z,\kappa,\ell)$. In the first two figures, we annotated each vertex $v$ with its label $\ell(v)$; for the probabilities, we omitted the dependency on $\kappa$ for better readability. The third and fourth figure include the dependency on $\kappa$ and list the heaviness thresholds for each node, each edge, and two of the four wedges.
• Figure 4: Example of a substructure with non-monotone heaviness. Here, we consider a node $v$ with label $\ell(v)=S_1$ which is contained in $2T^{3/4}/\delta^{1.5}$ edge-disjoint four-cycles. This makes $v$ heavy for $\kappa=T^{1/4}$ as $t(v,T^{1/4},\ell)>\theta_{(v,T^{1/4},\ell)}=T^{3/4}/\delta^{1.5}$. Additionally, $v$ is endpoint of $T^{1/8}/\delta^{1.5}$ onions of width $\kappa'=T^{3/8}$. Then all four-cycles in the onions do not count towards $t(v,\kappa'/2,\ell)$ but do count towards $t(v,\kappa',\ell)$. As $t(v,T^{3/8},\ell)>\theta_{(v,T^{3/8},\ell)}=T^{7/8}/\delta^{1.5}$, we observe that $v$ is heavy for $\kappa=T^{1/4}$ and $\kappa=\kappa'$ but light for $\kappa=\kappa'/2$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 5.1
• Definition 5.2
• Definition 5.3
• Proposition 5.6
• Lemma 5.7: mcgregor2020triangle
• Lemma 5.8
• proof
• Lemma 5.9: Combination lemma
• proof