__ Summary and Contributions__: Authors provide a new version of DBSCAN that is optimized for time complexity by sub-sampling edges of the epsilon-neighboring graph on which clustering partition is computed.
Proposed time complexity is O(sn**2) where:
-n is the number of data points
-s is the sampling rate of the edges

__ Strengths__: - Authors support the approach with theoretical guarantees: under some assumptions, s ~ (log n) / n is proven to be enough for ensuring statistical recovery of the clustering partition.
Demonstrations are well conducted: interpretation of each assumption makes it easy to follow.
- Empirical evaluation show that competitive performance (quality of the clustering and runtime) for big and even small datasets.

__ Weaknesses__: - Empirical evaluation is a bit too weak for me. The comparison with DBSCAN++ is not fair. Even if DBSCAN++ and approach proposed by the authors come up with the same bound for runtime complexity O(sn**2) with n the number of nodes and s the sampling rate, this is definitely not fair to use the same sampling rate for edges (authors' approach) and DBSCAN++.
Hence could the authors provide experiments for which sampling rate is optimized for both competitors?
- Authors DBSCAN++ show cases where DBSCAN++ is even providing better clustering partition than for DBSCAN. Could the authors perform a similar analysis here with SNG-DBSCAN?
- DBSCAN++ is also applied to outlier detection. How is performing SNG-DBSCAN compared to DBSCAN++ in this context?
- Which parameters are used with DBSCAN for time and space consumption comparison?
- Would it be possible to have empirical results on datasets for which DBSCAN is actually working? It seems that DBSCAN is performing well only for Still. Hence DBSCAN and SNG-DBSCAN are equally performing bad on the remaining sets...

__ Correctness__: My main concern is the fact that SNG-DBSCAN and DBSCAN++ are empirically compared by using the same sampling rate while for SNG-DBSCAN we sample edges and for DBSCAN++ we sample nodes. Hence, SNG-DBSCAN has naturally an advantage over DBSCAN regarding clustering accuracy which is not fair. Authors should present the results by tuning the sample rate for both methods.

__ Clarity__: The paper is undoubtedly well written and easy to follow.
Each theoretical result is well presented before the proof.

__ Relation to Prior Work__: Yes. Main previous contribution is mainly DBSCAN ++ for the last optimization of DBSCAN. SNG-DBSCAN is about sampling edges rather than nodes.

__ Reproducibility__: Yes

__ Additional Feedback__: As it is about optimization of a well-established algorithms and execution times are compared with scikitlearn implementation, it could be very valuable the authors provide actually access to the code.
=== AFTER AUTHOR FEEDBACK ===
My main concern about the mise-use of the same sampling rate for edges vs nodes has been correctly addressed by the authors and I trust them to do the appropriate changes in the revised version of the paper. Hence I upgrade my rating to a weak accept.

__ Summary and Contributions__: The paper proposes a new approach for speeding up the popular DBSCAN algorithm for density based clustering. The speed up is based on sampling edges in the e-neighbourhood graph, where previous sampling based speed ups arise from sampling from the data instead. The authors provide a theoretical guarantee for the correct recovery of the level set of a density with high probability, provided assumptions on the density hold. The experiments provided illustrate potential for practical relevance when clustering large data sets

__ Strengths__: The paper is concisely and clearly written. The proposed idea is simple and intuitive, but appears to be novel. It also seems to work very well practically. DBSCAN is a highly influential clustering algorithm, and improvements thereon have potential for substantial impact.

__ Weaknesses__: The paper includes a theoretical guarantee for the algorithm which ensures correct recovery of the level set components of the underlying density with high probability, however the assumptions required for this to hold seem to be very restrictive. Aside from being far more restrictive than in studies such as those in [1] and [2] which also deal with the estimation of level sets and their components, the assumptions are not very natural. In particular, as far as it seems, the density must be discontinuous at the boundaries of the level set and the noise region must be separated from the clusters by a regions of zero probability. While this is not stated explicilty, it can be inferred from the assumptions and the conclusion in Theorem 1 which states that the exact clusters are recovered, without the potential for points just beyond the boundary of the level set, which with a continuous density would have density arbitrarily close to the height of the density inside the level set.
References:
[1] Steinwart, Ingo. "Fully adaptive density-based clustering." The Annals of Statistics 43.5 (2015): 2132-2167.
[2] Hofmeyr, David P. "Connecting Spectral Clustering to Maximum Margins and Level Sets." Journal of Machine Learning Research 21.18 (2020): 1-35.

__ Correctness__: The results and claims appear to be correct (given the assumptions), and the empirical methodology is straightforward but compelling.

__ Clarity__: See above in the strengths section

__ Relation to Prior Work__: Given the limited space the authors have done well to mention a large number of related prior works. Under some circumstances the lack of detail would be a concern, but given the relative simplicity of the DBSCAN algorithm and sampling based speed ups, I think the coverage is sufficient.

__ Reproducibility__: Yes

__ Additional Feedback__: EDIT: After reading the authors' response and discussing with other reviewers, I am satisfied that the authors can appropriately address my concerns.
Additional to the concern mentioned above regarding the assumptions on the underlying density, unless I am mistaken the conditions on \rho are also very restrictive and could even have been written incorrectly? In particular, even in the ideal scenario of a very "nicely" shaped cluster, a point x near the boundary of the cluster C would result in Vol(B(x, r) ^ C) \approx Vol(B(x, r)) / 2, meaning that \rho can never be greater than 1/2. Applying this in the first point in Assumption 1 suggests that we can't use this result if any of the clusters has a density only \approx twice that of the height of the noise density.
I would be surprised if the assumptions cannot be relaxed to a large degree to ensure the same or a very similar result will hold under far more general and natural conditions. I would be very willing to recommend acceptance if the authors can address my concerns regarding the relevance of the theory, as the empirical performance of the method is very persuasive.
In addition, I found a few typos/curiosities:
1. kd-trees (line 113) and KD-trees (line 40) are both used
2. line 123 repetition of the word "degree" in the parentheses
3. In the statement of Lemma 1 the first instance of C_{D, p} also has a \delta in the subscript

__ Summary and Contributions__: 1. DBSCAN is a popular density-based clustering algorithm. This paper proposes a faster variant of DBSCAN called SNG-DBSCAN by clustering the subsampled -neighborhood graph instead of computing the entire graph. They show that under some natural assumptions the algorithm provides comparable clustering performance at lower computational costs. The theoretical work depends heavily on statistical as well as graph theory lemmas and theorems. They compare the performance with DBSCAN and DBSCAN++ for some datasets.

__ Strengths__: The paper converts the epsilon neighborhood of points as a graph of vertices and the distance between them as edges and applies principles of graph theory as well as statistical sampling theory to prove the claim that the cluster quality is ensured. The paper also relies on previous work of sampling in graph theory and applies it for epsilon neighborhood graph. The paper provides extensive results against two previous techniques, DBSCAN and DBSCAN++. They also provide memory usage metric.

__ Weaknesses__: The already existing work of DBSCAN++ subsamples the vertices of the graph while the proposed SNG-DBSCAN subsamples the edges for better time complexity, as such the idea is not very novel and draws from previous works.

__ Correctness__: The claims and method are correct.

__ Clarity__: The theoretical work in the paper needs to be written more clearly. It is missing out on explaining basic symbols and terminology used and as such makes the work difficult to understand without the supplementary material. The theory written appears vague and would lead one to think that there are several missed points till we dig deeper into supplementary material. The authors should have spent more space on presenting their robust theoretical work rather than providing long explanations of experiments.

__ Relation to Prior Work__: The paper briefly discusses how it differs from previous contributions but fails to draw our attention to the novelty of their approach.

__ Reproducibility__: Yes

__ Additional Feedback__: N/A

__ Summary and Contributions__:
The paper proposes an edge subsampling approximation to DBSCAN
along with theoretical conditions under which it is likely to
recover identical results to full DBSCAN, and provides some
experimentation over clustering accuracy, time taken, and
memory consumed.

__ Strengths__:
DBSCAN is widely used clustering algorithm with known
scalability problems, so this is a valuable problem to
consider. The approach is pleasingly simple, the theoretical
analysis provides some explanation of why it should work, and
the experimental results confirm the improved scaling
performance without degradation of quality or accuracy.

__ Weaknesses__:
The connection between the theoretical analysis and the
experimental results was not very strongly established: what
are the constants hiding in the Big Theta, can we take any
guidance for how to set s from this? It was also somewhat
surprising to see SNG actually doing better than base DBSCAN
in Figure 3, when we should expect identical results under the
theory? How do results degrade if the conditions are not met
exactly?
Without some tighter connection, the theory would seem to
serve as a somewhat hand-wavy justification for a very clever
and effective implementation trick. This is not without value,
but a more direct connection would strengthen the submission.

__ Correctness__:
I followed but did not rigorously check the derivations. The
experiments seemed reasonable modulo my other
questions/comments.

__ Clarity__:
The paper is clear overall and in particular the structure of
the theoretical analysis was nicely laid out in terms of the
necessary assumptions and the resulting conclusion. The plain
language parenthetical explanations of the different
conditions in L151-158 were helpful, as was the overview of
the approach starting on L159.

__ Relation to Prior Work__:
As admitted by the authors, the DBSCAN literature is vast. The
related work is generally well-contextualized. One question
area was around how this work compares to the data-parallel
(eg, Map-Reduce) class of modifications: how exactly could
those apporaches be easily composed with SNG-DBSCAN? Or, what
are the tradeoffs of SNG-DBSCAN versus these?
The comparison with DBSCAN++ and conceptual framing as
node-vs-edge sampling was useful, although it is not obvious
to me that using the same sampling rate s is really an
appropriate comparison when one algorithm is sampling nodes while
the other is sampling edges?

__ Reproducibility__: Yes

__ Additional Feedback__:
It wasn't obvious to me what the "true clusters" meant
exactly, in terms of whether we are talking about "the exact
results standard DBSCAN would be expected to return" or "the
true clustering we would expect based on the underlying
density P"? Or is this some intrinsic property of the
assumptions set up in terms of C1, ...? Do the sets
C1,... need to entirely cover the support of P? Do specific
steps of the proof depend on them being compact and connected,
or are these just some general "sanity" conditions?
Figure 4: I didn't see the standard errors on these charts?
In several cases comparisons to other DBSCAN implementations
are explained by the fact that the other implementations
exploit spatially-aware data structures such as kd trees. Are
there any potential future extensions to leverage those in
this sampling framework?
I have read and considered the author feedback.