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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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**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 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 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 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

We present a framework for symbolically executing and model checking higher-order programs with external (open) methods. We focus on the client-library paradigm and in particular we aim to check libraries with respect to any definable client. We combine traditional symbolic execution techniques with operational game semantics to build a symbolic execution semantics that captures arbitrary external behaviour. We prove the symbolic semantics to be sound and complete. This yields a bounded technique by imposing bounds on the depth of recursion and callbacks. We provide an implementation of our technique in the 𝕂 framework and showcase its performance on a custom benchmark based on higher-order coding errors such as reentrancy bugs.

Yu-Yang Lin and Nikos Tzevelekos. Symbolic Execution Game Semantics. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 27:1-27:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lin_et_al:LIPIcs.FSCD.2020.27, author = {Lin, Yu-Yang and Tzevelekos, Nikos}, title = {{Symbolic Execution Game Semantics}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {27:1--27:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.27}, URN = {urn:nbn:de:0030-drops-123493}, doi = {10.4230/LIPIcs.FSCD.2020.27}, annote = {Keywords: game semantics, symbolic execution, higher-order open programs} }

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