Sublinear-Time Algorithms for Approximating Graph Parameters

Ron, Dana

doi:10.1007/978-3-319-91908-9_7

Cited by 5 publications

(5 citation statements)

References 31 publications

(62 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since the graphs appearing in modern applications are massive, it is also often desirable to design sublineartime algorithms that approximate natural combinatorial properties of the graph, such as the average degree, the number of connected components, the cost of a minimum spanning tree, the number of triangles, the size of a maximum matching, the size of a minimum vertex cover, etc. For an excellent survey on sublinear-time algorithms for approximating graph parameters, we refer the reader to [24].…”

Section: Introductionmentioning

confidence: 99%

Privately Estimating Graph Parameters in Sublinear time

Blocki¹,

Grigorescu²,

Mukherjee³

2022

Preprint

View full text Add to dashboard Cite

We initiate a systematic study of algorithms that are both differentially-private and run in sublinear time for several problems in which the goal is to estimate natural graph parameters. Our main result is a differentiallyprivate (1 + ρ)-approximation algorithm for the problem of computing the average degree of a graph, for every ρ > 0. The running time of the algorithm is roughly the same as its non-private version proposed by Goldreich and Ron (Sublinear Algorithms, 2005). We also obtain the first differentially-private sublinear-time approximation algorithms for the maximum matching size and the minimum vertex cover size of a graph.An overarching technique we employ is the notion of coupled global sensitivity of randomized algorithms. Related variants of this notion of sensitivity have been used in the literature in ad-hoc ways. Here we formalize the notion and develop it as a unifying framework for privacy analysis of randomized approximation algorithms. J. B. and T.M were supported in part by NSF CNS-1931443 and NSF CCF-1910659. E.G and T. M. were supported in part by NSF CCF-1910659 and NSF CCF-1910411. 1 GraphsGraphs G 1 and G 2 are edge-neighboring i.e., G 1 ∼ e G 2 if there exists an edge e such that E 1 \ {e} = E 2 \ {e}. 1 algorithms. They showed how to estimate the cost of a minimum spanning tree and the number of triangles in a graph by calibrating noise to a local variant of sensitivity called smooth sensitivity. Subsequent works in designing edge differentially-private algorithms for computing graph statistics include [16,15,18,31]. Gupta, Ligett, McSherry, Roth and Talwar [14] gave the first edge differentially-private algorithms for classical graph optimization problems, such as vertex cover, and minimum s-t cut, by making clever use of the exponential mechanism in existing non-private algorithms that solve the same problem.An even more desirable notion of privacy in graphs is the notion of node differential privacy i.e., neighboring graphs that differ by a single node and edges incident to it in Definition 1. The concept of node differentiallyprivate algorithms for 1-dimensional functions (functions that output a single real value) on graphs was first rigorously studied independently by Kasiviswanathan, Nissim, Raskhodnikova and Smith [17], as well as, Blocki, Blum, Datta, and Sheffet [2], and Chen and Zhou [6]. Their techniques were later extended to higher-dimensional functions on graphs [23,3]. Subsequent works have focused on developing node differentially-private algorithms for a family of network models: stochastic block models and graphons [4,25]. A more recent line of work has focused on the continual release of graph statistics such as degree-distributions and subgraph counts in an online setting [28,11]. Gehrke, Lui, and Pass [12] introduce a more robust notion of differential privacy called Zero-Knowledge Differential Privacy (ZKDP), which tackles the problem of auxiliary information in social networks. This work uses existing results from sublinear-time algorithms as a building...

show abstract

Section: Introductionmentioning

confidence: 99%

Privately Estimating Graph Parameters in Sublinear time

Blocki¹,

Grigorescu²,

Mukherjee³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In the literature, the first two types of queries form the adjacency list query model, while all three types of queries form the adjacency matrix query model. Under these models, a variety of graph estimation problems have been well studied, including edge counting and sampling [ER18, GR08, Ses, TT22], subgraph counting [ABG + 18, BER21, ERS20], vertex cover [Beh22,ORRR12], and beyond [Ron19].…”

Section: Introductionmentioning

confidence: 99%

Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

Addanki¹,

McGregor²,

Musco³

2022

Preprint

View full text Add to dashboard Cite

We study the problem of estimating the number of edges in an n-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (ITCS '18). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a (1 ± ǫ) relative error approximation to the number of edges, with query complexity O(ǫ −5 log 5 n), where O(•) hides poly(log log n) dependencies. This is the first non-adaptive algorithm in this setting achieving poly(1/ǫ, log n) query complexity. Prior work requires Ω(log 2 n) rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant ǫ, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires O(ǫ −2 log 11 n) queries. Building on our edge estimation result, we give the first non-adaptive algorithm for outputting a nearly uniformly sampled edge with query complexity O(ǫ −6 log 6 n), improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require Ω(log 3 n) rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a O(n log 8 n) query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making o(n 2 ) queries.

show abstract

“…For any specified vertex pair 𝑠, 𝑡, our algorithms output an estimate of 𝑅 𝐺 (𝑠, 𝑡) with an arbitrarily small constant additive error, while exploring a small portion of the graph. To formally state our results, we utilize the well-known adjacency list model [33], which assumes query access to the input graph 𝐺 and supports the following types of queries in constant time:…”

Section: Introductionmentioning

confidence: 99%

Local Algorithms for Estimating Effective Resistance

Pan¹,

Lopatta²,

Yoshida³

et al. 2021

Preprint

View full text Add to dashboard Cite

Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs.In this work, we design several local algorithms for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair 𝑠, 𝑡 with an arbitrarily small additive error 𝜀 in time 𝑂 (poly(log 𝑛/𝜀)), whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms. CCS CONCEPTS• Mathematics of computing → Graph algorithms; • Theory of computation → Sketching and sampling.

show abstract

Sublinear-Time Algorithms for Approximating Graph Parameters

Cited by 5 publications

References 31 publications

Privately Estimating Graph Parameters in Sublinear time

Privately Estimating Graph Parameters in Sublinear time

Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

Local Algorithms for Estimating Effective Resistance

Contact Info

Product

Resources

About