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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency.
Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space.
Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language).
Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself.

Amartya Shankha Biswas, Ronitt Rubinfeld, and Anak Yodpinyanee. Local Access to Huge Random Objects Through Partial Sampling. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 27:1-27:65, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{biswas_et_al:LIPIcs.ITCS.2020.27, author = {Biswas, Amartya Shankha and Rubinfeld, Ronitt and Yodpinyanee, Anak}, title = {{Local Access to Huge Random Objects Through Partial Sampling}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {27:1--27:65}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.27}, URN = {urn:nbn:de:0030-drops-117126}, doi = {10.4230/LIPIcs.ITCS.2020.27}, annote = {Keywords: sublinear time algorithms, random generation, local computation} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently.
Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store.
To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge (u,v) in E belongs to the output (sparse) spanner or not. Such LCAs give the user the "illusion" that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present several results for this setting, including:
- For general n-vertex graphs and for parameter r in {2,3}, there exists an LCA for (2r-1)-spanners with O~(n^{1+1/r}) edges and sublinear probe complexity of O~(n^{1-1/2r}). These size/stretch trade-offs are best possible (up to polylogarithmic factors).
- For every k >= 1 and n-vertex graph with maximum degree Delta, there exists an LCA for O(k^2) spanners with O~(n^{1+1/k}) edges, probe complexity of O~(Delta^4 n^{2/3}), and random seed of size polylog(n). This improves upon, and extends the work of [Lenzen-Levi, ICALP'18].
We also complement these constructions by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with o(m) edges.
To the best of our knowledge, our results on 3 and 5-spanners are the first LCAs with sublinear (in Delta) probe-complexity for Delta = n^{Omega(1)}.

Merav Parter, Ronitt Rubinfeld, Ali Vakilian, and Anak Yodpinyanee. Local Computation Algorithms for Spanners. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 58:1-58:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{parter_et_al:LIPIcs.ITCS.2019.58, author = {Parter, Merav and Rubinfeld, Ronitt and Vakilian, Ali and Yodpinyanee, Anak}, title = {{Local Computation Algorithms for Spanners}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {58:1--58:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.58}, URN = {urn:nbn:de:0030-drops-101510}, doi = {10.4230/LIPIcs.ITCS.2019.58}, annote = {Keywords: Local Computation Algorithms, Sub-linear Algorithms, Graph Spanners} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We study the Fractional Set Cover problem in the streaming model. That is, we consider the relaxation of the set cover problem over a universe of n elements and a collection of m sets, where each set can be picked fractionally, with a value in [0,1]. We present a randomized (1+a)-approximation algorithm that makes p passes over the data, and uses O(polylog(m,n,1/a) (mn^(O(1/(pa)))+n)) memory space. The algorithm works in both the set arrival and the edge arrival models. To the best of our knowledge, this is the first streaming result for the fractional set cover problem. We obtain our results by employing the multiplicative weights update framework in the streaming settings.

Piotr Indyk, Sepideh Mahabadi, Ronitt Rubinfeld, Jonathan Ullman, Ali Vakilian, and Anak Yodpinyanee. Fractional Set Cover in the Streaming Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{indyk_et_al:LIPIcs.APPROX-RANDOM.2017.12, author = {Indyk, Piotr and Mahabadi, Sepideh and Rubinfeld, Ronitt and Ullman, Jonathan and Vakilian, Ali and Yodpinyanee, Anak}, title = {{Fractional Set Cover in the Streaming Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {12:1--12:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.12}, URN = {urn:nbn:de:0030-drops-75613}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.12}, annote = {Keywords: Streaming Algorithms, Fractional Set Cover, LP relaxation, Multiplicative Weight Update} }

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