SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma 3.4. Suppose  $G = (V, E)$  is any labeled graph and OPT be the optimal correlation clustering cost, and let  $\varepsilon \leq \frac{1}{32}$  and  $\beta \leq \frac{1}{2\varepsilon}$ , then  $m_{\beta \varepsilon \text{-sparse}}^{+} \leq \frac{2}{\beta \varepsilon} \cdot \text{OPT}$  and  $\widehat{m}_{\beta \varepsilon \text{-dense}}^{-} \leq 8 \cdot \text{OPT}$ .
By Lemmas 3.3 and 3.4 if we are able to exactly recover  $m_{\varepsilon\text{-sparse}}^{+}$  and  $\widehat{m}_{\varepsilon\text{-dense}}^{-}$  then we would get an  $O(1)$  approximation of OPT. Such a task seems difficult, however, using Lemmas 3.1 and 3.2 we design our tools for estimating  $m_{\varepsilon\text{-sparse}}^{+}$  and  $\widehat{m}_{\varepsilon\text{-dense}}^{-}$  with an additive error.
Lemma 3.5. There exists a dynamic streaming algorithm with parameters  $\varepsilon, \delta$  that given any graph  $G = (V, E)$  in a stream returns a value  $Z_{sp}$  that is at least  $m_{\varepsilon\text{-sparse}}^{+}$  and at most  $m_{\varepsilon/8\text{-sparse}}^{+} + \delta n^{2}$  with high probability and uses  $O(\log n / \varepsilon^{2}\delta^{3})$  words of space.
Lemma 3.6. There exists a dynamic streaming algorithm with parameters  $\varepsilon, \delta$  that given any graph  $G = (V, E)$  in a stream returns a value  $Z_{den}$  that is at least  $\widehat{m}_{\varepsilon\text{-dense}}^{-}$  and at most  $\widehat{m}_{8\varepsilon\text{-dense}}^{-} + \delta n^{2}$  with high probability and uses  $O(\log^{2}n / \varepsilon^{2}\delta^{5})$  words of space.
Finalizing the proof sketch of Theorem 1. We are given a parameter  $\delta$  as input and we want the additive error to be at most  $\delta n^2$ . We fix  $\varepsilon = 1/360$ ,  $\eta_0 = 1/20$  and  $\delta' = \delta \cdot (2 + \frac{2}{\eta_0\varepsilon})^{-1}$ .
Run the algorithm in Lemma 3.5 with parameters  $\varepsilon$  and  $\delta'$ , and let the output be  $Z_{sp}^{\varepsilon}$ . Also, run the algorithm in Lemma 3.5 with parameters  $\eta_0\varepsilon$  and  $\delta'$ , and let the output be  $Z_{sp}^{\eta_0\varepsilon}$ . Run the algorithm in Lemma 3.6 with parameters  $4\varepsilon$  and  $\delta'$ , and let the output be  $Z_{den}$ .
Lemmas 3.3 to 3.6 imply that the cost  $Z_{CC}$  we output satisfies:
$$
\mathsf {O P T} \leq Z _ {C C} := Z _ {s p} ^ {\varepsilon} + (2 / \eta_ {0} \varepsilon) \cdot Z _ {s p} ^ {\eta_ {0} \varepsilon} + Z _ {d e n} \leq O (1) \cdot \mathsf {O P T} + \delta n ^ {2}.
$$
The space taken is  $O(\log n / \delta^3)$  for both copies of Lemma 3.5 and  $O(\log^2 n / \delta^5)$  for Lemma 3.6 giving a total space of  $O(\log^2 n / \delta^5)$  words. This proves Theorem 1.
4 An Algorithm based on Pivot
In this section, we give our second streaming algorithm that is a  $(3 + \gamma, \delta n^2)$ -approximation for the correlation clustering value for any choice of  $\delta$  and  $\gamma < 1/2$ . The algorithm works in  $O(\log \log (n))$  space as long as  $\delta \geq \Omega \left( \frac{1}{\log \log(n)} \right)$ . Consider the following formal statement:
Theorem 2. There is a (dynamic) streaming algorithm that with high probability gives a  $(3 + \gamma, \delta n^{2})$ -approximation for the correlation clustering value and takes space  $O\left(\frac{2^{7/6\delta} \cdot \log^{2} n}{\gamma \cdot \delta^{5}}\right)$  words.
Our algorithm in Theorem 2 is inspired by the Local Cluster algorithm from [BGK13]. The Local Cluster algorithm samples  $1 / \delta$  random vertices in a set  $U$  and computes the greedy maximal independent set (MIS)  $M$  of  $U$  to get the cluster centers  $p_1, p_2, \ldots, p_t$ . The clusters generated then are  $N^+ [p_i] - \cup_{j=1}^{i-1} N^+ [p_j]$  and all the remaining vertices (called unclustered vertices) are clustered in their own singleton cluster. [BGK13] proved that the expected cost of this clustering is at most  $3 \cdot \mathsf{OPT} + \frac{\delta}{2} n^2$ . More formally, they showed the following:
Proposition 4.1 ([BGK13]). The expected cost of Local Cluster is at most  $3 \cdot \mathsf{OPT} + \frac{\delta}{2} n^2$ .
To simulate the Local Cluster algorithm in a streaming manner, while forming a cluster we need to know which of its neighbors are already clustered in previous clusters which requires new sketching tools (on  $G^{+}$ ) that are different from the ones we used in Section 3. These tools estimate the number of non-edges within or the number of edges going out of the neighborhood of a vertex  $(u)$  outside of the (+) neighborhood of a known set of vertices  $(S)$ . In other words, fix a graph  $G = (V, E)$ , a vertex  $u$ , and a set  $S$ , we want to estimate the number of non-edges within or the number of edges going out of  $N[u] - N[S]$ . We again use the generic form to present the sketching tools and do not specify (+) edges. The formal definitions can be given as follows.
Definition 3. A non-edge  $(x,y)$  is within  $N[u] - N[S]$  iff: i)  $x \notin N[S]$  and ii)  $y \notin N[S]$  and iii)  $x \in N[u]$  and  $y \in N[u]$ .
Definition 4. An edge  $(x,y)$  is going out of  $N[u] - N[S]$  iff  $i)$ $x\notin N[S]$  and  $ii)$ $y\notin N[S]$  and  $iii)$ $x\in N[u]$  or  $y\in N[u]$  but not both.
Definition 5. An edge  $(x,y)$  is unclustered w.r.t.  $S$  iff  $i) x \notin N[S]$  and  $ii) y \notin N[S]$ .
We also define  $m_{ne}(u, S)$  as the number of non-edges within  $N[u] - N[S]$ ,  $m_e(u, S)$  as the number of edges going out of  $N[u] - N[S]$ , and  $m_u(S)$  as the number of unclustered edges w.r.t.  $S$ . We write  $m_{ne}, m_e$ , and  $m_u$  when  $u$  and  $S$  are clear from context.
Exactly recovering  $m_u, m_{ne}$  and  $m_e$  is difficult, so we have the following lemma about estimating the number of non-edges within  $N[u] - N[S]$ , the edges going out of  $N[u] - N[S]$ , and the unclustered edges with respect to  $S$ :
Lemma 4.2. There exist streaming algorithms called NE-Tool, E-Tool, and U-Tool that given  $u$  and  $S$  before the stream, respectively compute with high probability  $i$ ). The number of non-edges within; ii). the number of edges going out of  $N[u] - N[S]$ ; and iii). the number of edges with unclustered w.r.t.  $S$  with an overestimation of at most  $\delta n^2$  and take space  $O(1 / \delta^2)$  words.
Using the tools we have built so far, we present a  $(3,\delta n^2)$  approximation algorithm in expectation.
Algorithm 1. Simulation of the Local Cluster algorithm
Input:  $G = (V, E^{+} \cup E^{-})$  in a (dynamic) stream
Output:  $(3,\delta n^{2})$ -approximation to the correlation clustering value in expectation
Pre-Processing:
1. Sample a set  $U$  of  $1 / \delta$  random vertices
2. Let  $\pi$  be a random permutation of the vertices in  $U$
During the Stream:
1. Store  $(+)$  all edges between vertices of  $U$
2. For all  $u \in U$  and  $S \subseteq U$  compute NE-Too1(u, S), E-Too1(u, S), U-Too1(S) with parameter  $\delta^2 / 6$  using the  $G^{+}$  subgraph.
Post-Processing:
1. Compute the greedy MIS  $M \coloneqq p_1, p_2, \ldots, p_t$  of  $U$  in the order of  $\pi$  (using edges stored between vertices of  $U$ ).
2.  $S_0 = \emptyset$ .  $\tilde{Z} = 0$
3. For  $i = 1$  to  $t$ :
$$
\begin{array}{l} \cdot \tilde {Z} = \tilde {Z} + \mathrm {N E - T o o l} \left(p _ {i}, S _ {i - 1}\right) + \mathrm {E - T o o l} \left(p _ {i}, S _ {i - 1}\right). \\ \bullet S _ {i} := S _ {i - 1} \cup \left\{p _ {i} \right\} \\ \end{array}
$$
4. Output  $Z \coloneqq \tilde{Z} + \mathrm{U - Tool}(M)$
Algorithm 1 estimates  $m_{u}, m_{ne}$  and  $m_{e}$  with at most  $\delta n^{2}$  additive error. We now show that with the exact values of  $m_{u}, m_{ne}$  and  $m_{e}$ , we get the exact clustering value of the Local Cluster algorithm.
Claim 4.3. The clustering value of Local Cluster is equal to  $m_u + \sum_i m_{ne}(p_i, S_{i-1}) + m_e(p_i, S_{i-1})$ .
Combining Lemma 4.2 and Claim 4.3 gives us the following performance guarantees for Algorithm 1.
Lemma 4.4. The output  $Z$  of Algorithm 1 is at least OPT with high probability and is at most  $3\mathrm{OPT} + \delta n^2$  in expectation.
Proof. Using Claim 4.3 we know that if the tools worked with no error, Algorithm 1 would give the exact clustering cost of Local Cluster. Also, we know that the tools do not underestimate and overestimate by  $\delta^2 n^2 /6$  with high probability (Lemma 4.2). We first note that  $Z$  is at least OPT because of the above conditioning and the fact that the clustering cost of Local Cluster is at least OPT. Using Proposition 4.1 we know that the expected cost of the clustering when choosing  $\frac{1}{\delta}$  random pivots is at most  $3\mathsf{OPT} + \frac{\delta}{2} n^{2}$ . Therefore, the clustering cost of Algorithm 1 is between OPT and  $3\mathsf{OPT} + \frac{\delta}{2} n^{2}$  plus the overestimate.
We now calculate the overestimate. We use parameter  $\delta^2 /6$  for the tools thus the additive error in each tool is at most  $\delta^2 n^2 /6$ . There are  $1 / \delta$  pivots implying a total additive error of at most  $\delta n^2 /3$  over all the copies of NE-Tool and E-Tool. U-Tool has an error of at most  $\delta^2 n^2 /6$  implying that the overall error is at most  $\delta n^2 /2 + \delta n^2 /3 + \delta^2 n^2 /6\leq \delta n^2$  giving a  $(3,\delta n^{2})$  approximation in expectation.
We note that we condition on the high probability events for all copies of the tools and union bound over the failure probabilities. Thus, for the overall failure probability to be small we need  $2^{1/\delta} \leq \mathrm{poly}(n)$ . This requirement is trivially satisfied by the fact that  $2^{1/\delta}$  is on the space bound (Lemma 4.5), and the space bound for any streaming algorithm is at most  $O(n^2)$ .
We now show the space bound of Algorithm 1.
Lemma 4.5. The space of Algorithm 1 is  $O\left(\frac{2^{1/\delta} \cdot \log n}{\delta^5}\right)$  words.
Proof. Each copy of the tools with parameter  $\delta^2 / 6$  takes  $O(\log n / \delta^4)$  words of space, and we compute the tools for all  $v \in U$  and  $S \subseteq U$ . Thus, the space used is  $O(2^{1 / \delta} \cdot \log n / \delta^5)$  words. Storing edges between vertices in  $U$  takes space at most  $1 / \delta^2$  words which is a lower order term.
Lemmas 4.4 and 4.5 together give us a  $(3, \delta n^2)$  approximation in expectation using  $O\left(\frac{2^{1/\delta} \cdot \log n}{\delta^5}\right)$  space. We now prove Theorem 2. We run Algorithm 1  $\frac{60 \log n}{\gamma}$  times in parallel and let  $Z_{min}$  be the minimum cost over all iterations.  $Z_{min}$  is a  $(3 + \gamma, \frac{7}{6} \delta n^2)$ -approximation with high probability.
Claim 4.6. OPT  $\leq Z_{min} \leq (3 + \gamma)\mathrm{OPT} + \frac{7}{6}\delta n^2$  with high probability.
Thus, we get a  $(3 + \gamma, \frac{7}{6}\delta n^{2})$ -approximation with high probability. Each parallel repetition of the algorithm takes  $O\left(\frac{2^{1/\delta} \cdot \log n}{\delta^{5}}\right)$  words of space. Repeating  $O\left(\frac{\log n}{\gamma}\right)$  times and re-scaling  $\delta$  by a factor of  $\frac{7}{6}$  gives a total space bound of  $O\left(\frac{2^{7/6\delta} \cdot \log^{2} n}{\gamma \cdot \delta^{5}}\right)$  words. This completes the proof of Theorem 2.
5 A Lower Bound for  $O(n^{2 - \varepsilon})$  Additive error
In this section, we show that if we only allow additive error, any streaming algorithm with polylogarithm memory cannot cross an error barrier of  $\Omega(n^{2 - \varepsilon})$  for some  $\varepsilon = o(1)$ . Here, and throughout, we will refer this lower bound as the almost-quadratic lower bound. The lower bound is weaker than the linear lower bound of Section 6 in terms of the multiplicative factor since it only works for  $c = 1$ . However, it is much stronger in the additive sense: the upper bounds obtained by our algorithms are  $O(n^2)$ , and the almost-quadratic lower bound matches this term up to an  $O(n^{\varepsilon})$  factor – this provides a strong justification of the additive error in our algorithms.
The formal statement of the almost-quadratic lower bound is as follows.
Theorem 3. There exists a constant  $C$ , such that any single-pass streaming algorithm that estimates the optimal value  $\mathsf{OPT}$  of correlation clustering by a  $C \cdot n^{2 - \varepsilon}$  purely additive error (i.e., an estimated value that is at most  $(\mathsf{OPT} + C \cdot n^{2 - \varepsilon})$ ) with probability at least  $\frac{99}{100}$  has to use a memory of  $\Omega(n^{\varepsilon})$  bits, even on labeled complete graphs.
Note that Theorem 3 does not require the stream to be dynamic, which is in contrast to our upper bound results that work for dynamic streams. We obtain the almost-quadratic lower bound by a new reduction from the INDEX problem. On the high level, the instance we construct 'hides' an  $\Omega(n^{2 - \varepsilon})$  gap between the yes and no cases inside a case-invariant  $\Omega(n^2)$  cost. The reduction can be viewed as a more involved variant of the space lower bound for the exact correlation clustering in a very recent work [AAD+23]. In a nutshell, we modify their construction to 'boost' the gap between yes and no cases, and apply a new trick to separate the values of clustering. Due to space limits, we defer the proof of Theorem 3 to the supplementary material.
6 A Lower Bound for  $O(n)$  Additive error
In this section, we show that any dynamic streaming algorithm that gets a  $(c,\varepsilon n)$ -approximation for  $c < 1.2$  and  $O(n)$  additive error needs  $\Omega (\sqrt{n})$  bits of space. Here, and throughout, we will call this lower bound the linear lower bound. Formally, we have:
Theorem 4. Let  $c \in [1, \frac{6}{5})$  and  $\varepsilon \in (0, \frac{6}{5} - c)$ . Any single-pass streaming algorithm that estimates the optimal value  $\mathsf{OPT}$  of correlation clustering by a  $(c, \varepsilon n)$ -approximation (i.e., an estimated value that is at most  $(c \cdot \mathsf{OPT} + \varepsilon \cdot n)$ ) with probability at least  $\frac{99}{100}$  has to use a memory of  $\Omega(\sqrt{n})$  bits, even on labeled (complete) graphs.
Similar to the case in Theorem 3, Theorem 4 does not require the stream to be dynamic. Comparing with the almost-quadratic lower bound we show in Section 5, the lower bound in Theorem 4 is weaker in the additive sense. However, it allows the multiplicative approximation of the algorithm to be  $>1$ , while the lower bound in Section 5 only rules out algorithms with purely additive errors.
Our lower bound uses the celebrated machinery from Boolean Hidden Hypermatching (BHH) and Gap Cycle Counting (GCC) pioneered by [VY11]. The Gap Cycle Counting (GCC) lower bound states that for any algorithm to distinguish whether a graph consists of cycles with length  $2t$  or cycles with length  $4t$  for some  $t \geq 2$ , a memory of  $\Omega\left(n^{1 - \frac{1}{t}}\right)$  bits is necessary. On the high level, our plan is to show that for graphs similar to the ones prescribed in the GCC problem, the values of correlation clustering are different by an additive gap of  $O(n)$ . Therefore, by a reduction argument, any algorithm that breaks this barrier of additive gap requires  $\omega$  ( $\mathrm{polylog}(n)$ ) memory.
Due to space limits, we defer the lower bound construction to the supplementary material.
We describe in this section the experiments of algorithms on the Stochastic Block Model and the Erdos-Renyi random graphs. These experiments show that for a very natural family of graphs, our algorithms can achieve a very competitive performance on the estimated cost values, and the performances are often much better than the worst-case theoretical analysis. Furthermore, our algorithms are capable of separating "well-clusterable" vs. "badly-clusterable" instances.
Experiment Settings Limited by space, we sketch the key settings of our experiment, and defer the full discussion to the full version in the supplementary material. As we have discussed, we perform our experiments on the data generated from the well-studied Stochastic Block Model (SBM) that plants ground-truth clusters with sizes  $\Omega(n)$ , samples (+) edge between vertex pairs  $(x_i, x_j)$  in the same planted cluster with probability  $p > 0.5$ , and samples (+) edges between vertex pairs  $(x_i, x_j)$  in different clusters with probability  $1 - p$ . The SBM captures a lot of real-world scenarios, including social networks [HLL83], community detection [Abb17b], graph clustering [LW19], and Bioinformatics [MGC21], to name a few.
We test SBM instances that are reasonably "clUSTERABLE", i.e., we set  $p = 0.8$  with vertices  $n = 500$ ,  $n = 1000$ , and  $n = 2000$ . Furthermore, we compare the costs estimated by algorithms for the SBM and the Erdos-Renyi random graphs  $G(n,p)$  with  $p = 0.5$ . In this regime, the Erdos-Renyi graph does not appear to have any clustering property, and the cost is high. Under the  $n = 1000$  setting, we tested whether our algorithms can distinguish the costs between "good" (SBM with  $p = 0.95$ ) and "bad" (Erdos-Renyi with  $p = 0.5$ ) instances - a property that can be extremely useful in practice.
We implement our algorithms based on our descriptions in Section 3 and Section 4 with some relaxation of parameters. Notably, for the pivot-based algorithm, we slightly relax the requirement to allow 2-pass over the stream; as such, we can use the first pass to perform greedy MIS on the sampled vertex set, so we do not have to pay the  $2^{O(1/\delta)}$  factor in the space.
We evaluate the performances mainly based on two metrics: the multiplicative factor of the cost estimation (which we call the "competitive ratio") and the fraction of the edges used. To overcome the possible effects of random seed, we fix our random seed from 0 to 14, and run 15 experiments. We include the error bars and the curves of the competitive ratios and fraction of the edges. For the experiment to distinguish SBM and Erdos-Renyi graphs, we simply plot the two types of costs w.r.t. experiment runs, and give the distributions of the costs. All the experiments were conducted on two macbooks and public Colab clusters.
Experimental Results We first show that our algorithms are insensitive to the choice of parameters - this property is evident by the figures we used in the full version in the supplementary material. As such, we focus on the settings with  $\varepsilon = 0.01$  for the SDD-based algorithm and  $\delta = 0.1$  for the pivot-based algorithm for the rest of this section.
Due to the space limit, we only discuss our algorithm based on sparse dense decomposition (SDD). The left 3 plots in Figure 1 give the plots of the cost estimation for the SDD-based algorithm on  $n = 2000$ . The figures show that the approximation factor for this algorithm is roughly between 1 and 2, our algorithm consistently uses less than  $4\%$  of the edges in the graph.
The results for our SDD-based algorithm to distinguish between SBM instances with  $p = 0.95$  and ER instances with  $p = 0.5$  with the SDD-based algorithm is shown in the right two plots of Figure 1. From the figure, it can be observed that the SDD-based algorithm outputs drastically different clustering costs of the SBM instances vs. the Erdos-Renyi instances. We can observe from the left plot that the supports of the costs are disjoint. As such, by a simple linear threshold, our SDD algorithm is able to perfectly distinguish between both types of instances while using less than  $10\%$  of the edges ( $n = 1000$  case).
Figure 1: The performance of the SDD-based algorithm for cost estimation and instance clusterability distinguishing. Top 3 plots: the cost estimation of the SDD-based algorithm over 15 runs; Bottom 2 plots: the costs of SBM and ER graphs estimated by the SDD-based algorithm over 15 runs.
Research supported in part by the Alfred P. Sloan Research Fellowship, a Faculty of Mathematics Research Chair Award from University of Waterloo, an NSF CAREER Grant CCF-2047061, a gift from Google Research, and a Rutgers Research Council Fulcrum Award.
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