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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Effective resistances are ubiquitous in graph algorithms and network analysis. For an undirected graph G, its effective resistance R_G(s,t) between two vertices s and t is defined as the equivalent resistance between s and t if G is thought of as an electrical network with unit resistance on each edge. If we use L_G to denote the Laplacian matrix of G and L_G^† to denote its pseudo-inverse, we have R_G(s,t) = (𝟏_s-𝟏_t)^⊤ L^† (𝟏_s-𝟏_t) such that classical Laplacian solvers [Daniel A. Spielman and Shang{-}Hua Teng, 2014] provide almost-linear time algorithms to approximate R_G(s,t).
In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair s and t. We consider the classical adjacency list model [Ron, 2019] for local algorithms. While recent works [Andoni et al., 2018; Peng et al., 2021; Li and Sachdeva, 2023] have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of n vertices and m edges:
1) It needs Ω(n) queries to obtain 1.01-approximations of the effective resistance of an adjacent pair s and t, even for graphs of degree at most 3 except s and t.
2) For graphs of degree at most d and any parameter 𝓁, it needs Ω(m/𝓁) queries to obtain c ⋅ min{d,𝓁}-approximations where c > 0 is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time (1+ε)-approximation algorithm for graphs of degree 2 except the pair s and t.
One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.

Dongrun Cai, Xue Chen, and Pan Peng. Effective Resistances in Non-Expander Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 29:1-29:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cai_et_al:LIPIcs.ESA.2023.29, author = {Cai, Dongrun and Chen, Xue and Peng, Pan}, title = {{Effective Resistances in Non-Expander Graphs}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {29:1--29:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.29}, URN = {urn:nbn:de:0030-drops-186823}, doi = {10.4230/LIPIcs.ESA.2023.29}, annote = {Keywords: Sublinear Time Algorithm, Effective Resistance, Leverage Scores, Matrix Perturbation} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Learning the community structure of a large-scale graph is a fundamental problem in machine learning, computer science and statistics. Among others, the Stochastic Block Model (SBM) serves a canonical model for community detection and clustering, and the Massively Parallel Computation (MPC) model is a mathematical abstraction of real-world parallel computing systems, which provides a powerful computational framework for handling large-scale datasets. We study the problem of exactly recovering the communities in a graph generated from the SBM in the MPC model. Specifically, given kn vertices that are partitioned into k equal-sized clusters (i.e., each has size n), a graph on these kn vertices is randomly generated such that each pair of vertices is connected with probability p if they are in the same cluster and with probability q if not, where p > q > 0.
We give MPC algorithms for the SBM in the (very general) s-space MPC model, where each machine is guaranteed to have memory s = Ω(log n). Under the condition that (p-q)/√p ≥ Ω̃(k^{1/2} n^{-1/2+1/(2(r-1))}) for any integer r ∈ [3,O(log n)], our first algorithm exactly recovers all the k clusters in O(kr log_s n) rounds using Õ(m) total space, or in O(rlog_s n) rounds using Õ(km) total space. If (p-q)/√p ≥ Ω̃(k^{3/4} n^{-1/4}), our second algorithm achieves O(log_s n) rounds and Õ(m) total space complexity. Both algorithms significantly improve upon a recent result of Cohen-Addad et al. [PODC'22], who gave algorithms that only work in the sublinear space MPC model, where each machine has local memory s = O(n^δ) for some constant δ > 0, with a much stronger condition on p,q,k. Our algorithms are based on collecting the r-step neighborhood of each vertex and comparing the difference of some statistical information generated from the local neighborhoods for each pair of vertices. To implement the clustering algorithms in parallel, we present efficient approaches for implementing some basic graph operations in the s-space MPC model.

Zelin Li, Pan Peng, and Xianbin Zhu. Massively Parallel Algorithms for the Stochastic Block Model. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 78:1-78:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{li_et_al:LIPIcs.ESA.2023.78, author = {Li, Zelin and Peng, Pan and Zhu, Xianbin}, title = {{Massively Parallel Algorithms for the Stochastic Block Model}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {78:1--78:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.78}, URN = {urn:nbn:de:0030-drops-187313}, doi = {10.4230/LIPIcs.ESA.2023.78}, annote = {Keywords: Massively Parallel Computation, Stochastic Block Model, Graph Algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by d, any property that can be tested with query complexity O_{ε,d}(1) in the bidirectional model can be tested with n^{1-Ω_{ε,d}(1)} queries in the unidirectional model. In particular, {if the proximity parameter ε approaches 0, then the query complexity of the transformed tester in the unidirectional model approaches n}. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation.
We prove that testing subgraph-freeness in which the subgraph contains k source components, requires Ω(n^{1-1/k}) queries in the unidirectional model. This directly gives the first explicit properties that exhibit an O_{ε,d}(1) vs Ω(n^{1-f(ε,d)}) separation of the query complexities between the bidirectional model and unidirectional model, where f(ε,d) is a function that approaches 0 as ε approaches 0. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing k-star-freeness.

Pan Peng and Yuyang Wang. An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 96:1-96:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{peng_et_al:LIPIcs.ICALP.2023.96, author = {Peng, Pan and Wang, Yuyang}, title = {{An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {96:1--96:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.96}, URN = {urn:nbn:de:0030-drops-181480}, doi = {10.4230/LIPIcs.ICALP.2023.96}, annote = {Keywords: Graph property testing, Directed graphs, Lower bound, Subgraph-freeness} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

In Property Testing, proximity-oblivious testers (POTs) form a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter.
In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that allow constant-query proximity-oblivious testing in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.

Isolde Adler, Noleen Köhler, and Pan Peng. GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 34:1-34:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{adler_et_al:LIPIcs.CCC.2021.34, author = {Adler, Isolde and K\"{o}hler, Noleen and Peng, Pan}, title = {{GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {34:1--34:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.34}, URN = {urn:nbn:de:0030-drops-143082}, doi = {10.4230/LIPIcs.CCC.2021.34}, annote = {Keywords: Property testing, proximity-oblivous testing, locality, first-order logic, lower bound} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

For any undirected graph G = (V,E) and a set E_W of candidate edges with E ∩ E_W = ∅, the (k,γ)-spectral augmentability problem is to find a set F of k edges from E_W with appropriate weighting, such that the algebraic connectivity of the resulting graph H = (V, E ∪ F) is least γ. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph’s conductance and the mixing time of random walks in a graph, maximising the resulting graph’s algebraic connectivity by adding a small number of edges has been studied over the past 15 years, and has many practical applications in network optimisation.
In this work we present an approximate and efficient algorithm for the (k,γ)-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the (k,γ)-spectral augmentability problem:
- We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [Ghosh and Boyd, 2006]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly.
- We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least Ω(n²mk) time [Kolla et al., 2010]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces.

Bogdan-Adrian Manghiuc, Pan Peng, and He Sun. Augmenting the Algebraic Connectivity of Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 70:1-70:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{manghiuc_et_al:LIPIcs.ESA.2020.70, author = {Manghiuc, Bogdan-Adrian and Peng, Pan and Sun, He}, title = {{Augmenting the Algebraic Connectivity of Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {70:1--70:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.70}, URN = {urn:nbn:de:0030-drops-129367}, doi = {10.4230/LIPIcs.ESA.2020.70}, annote = {Keywords: Graph sparsification, Algebraic connectivity, Semidefinite programming} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We consider the fundamental question of understanding the relative power of two important computational models: property testing and data streaming. We present a novel framework closely linking these areas in the setting of general graphs in the context of constant-query complexity testing and constant-space streaming. Our main result is a generic transformation of a one-sided error property tester in the random-neighbor model with constant query complexity into a one-sided error property tester in the streaming model with constant space complexity. Previously such a generic transformation was only known for bounded-degree graphs.

Artur Czumaj, Hendrik Fichtenberger, Pan Peng, and Christian Sohler. Testable Properties in General Graphs and Random Order Streaming. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 16:1-16:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{czumaj_et_al:LIPIcs.APPROX/RANDOM.2020.16, author = {Czumaj, Artur and Fichtenberger, Hendrik and Peng, Pan and Sohler, Christian}, title = {{Testable Properties in General Graphs and Random Order Streaming}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {16:1--16:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.16}, URN = {urn:nbn:de:0030-drops-126190}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.16}, annote = {Keywords: Graph property testing, sublinear algorithms, graph streaming algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We present a simple sublinear-time algorithm for sampling an arbitrary subgraph H exactly uniformly from a graph G, to which the algorithm has access by performing the following types of queries: (1) uniform vertex queries, (2) degree queries, (3) neighbor queries, (4) pair queries and (5) edge sampling queries. The query complexity and running time of our algorithm are Õ(min{m, (m^ρ(H))/#H}) and Õ((m^ρ(H))/#H}), respectively, where ρ(H) is the fractional edge-cover of H and #H is the number of copies of H in G. For any clique on r vertices, i.e., H = K_r, our algorithm is almost optimal as any algorithm that samples an H from any distribution that has Ω(1) total probability mass on the set of all copies of H must perform Ω(min{m, (m^ρ(H))/(#H⋅(cr)^r)}) queries.
Together with the query and time complexities of the (1±ε)-approximation algorithm for the number of subgraphs H by Assadi et al. [Sepehr Assadi et al., 2018] and the lower bound by Eden and Rosenbaum [Eden and Rosenbaum, 2018] for approximately counting cliques, our results suggest that in our query model, approximately counting cliques is "equivalent to" exactly uniformly sampling cliques, in the sense that the query and time complexities of exactly uniform sampling and randomized approximate counting are within polylogarithmic factor of each other. This stands in interesting contrast to an analogous relation between approximate counting and almost uniformly sampling for self-reducible problems in the polynomial-time regime by Jerrum, Valiant and Vazirani [Jerrum et al., 1986].

Hendrik Fichtenberger, Mingze Gao, and Pan Peng. Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 45:1-45:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fichtenberger_et_al:LIPIcs.ICALP.2020.45, author = {Fichtenberger, Hendrik and Gao, Mingze and Peng, Pan}, title = {{Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.45}, URN = {urn:nbn:de:0030-drops-124526}, doi = {10.4230/LIPIcs.ICALP.2020.45}, annote = {Keywords: Graph sampling, Graph algorithms, Sublinear-time algorithms} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper (Δ+1)-vertex coloring of a graph with maximum degree at most Δ. This improves upon the previous O(log Δ)-time algorithm by Bhattacharya et al. (SODA 2018). Our algorithm uses an approach based on assigning random ranks to vertices and does not need to maintain a hierarchical graph decomposition. We show that our result does not only have optimal running time, but is also optimal in the sense that already deciding whether a Δ-coloring exists in a dynamically changing graph with maximum degree at most Δ takes Ω(log n) time per operation.

Monika Henzinger and Pan Peng. Constant-Time Dynamic (Δ+1)-Coloring. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 53:1-53:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.STACS.2020.53, author = {Henzinger, Monika and Peng, Pan}, title = {{Constant-Time Dynamic (\Delta+1)-Coloring}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {53:1--53:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.53}, URN = {urn:nbn:de:0030-drops-119145}, doi = {10.4230/LIPIcs.STACS.2020.53}, annote = {Keywords: Dynamic graph algorithms, Graph coloring, Random sampling} }

Document

**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an n^{c}-separator theorem for some c<1. We give a fully dynamic algorithm that maintains (1+epsilon)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with O~(sqrt{n}/epsilon^2) worst-case update time and O~(sqrt{n}/epsilon^2) worst-case query time, if G is guaranteed to be sqrt{n}-separable (i.e., it is taken from a class satisfying a sqrt{n}-separator theorem) and its separator can be computed in O~(n) time. Our algorithm is built upon a dynamic algorithm for maintaining approximate Schur complement that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest.
We complement our result by proving that for any two fixed vertices s and t, no incremental or decremental algorithm can maintain the s-t effective resistance for sqrt{n}-separable graphs with worst-case update time O(n^{1/2-delta}) and query time O(n^{1-delta}) for any delta>0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false.
We further show that for general graphs, no incremental or decremental algorithm can maintain the s-t effective resistance problem with worst-case update time O(n^{1-delta}) and query-time O(n^{2-delta}) for any delta >0, unless the OMv conjecture is false.

Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 40:1-40:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2018.40, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.40}, URN = {urn:nbn:de:0030-drops-95036}, doi = {10.4230/LIPIcs.ESA.2018.40}, annote = {Keywords: Dynamic graph algorithms, effective resistance, separable graphs, Schur complement, conditional lower bounds} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Graph Sparsification aims at compressing large graphs into smaller ones while (approximately) preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. Given a weighted graph G=(V,E), and a terminal set K with |K|=k, a quality-q vertex cut sparsifier of G is a graph H with K contained in V_H that preserves the value of minimum cuts separating any bipartition of K, up to a factor of q. We show that planar graphs with all the k terminals lying on the same face admit quality-1 vertex cut sparsifier of size O(k^2) that are also planar. Our result extends to vertex flow and distance sparsifiers. It improves the previous best known bound of O(k^2 2^(2k)) for cut and flow sparsifiers by an exponential factor, and matches an Omega(k^2) lower-bound for this class of graphs.
We also study vertex reachability sparsifiers for directed graphs. Given a digraph G=(V,E) and a terminal set K, a vertex reachability sparsifier of G is a digraph H=(V_H,E_H), K contained in V_H that preserves all reachability information among terminal pairs. We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G. Among others, for general planar digraphs, we construct reachability-preserving minors of size O(k^2 log^2 k). We complement our upper-bound by showing that there exists an infinite family of acyclic planar digraphs such that any reachability-preserving minor must have Omega(k^2) vertices.

Gramoz Goranci, Monika Henzinger, and Pan Peng. Improved Guarantees for Vertex Sparsification in Planar Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 44:1-44:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2017.44, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{Improved Guarantees for Vertex Sparsification in Planar Graphs}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {44:1--44:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.44}, URN = {urn:nbn:de:0030-drops-78337}, doi = {10.4230/LIPIcs.ESA.2017.44}, annote = {Keywords: Vertex Sparsification, Graph Sparsification, Planar Graphs, Metric Embedding, Reachability} }

Document

**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms.
Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1 + epsilon)-approximating the energy of electrical flows in n-vertex planar graphs with tilde{O}(r epsilon^{-2}) worst-case update time and tilde{O}((r + n / sqrt{r}) epsilon^{-2}) worst-case query time, for any r larger than some constant. For r=n^{2/3}, this gives tilde{O}(n^{2/3} epsilon^{-2}) update time and tilde{O}(n^{2/3} epsilon^{-2}) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification.
We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest.

Gramoz Goranci, Monika Henzinger, and Pan Peng. The Power of Vertex Sparsifiers in Dynamic Graph Algorithms. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2017.45, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{The Power of Vertex Sparsifiers in Dynamic Graph Algorithms}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.45}, URN = {urn:nbn:de:0030-drops-78460}, doi = {10.4230/LIPIcs.ESA.2017.45}, annote = {Keywords: Dynamic graph algorithms, electrical flow, minor-free graphs, max flow} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constant-query testable in the adjacency list model can be tested with constant space in a single-pass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space.
We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the adjacency list model: As an example, we obtain a constant-space single-pass random order streaming algorithms for approximating the size of a maximum matching with additive error epsilon n (n is the number of nodes).
Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while Omega(n) space is needed for many graph streaming problems for adversarial orders.

Morteza Monemizadeh, S. Muthukrishnan, Pan Peng, and Christian Sohler. Testable Bounded Degree Graph Properties Are Random Order Streamable. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 131:1-131:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{monemizadeh_et_al:LIPIcs.ICALP.2017.131, author = {Monemizadeh, Morteza and Muthukrishnan, S. and Peng, Pan and Sohler, Christian}, title = {{Testable Bounded Degree Graph Properties Are Random Order Streamable}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {131:1--131:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.131}, URN = {urn:nbn:de:0030-drops-74782}, doi = {10.4230/LIPIcs.ICALP.2017.131}, annote = {Keywords: Graph streaming algorithms, graph property testing, constant-time approximation algorithms} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require Omega(n) space, where n is the number of vertices, existing works mainly focused on designing ~O(n) space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. n is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present o(n) space algorithms for estimating the number of connected components with additive error epsilon*n and (1 + epsilon)-approximating the weight of minimum spanning tree. The latter improves previous ~O(n) space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are epsilon-far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly ~O(n^{1-epsilon}) space, which is o(n) for any constant epsilon. To complement our algorithms, we present Omega(n^{1-O(epsilon)}) space lower bounds for these problems, which show that such a dependence on epsilon is necessary.

Zengfeng Huang and Pan Peng. Dynamic Graph Stream Algorithms in o(n) Space. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 18:1-18:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{huang_et_al:LIPIcs.ICALP.2016.18, author = {Huang, Zengfeng and Peng, Pan}, title = {{Dynamic Graph Stream Algorithms in o(n) Space}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.18}, URN = {urn:nbn:de:0030-drops-62801}, doi = {10.4230/LIPIcs.ICALP.2016.18}, annote = {Keywords: dynamic graph streams, sketching, property testing, minimum spanning tree} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Let G=(V,E) be an undirected graph with maximum degree d. The k-disc of a vertex v is defined as the rooted subgraph that is induced by all vertices whose distance to v is at most k. The k-disc frequency vector of G, freq(G), is a vector indexed by all isomorphism types of k-discs. For each such isomorphism type Gamma, the k-disc frequency vector counts the fraction of vertices that have k-disc isomorphic to Gamma. Thus, the frequency vector freq(G) of G captures the local structure of G. A natural question is whether one can construct a much smaller graph H such that H has a similar local structure. N. Alon proved that for any epsilon>0 there always exists a graph H whose size is independent of |V| and whose frequency vector satisfies ||freq(G) - freq(G)||_1 <= epsilon. However, his proof is only existential and neither gives an explicit bound on the size of H nor an efficient algorithm. He gave the open problem to find such explicit bounds. In this paper, we solve this problem for the special case of high girth graphs. We show how to efficiently compute a graph H with the above properties when G has girth at least 2k+2 and we give explicit bounds on the size of H.

Hendrik Fichtenberger, Pan Peng, and Christian Sohler. On Constant-Size Graphs That Preserve the Local Structure of High-Girth Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 786-799, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fichtenberger_et_al:LIPIcs.APPROX-RANDOM.2015.786, author = {Fichtenberger, Hendrik and Peng, Pan and Sohler, Christian}, title = {{On Constant-Size Graphs That Preserve the Local Structure of High-Girth Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {786--799}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.786}, URN = {urn:nbn:de:0030-drops-53363}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.786}, annote = {Keywords: local graph structure, k-disc frequency vector, graph property testing} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We consider the problem of testing small set expansion for general graphs. A graph G is a (k,\phi)-expander if every subset of volume at most k has conductance at least \phi. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes.
We give testers with two-sided error and one-sided error in the \adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an n-vertex graph G, a volume bound k, an expansion bound \phi and a distance parameter \varepsilon>0. For the two-sided error tester, with probability at least 2/3, it accepts the graph if it is a (k,\phi)-expander and rejects the graph if it is \varepsilon-far from any (k^*,\phi^*)-expander, where k^*=\Theta(k\varepsilon) and \phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)}). The query complexity and running time of the tester are \widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2}), where m is the number of edges of the graph. For the one-sided error tester, it accepts every (k,\phi)-expander, and with probability at least 2/3, rejects every graph that is \varepsilon-far from (k^*,\phi^*)-expander, where k^*=O(k^{1-\xi}) and \phi^*=O(\xi\phi^2) for any 0<\xi<1. The query complexity and running time of this tester are \widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4}).
We also give a two-sided error tester in the \textit{rotation map} model that allows \textit{(neighbor, index)} queries and degree queries. This tester has asymptotically almost the same query complexity and running time as the two-sided error tester in the adjacency list model, but has a better performance: it can distinguish any $(k,\phi)$-expander from graphs that are $\varepsilon$-far from $(k^*,\phi^*)$-expanders, where $k^*=\Theta(k\varepsilon)$ and $\phi^*=\Theta(\frac{\phi^2}{\min\{\log(4m/k),\log n\}\cdot(\ln k)})$.
In our analysis, we introduce a new graph product called \textit{non-uniform replacement product} that transforms a general graph into a bounded degree graph, and approximately preserves the expansion profile as well as the corresponding spectral property.

Angsheng Li and Pan Peng. Testing Small Set Expansion in General Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 622-635, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{li_et_al:LIPIcs.STACS.2015.622, author = {Li, Angsheng and Peng, Pan}, title = {{Testing Small Set Expansion in General Graphs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {622--635}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.622}, URN = {urn:nbn:de:0030-drops-49461}, doi = {10.4230/LIPIcs.STACS.2015.622}, annote = {Keywords: graph property testing, small set expansion, random walks, spectral graph theory} }

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