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**Published in:** LIPIcs, Volume 48, 19th International Conference on Database Theory (ICDT 2016)

Provisioning is a technique for avoiding repeated expensive computations in what-if analysis. Given a query, an analyst formulates k hypotheticals, each retaining some of the tuples of a database instance, possibly overlapping, and she wishes to answer the query under scenarios, where a scenario is defined by a subset of the hypotheticals that are "turned on". We say that a query admits compact provisioning if given any database instance and any k hypotheticals, one can create a poly-size (in k) sketch that can then be used to answer the query under any of the 2^k possible scenarios without accessing the original instance.
In this paper, we focus on provisioning complex queries that combine relational algebra (the logical component), grouping, and statistics/analytics (the numerical component). We first show that queries that compute quantiles or linear regression (as well as simpler queries that compute count and sum/average of positive values) can be compactly provisioned to provide (multiplicative) approximate answers to an arbitrary precision. In contrast, exact provisioning for each of these statistics requires the sketch size to be exponential in k. We then establish that for any complex query whose logical component is a positive relational algebra query, as long as the numerical component can be compactly provisioned, the complex query itself can be compactly provisioned. On the other hand, introducing negation or recursion in the logical component again requires the sketch size to be exponential in k. While our positive results use algorithms that do not access the original instance after a scenario is known, we prove our lower bounds even for the case when, knowing the scenario, limited access to the instance is allowed.

Sepehr Assadi, Sanjeev Khanna, Yang Li, and Val Tannen. Algorithms for Provisioning Queries and Analytics. In 19th International Conference on Database Theory (ICDT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 48, pp. 18:1-18:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{assadi_et_al:LIPIcs.ICDT.2016.18, author = {Assadi, Sepehr and Khanna, Sanjeev and Li, Yang and Tannen, Val}, title = {{Algorithms for Provisioning Queries and Analytics}}, booktitle = {19th International Conference on Database Theory (ICDT 2016)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-002-6}, ISSN = {1868-8969}, year = {2016}, volume = {48}, editor = {Martens, Wim and Zeume, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2016.18}, URN = {urn:nbn:de:0030-drops-57877}, doi = {10.4230/LIPIcs.ICDT.2016.18}, annote = {Keywords: What-if Analysis, Provisioning, Data Compression, Approximate Query Answering} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

In this paper, we introduce a new model for sublinear algorithms called dynamic sketching. In this model, the underlying data is partitioned into a large static part and a small dynamic part and the goal is to compute a summary of the static part (i.e, a sketch) such that given any update for the dynamic part, one can combine it with the sketch to compute a given function. We say that a sketch is compact if its size is bounded by a polynomial function of the length of the dynamic data, (essentially) independent of the size of the static part.
A graph optimization problem P in this model is defined as follows. The input is a graph G(V,E) and a set T \subseteq V of k terminals; the edges between the terminals are the dynamic part and the other edges in G are the static part. The goal is to summarize the graph G into a compact sketch (of size poly(k)) such that given any set Q of edges between the terminals, one can answer the problem P for the graph obtained by inserting all edges in Q to G, using only the sketch.
We study the fundamental problem of computing a maximum matching and prove tight bounds on the sketch size. In particular, we show that there exists a (compact) dynamic sketch of size O(k^2) for the matching problem and any such sketch has to be of size \Omega(k^2). Our sketch for matchings can be further used to derive compact dynamic sketches for other fundamental graph problems involving cuts and connectivities. Interestingly, our sketch for matchings can also be used to give an elementary construction of a cut-preserving vertex sparsifier with space O(kC^2) for k-terminal graphs, which matches the best known upper bound; here C is the total capacity of the edges incident on the terminals. Additionally, we give an improved lower bound (in terms of C) of Omega(C/log{C}) on size of cut-preserving vertex sparsifiers, and establish that progress on dynamic sketching of the s-t max-flow problem (either upper bound or lower bound) immediately leads to better bounds for size of cut-preserving vertex sparsifiers.

Sepehr Assadi, Sanjeev Khanna, Yang Li, and Val Tannen. Dynamic Sketching for Graph Optimization Problems with Applications to Cut-Preserving Sketches. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 52-68, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{assadi_et_al:LIPIcs.FSTTCS.2015.52, author = {Assadi, Sepehr and Khanna, Sanjeev and Li, Yang and Tannen, Val}, title = {{Dynamic Sketching for Graph Optimization Problems with Applications to Cut-Preserving Sketches}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {52--68}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.52}, URN = {urn:nbn:de:0030-drops-56361}, doi = {10.4230/LIPIcs.FSTTCS.2015.52}, annote = {Keywords: Small-space Algorithms, Maximum Matchings, Vertex Sparsifiers} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness/non-uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree D. To this end, we first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature (corresponding to the region where the ordered phase "dominates") that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D.
The #BIS-hardness result uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent results establishing connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. In this paper we extend these connections to random regular graphs for all ferromagnetic models. Using these connections, we establish the Bethe prediction for every ferromagnetic spin system on random regular graphs, which says roughly that the expectation of the log of the partition function Z is the same as the log of the expectation of Z. As a further consequence of our results, we prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing (i.e., exponentially slow convergence to its stationary distribution) on random D-regular graphs at the critical temperature for sufficiently large q.

Andreas Galanis, Daniel Stefankovic, Eric Vigoda, and Linji Yang. Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 677-691, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{galanis_et_al:LIPIcs.APPROX-RANDOM.2014.677, author = {Galanis, Andreas and Stefankovic, Daniel and Vigoda, Eric and Yang, Linji}, title = {{Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {677--691}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.677}, URN = {urn:nbn:de:0030-drops-47319}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.677}, annote = {Keywords: Ferromagnetic Potts model, approximate counting, spin systems, phase transition, random regular graphs} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0.
For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved.

Diptarka Chakraborty, A. Pavan, Raghunath Tewari, N. V. Vinodchandran, and Lin Forrest Yang. New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 585-595, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2014.585, author = {Chakraborty, Diptarka and Pavan, A. and Tewari, Raghunath and Vinodchandran, N. V. and Yang, Lin Forrest}, title = {{New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {585--595}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.585}, URN = {urn:nbn:de:0030-drops-48730}, doi = {10.4230/LIPIcs.FSTTCS.2014.585}, annote = {Keywords: Reachability, Space complexity, Time-Space Efficient Algorithms, Graphs on Surfaces, Minor Free Graphs, Savitch's Algorithm, BBRS Bound} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

Let S and S' be two strings of the same length.We consider the following two variants of string matching.
* Parameterized Matching: The characters of S and S' are partitioned into static characters and parameterized characters.
The strings are parameterized match iff the static characters match exactly and there exists a one-to-one function which renames the parameterized characters in S to those in S'.
* Order-Preserving Matching: The strings are order-preserving match iff for any two integers i,j in [1,|S|], S[i] <= S[j] iff S'[i] <= S'[j].
Let P be a collection of d patterns {P_1, P_2, ..., P_d} of total length n characters, which are chosen from an alphabet Sigma.
Given a text T, also over Sigma, we consider the dictionary indexing problem under the above definitions of string matching.
Specifically, the task is to index P, such that we can report all positions j where at least one of the patterns P_i in P is a parameterized-match (resp. order-preserving match) with the same-length substring of $T$ starting at j. Previous best-known indexes occupy O(n * log(n)) bits and can report all occ positions in O(|T| * log(|Sigma|) + occ) time. We present space-efficient indexes that occupy O(n * log(|Sigma|+d) * log(n)) bits and reports all occ positions in O(|T| * (log(|Sigma|) + log_{|Sigma|}(n)) + occ) time for parameterized matching and in O(|T| * log(n) + occ) time for order-preserving matching.

Arnab Ganguly, Wing-Kai Hon, Kunihiko Sadakane, Rahul Shah, Sharma V. Thankachan, and Yilin Yang. Space-Efficient Dictionaries for Parameterized and Order-Preserving Pattern Matching. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 2:1-2:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganguly_et_al:LIPIcs.CPM.2016.2, author = {Ganguly, Arnab and Hon, Wing-Kai and Sadakane, Kunihiko and Shah, Rahul and Thankachan, Sharma V. and Yang, Yilin}, title = {{Space-Efficient Dictionaries for Parameterized and Order-Preserving Pattern Matching}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {2:1--2:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.2}, URN = {urn:nbn:de:0030-drops-60736}, doi = {10.4230/LIPIcs.CPM.2016.2}, annote = {Keywords: Parameterized Matching, Order-preserving Matching, Dictionary Indexing, Aho-Corasick Automaton, Sparsification} }

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**Published in:** LIPIcs, Volume 88, 17th International Workshop on Algorithms in Bioinformatics (WABI 2017)

Motivation: Many common clustering algorithms require a two-step process that limits their efficiency. The algorithms need to be performed repetitively and need to be implemented together with a model selection criterion, in order to determine both the number of clusters present in the data and the corresponding cluster memberships. As biomedical datasets increase in size and prevalence, there is a growing need for new methods that are more convenient to implement and are more computationally efficient. In addition, it is often essential to obtain clusters of sufficient sample size to make the clustering result meaningful and interpretable for subsequent analysis.
Results: We introduce Shrinkage Clustering, a novel clustering algorithm based on matrix factorization that simultaneously finds the optimal number of clusters while partitioning the data. We report its performances across multiple simulated and actual datasets, and demonstrate its strength in accuracy and speed in application to subtyping cancer and brain tissues. In addition, the algorithm offers a straightforward solution to clustering with cluster size constraints. Given its ease of implementation, computing efficiency and extensible structure, we believe Shrinkage Clustering can be applied broadly to solve biomedical clustering tasks especially when dealing with large datasets.

Chenyue W. Hu, Hanyang Li, and Amina A. Qutub. Shrinkage Clustering: A Fast and Size-Constrained Algorithm for Biomedical Applications. In 17th International Workshop on Algorithms in Bioinformatics (WABI 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 88, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hu_et_al:LIPIcs.WABI.2017.11, author = {Hu, Chenyue W. and Li, Hanyang and Qutub, Amina A.}, title = {{Shrinkage Clustering: A Fast and Size-Constrained Algorithm for Biomedical Applications}}, booktitle = {17th International Workshop on Algorithms in Bioinformatics (WABI 2017)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-050-7}, ISSN = {1868-8969}, year = {2017}, volume = {88}, editor = {Schwartz, Russell and Reinert, Knut}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2017.11}, URN = {urn:nbn:de:0030-drops-76556}, doi = {10.4230/LIPIcs.WABI.2017.11}, annote = {Keywords: Clustering, Matrix Factorization, Cancer Subtyping, Gene Expression} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We resolve the randomized one-way communication complexity of Dynamic Time Warping (DTW) distance. We show that there is an efficient one-way communication protocol using O~(n/alpha) bits for the problem of computing an alpha-approximation for DTW between strings x and y of length n, and we prove a lower bound of Omega(n / alpha) bits for the same problem. Our communication protocol works for strings over an arbitrary metric of polynomial size and aspect ratio, and we optimize the logarithmic factors depending on properties of the underlying metric, such as when the points are low-dimensional integer vectors equipped with various metrics or have bounded doubling dimension. We also consider linear sketches of DTW, showing that such sketches must have size Omega(n).

Vladimir Braverman, Moses Charikar, William Kuszmaul, David P. Woodruff, and Lin F. Yang. The One-Way Communication Complexity of Dynamic Time Warping Distance. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 16:1-16:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{braverman_et_al:LIPIcs.SoCG.2019.16, author = {Braverman, Vladimir and Charikar, Moses and Kuszmaul, William and Woodruff, David P. and Yang, Lin F.}, title = {{The One-Way Communication Complexity of Dynamic Time Warping Distance}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {16:1--16:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.16}, URN = {urn:nbn:de:0030-drops-104203}, doi = {10.4230/LIPIcs.SoCG.2019.16}, annote = {Keywords: dynamic time warping, one-way communication complexity, tree metrics} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Given a finite set of points P subseteq R^d, we would like to find a small subset S subseteq P such that the convex hull of S approximately contains P. More formally, every point in P is within distance epsilon from the convex hull of S. Such a subset S is called an epsilon-hull. Computing an epsilon-hull is an important problem in computational geometry, machine learning, and approximation algorithms.
In many applications, the set P is too large to fit in memory. We consider the streaming model where the algorithm receives the points of P sequentially and strives to use a minimal amount of memory. Existing streaming algorithms for computing an epsilon-hull require O(epsilon^{(1-d)/2}) space, which is optimal for a worst-case input. However, this ignores the structure of the data. The minimal size of an epsilon-hull of P, which we denote by OPT, can be much smaller. A natural question is whether a streaming algorithm can compute an epsilon-hull using only O(OPT) space.
We begin with lower bounds that show, under a reasonable streaming model, that it is not possible to have a single-pass streaming algorithm that computes an epsilon-hull with O(OPT) space. We instead propose three relaxations of the problem for which we can compute epsilon-hulls using space near-linear to the optimal size. Our first algorithm for points in R^2 that arrive in random-order uses O(log n * OPT) space. Our second algorithm for points in R^2 makes O(log(epsilon^{-1})) passes before outputting the epsilon-hull and requires O(OPT) space. Our third algorithm, for points in R^d for any fixed dimension d, outputs, with high probability, an epsilon-hull for all but delta-fraction of directions and requires O(OPT * log OPT) space.

Avrim Blum, Vladimir Braverman, Ananya Kumar, Harry Lang, and Lin F. Yang. Approximate Convex Hull of Data Streams. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 21:1-21:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blum_et_al:LIPIcs.ICALP.2018.21, author = {Blum, Avrim and Braverman, Vladimir and Kumar, Ananya and Lang, Harry and Yang, Lin F.}, title = {{Approximate Convex Hull of Data Streams}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {21:1--21:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.21}, URN = {urn:nbn:de:0030-drops-90254}, doi = {10.4230/LIPIcs.ICALP.2018.21}, annote = {Keywords: Convex Hulls, Streaming Algorithms, Epsilon Kernels, Sparse Coding} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We revisit one of the classic problems in the data stream literature, namely, that of estimating the frequency moments F_p for 0 < p < 2 of an underlying n-dimensional vector presented as a sequence of additive updates in a stream. It is well-known that using p-stable distributions one can approximate any of these moments up to a multiplicative (1+epsilon)-factor using O(epsilon^{-2} log n) bits of space, and this space bound is optimal up to a constant factor in the turnstile streaming model. We show that surprisingly, if one instead considers the popular random-order model of insertion-only streams, in which the updates to the underlying vector arrive in a random order, then one can beat this space bound and achieve O~(epsilon^{-2} + log n) bits of space, where the O~ hides poly(log(1/epsilon) + log log n) factors. If epsilon^{-2} ~~ log n, this represents a roughly quadratic improvement in the space achievable in turnstile streams. Our algorithm is in fact deterministic, and we show our space bound is optimal up to poly(log(1/epsilon) + log log n) factors for deterministic algorithms in the random order model. We also obtain a similar improvement in space for p = 2 whenever F_2 >~ log n * F_1.

Vladimir Braverman, Emanuele Viola, David P. Woodruff, and Lin F. Yang. Revisiting Frequency Moment Estimation in Random Order Streams. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 25:1-25:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{braverman_et_al:LIPIcs.ICALP.2018.25, author = {Braverman, Vladimir and Viola, Emanuele and Woodruff, David P. and Yang, Lin F.}, title = {{Revisiting Frequency Moment Estimation in Random Order Streams}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.25}, URN = {urn:nbn:de:0030-drops-90294}, doi = {10.4230/LIPIcs.ICALP.2018.25}, annote = {Keywords: Data Stream, Frequency Moments, Random Order, Space Complexity, Insertion Only Stream} }

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Short Paper

**Published in:** LIPIcs, Volume 114, 10th International Conference on Geographic Information Science (GIScience 2018)

Text often includes references to places by name; in prior work, more than 20% of a sample of event-related tweets were found to include place names. Research has addressed the challenge of leveraging the geographic data reflected in text statements, with well-developed methods to recognize location mentions in text and related work on automated toponym resolution (deciding which place in the world is meant by a place name). A core issue that remains is to distinguish between text that mentions a place or places and text that is about a place or places. This paper presents the first step in research to address this challenge. The research reported here sets the conceptual and practical groundwork for subsequent supervised machine learning research; that research will leverage human-produced training data, for which a judgment is made about whether a statement is or is not about a place (or places), to train computational methods to do this classification for large volumes of text. The research step presented here focuses on three questions: (1) what kinds of entities are typically conceptualized as places, (2) what features of a statement prompt the reader to judge a statement to be about a place (or not about a place) and (3) how do judgments of whether or not a statement is about a place compare between a group of experts who have studied the concept of "place" from a geographic perspective and a cross-section of individuals recruited through a crowdsourcing platform to make these judgments.

Alan M. MacEachren, Richard Caneba, Hanzhou Chen, Harrison Cole, Emily Domanico, Nicholas Triozzi, Fangcao Xu, and Liping Yang. Is This Statement About A Place? Comparing two perspectives (Short Paper). In 10th International Conference on Geographic Information Science (GIScience 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 114, pp. 44:1-44:6, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{maceachren_et_al:LIPIcs.GISCIENCE.2018.44, author = {MacEachren, Alan M. and Caneba, Richard and Chen, Hanzhou and Cole, Harrison and Domanico, Emily and Triozzi, Nicholas and Xu, Fangcao and Yang, Liping}, title = {{Is This Statement About A Place? Comparing two perspectives}}, booktitle = {10th International Conference on Geographic Information Science (GIScience 2018)}, pages = {44:1--44:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-083-5}, ISSN = {1868-8969}, year = {2018}, volume = {114}, editor = {Winter, Stephan and Griffin, Amy and Sester, Monika}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GISCIENCE.2018.44}, URN = {urn:nbn:de:0030-drops-93720}, doi = {10.4230/LIPIcs.GISCIENCE.2018.44}, annote = {Keywords: geographic information retrieval, spatial language, crowdsourcing} }

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

Let G = (V,w) be a weighted undirected graph with m edges. The cut dimension of G is the dimension of the span of the characteristic vectors of the minimum cuts of G, viewed as vectors in {0,1}^m. For every n ≥ 2 we show that the cut dimension of an n-vertex graph is at most 2n-3, and construct graphs realizing this bound.
The cut dimension was recently defined by Graur et al. [Andrei Graur et al., 2020], who show that the maximum cut dimension of an n-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on n-vertex graphs. For every n ≥ 2, Graur et al. exhibit a graph on n vertices with cut dimension at least 3n/2 -2, giving the first lower bound larger than n on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of linear queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector x ∈ ℝ^{binom(n,2)} and receives the answer w^T x. Our results thus show a lower bound of 2n-3 on the number of linear queries needed by a deterministic algorithm to solve minimum cut on n-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension.
We further introduce a generalization of the cut dimension which we call the 𝓁₁-approximate cut dimension. The 𝓁₁-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on n = 3k+1 vertices with 𝓁₁-approximate cut dimension 2n-2, showing that it can be strictly larger than the cut dimension.

Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang. On the Cut Dimension of a Graph. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 15:1-15:35, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lee_et_al:LIPIcs.CCC.2021.15, author = {Lee, Troy and Li, Tongyang and Santha, Miklos and Zhang, Shengyu}, title = {{On the Cut Dimension of a Graph}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {15:1--15:35}, 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.2021.15}, URN = {urn:nbn:de:0030-drops-142890}, doi = {10.4230/LIPIcs.CCC.2021.15}, annote = {Keywords: Query complexity, submodular function minimization, cut dimension} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

Andrew M. Childs, Shih-Han Hung, and Tongyang Li. Quantum Query Complexity with Matrix-Vector Products. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 55:1-55:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{childs_et_al:LIPIcs.ICALP.2021.55, author = {Childs, Andrew M. and Hung, Shih-Han and Li, Tongyang}, title = {{Quantum Query Complexity with Matrix-Vector Products}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {55:1--55:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.55}, URN = {urn:nbn:de:0030-drops-141242}, doi = {10.4230/LIPIcs.ICALP.2021.55}, annote = {Keywords: Quantum algorithms, quantum query complexity, matrix-vector products} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with m constraint matrices, each of dimension n and rank r, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time O(m⋅poly(log n,r,1/ε)) given access to a sampling-based low-overhead data structure for the constraint matrices, where ε is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest:
- Weighted sampling: assuming sampling access to each individual constraint matrix A₁,…,A_τ, we propose a procedure that gives a good approximation of A = A₁+⋯+A_τ.
- Symmetric approximation: we propose a sampling procedure that gives the spectral decomposition of a low-rank Hermitian matrix A. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chia_et_al:LIPIcs.MFCS.2020.23, author = {Chia, Nai-Hui and Li, Tongyang and Lin, Han-Hsuan and Wang, Chunhao}, title = {{Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.23}, URN = {urn:nbn:de:0030-drops-126919}, doi = {10.4230/LIPIcs.MFCS.2020.23}, annote = {Keywords: Spectral decomposition, Semi-definite programming, Quantum-inspired algorithm, Sublinear algorithm} }

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

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling.

András Gilyén and Tongyang Li. Distributional Property Testing in a Quantum World. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 25:1-25:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gilyen_et_al:LIPIcs.ITCS.2020.25, author = {Gily\'{e}n, Andr\'{a}s and Li, Tongyang}, title = {{Distributional Property Testing in a Quantum World}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {25:1--25:19}, 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.2020.25}, URN = {urn:nbn:de:0030-drops-117100}, doi = {10.4230/LIPIcs.ITCS.2020.25}, annote = {Keywords: distributional property testing, quantum algorithms, quantum query complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r.
We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics.
As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.

Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 27:1-27:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brandao_et_al:LIPIcs.ICALP.2019.27, author = {Brand\~{a}o, Fernando G. S. L. and Kalev, Amir and Li, Tongyang and Lin, Cedric Yen-Yu and Svore, Krysta M. and Wu, Xiaodi}, title = {{Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.27}, URN = {urn:nbn:de:0030-drops-106036}, doi = {10.4230/LIPIcs.ICALP.2019.27}, annote = {Keywords: quantum algorithms, semidefinite program, convex optimization} }

Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We present a framework for speeding up the time it takes to sample from discrete distributions μ defined over subsets of size k of a ground set of n elements, in the regime where k is much smaller than n. We show that if one has access to estimates of marginals P_{S∼ μ} {i ∈ S}, then the task of sampling from μ can be reduced to sampling from related distributions ν supported on size k subsets of a ground set of only n^{1-α}⋅ poly(k) elements. Here, 1/α ∈ [1, k] is the parameter of entropic independence for μ. Further, our algorithm only requires sparsified distributions ν that are obtained by applying a sparse (mostly 0) external field to μ, an operation that for many distributions μ of interest, retains algorithmic tractability of sampling from ν. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of μ, and in return reduce the amortized cost needed to produce many samples from the distribution μ, as is often needed in upstream tasks such as counting and inference.
For a wide range of distributions where α = Ω(1), our result reduces the domain size, and as a corollary, the cost-per-sample, by a poly(n) factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partition-constrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to α = 1). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over O(1/k) relative error established in prior work.

Nima Anari, Michał Dereziński, Thuy-Duong Vuong, and Elizabeth Yang. Domain Sparsification of Discrete Distributions Using Entropic Independence. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{anari_et_al:LIPIcs.ITCS.2022.5, author = {Anari, Nima and Derezi\'{n}ski, Micha{\l} and Vuong, Thuy-Duong and Yang, Elizabeth}, title = {{Domain Sparsification of Discrete Distributions Using Entropic Independence}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {5:1--5:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.5}, URN = {urn:nbn:de:0030-drops-156013}, doi = {10.4230/LIPIcs.ITCS.2022.5}, annote = {Keywords: Domain Sparsification, Markov Chains, Sampling, Entropic Independence} }

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

We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders.
In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [Jerrum et al., 2004].

Siqi Liu, Sidhanth Mohanty, and Elizabeth Yang. High-Dimensional Expanders from Expanders. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 12:1-12:32, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{liu_et_al:LIPIcs.ITCS.2020.12, author = {Liu, Siqi and Mohanty, Sidhanth and Yang, Elizabeth}, title = {{High-Dimensional Expanders from Expanders}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {12:1--12:32}, 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.2020.12}, URN = {urn:nbn:de:0030-drops-116974}, doi = {10.4230/LIPIcs.ITCS.2020.12}, annote = {Keywords: High-Dimensional Expanders, Markov Chains} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We present a variant of the quantum relational Hoare logic from (Unruh, POPL 2019) that allows us to use "expectations" in pre- and postconditions. That is, when reasoning about pairs of programs, our logic allows us to quantitatively reason about how much certain pre-/postconditions are satisfied that refer to the relationship between the programs inputs/outputs.

Yangjia Li and Dominique Unruh. Quantum Relational Hoare Logic with Expectations. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 136:1-136:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{li_et_al:LIPIcs.ICALP.2021.136, author = {Li, Yangjia and Unruh, Dominique}, title = {{Quantum Relational Hoare Logic with Expectations}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {136:1--136:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.136}, URN = {urn:nbn:de:0030-drops-142058}, doi = {10.4230/LIPIcs.ICALP.2021.136}, annote = {Keywords: Quantum cryptography, Hoare logics, formal verification} }

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**Published in:** LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

We prove two classes of lower bounds on the communication complexity of information-theoretically secure multiparty computation. The first lower bound applies to perfect passive secure multiparty computation in the standard model with n = 2t+1 parties of which t are corrupted. We show a lower bound that applies to secure evaluation of any function, assuming that each party can choose to learn or not learn the output. Specifically, we show that there is a function H^* such that for any protocol that evaluates y_i = b_i ⋅ f(x₁,...,x_n) with perfect passive security (where b_i is a private boolean input), the total communication must be at least 1/2 ∑_{i = 1}ⁿ H_f^*(x_i) bits of information.
The second lower bound applies to the perfect maliciously secure setting with n = 3t+1 parties. We show that for any n and all large enough S, there exists a reactive functionality F_S taking an S-bit string as input (and with short output) such that any protocol implementing F_S with perfect malicious security must communicate Ω(nS) bits. Since the functionalities we study can be implemented with linear size circuits, the result can equivalently be stated as follows: for any n and all large enough g ∈ ℕ there exists a reactive functionality F_C doing computation specified by a Boolean circuit C with g gates, where any perfectly secure protocol implementing F_C must communicate Ω(n g) bits. The results easily extends to constructing similar functionalities defined over any fixed finite field. Using known techniques, we also show an upper bound that matches the lower bound up to a constant factor (existing upper bounds are a factor lg n off for Boolean circuits).
Both results also extend to the case where the threshold t is suboptimal. Namely if n = kt+s the bound is weakened by a factor O(s), which corresponds to known optimizations via packed secret-sharing.

Ivan Bjerre Damgård, Boyang Li, and Nikolaj Ignatieff Schwartzbach. More Communication Lower Bounds for Information-Theoretic MPC. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 2:1-2:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{damgard_et_al:LIPIcs.ITC.2021.2, author = {Damg\r{a}rd, Ivan Bjerre and Li, Boyang and Schwartzbach, Nikolaj Ignatieff}, title = {{More Communication Lower Bounds for Information-Theoretic MPC}}, booktitle = {2nd Conference on Information-Theoretic Cryptography (ITC 2021)}, pages = {2:1--2:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-197-9}, ISSN = {1868-8969}, year = {2021}, volume = {199}, editor = {Tessaro, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.2}, URN = {urn:nbn:de:0030-drops-143211}, doi = {10.4230/LIPIcs.ITC.2021.2}, annote = {Keywords: Multiparty Computation, Lower bounds} }

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