10 Search Results for "Ben-Aroya, Avraham"


Document
The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs

Authors: Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir

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


Abstract
We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of n disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter r. The case of intersection graphs is a special case with r = 0. We give an algorithm that, given a target length k, computes the smallest value of r for which there is a path of length at most k between some given pair of disks in the proximity graph. Our algorithm runs in O^*(n^{5/4}) randomized expected time, which improves to O^*(n^{6/5}) for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and k is replaced by a target weight w. In other variants, we want to optimize a parameter different from r, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given r has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra’s algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [R. Ben Avraham et al., 2015].

Cite as

Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir. The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 67:1-67:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{kaplan_et_al:LIPIcs.ESA.2023.67,
  author =	{Kaplan, Haim and Katz, Matthew J. and Saban, Rachel and Sharir, Micha},
  title =	{{The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{67:1--67:14},
  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.2023.67},
  URN =		{urn:nbn:de:0030-drops-187208},
  doi =		{10.4230/LIPIcs.ESA.2023.67},
  annote =	{Keywords: Computational geometry, geometric optimization, disk graphs, BFS, Dijkstra’s algorithm, reverse shortest path}
}
Document
Near-Optimal Erasure List-Decodable Codes

Authors: Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)


Abstract
A code 𝒞 ⊆ {0,1}^n̅ is (s,L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n̅-s symbols have at most L possible completions into a codeword of 𝒞. Non-explicitly, there exist binary ((1-τ)n̅,L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log 1/(τ)). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [Guruswami and Indyk, 2002; Guruswami, 2003; Guruswami, 2004]). While partial progress on the problem has been achieved, no prior nontrivial explicit construction achieved rate better than Ω(τ²) or list-size smaller than Ω(1/τ). Furthermore, Guruswami showed no linear code can have list-size smaller than Ω(1/τ) [Guruswami, 2003]. We construct an explicit binary ((1-τ)n̅,L) erasure list-decodable code having rate τ^(1+γ) (for any constant γ > 0 and small τ) and list-size poly(log 1/τ), answering simultaneously both questions, and exhibiting an explicit non-linear code that provably beats the best possible linear code. The binary erasure list-decoding problem is equivalent to the construction of explicit, low-error, strong dispersers outputting one bit with minimal entropy-loss and seed-length. For error ε, no prior explicit construction achieved seed-length better than 2log(1/ε) or entropy-loss smaller than 2log(1/ε), which are the best possible parameters for extractors. We explicitly construct an ε-error one-bit strong disperser with near-optimal seed-length (1+γ)log(1/ε) and entropy-loss O(log log1/ε). The main ingredient in our construction is a new (and almost-optimal) unbalanced two-source extractor. The extractor extracts one bit with constant error from two independent sources, where one source has length n and tiny min-entropy O(log log n) and the other source has length O(log n) and arbitrarily small constant min-entropy rate. When instantiated as a balanced two-source extractor, it improves upon Raz’s extractor [Raz, 2005] in the constant error regime. The construction incorporates recent components and ideas from extractor theory with a delicate and novel analysis needed in order to solve dependency and error issues that prevented previous papers (such as [Li, 2015; Chattopadhyay and Zuckerman, 2019; Cohen, 2016]) from achieving the above results.

Cite as

Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. Near-Optimal Erasure List-Decodable Codes. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 1:1-1:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{benaroya_et_al:LIPIcs.CCC.2020.1,
  author =	{Ben-Aroya, Avraham and Doron, Dean and Ta-Shma, Amnon},
  title =	{{Near-Optimal Erasure List-Decodable Codes}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{1:1--1:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.1},
  URN =		{urn:nbn:de:0030-drops-125531},
  doi =		{10.4230/LIPIcs.CCC.2020.1},
  annote =	{Keywords: Dispersers, Erasure codes, List decoding, Ramsey graphs, Two-source extractors}
}
Document
RANDOM
Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions

Authors: Avraham Ben-Aroya, Gil Cohen, Dean Doron, and Amnon Ta-Shma

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)


Abstract
In their seminal work, Chattopadhyay and Zuckerman (STOC'16) constructed a two-source extractor with error epsilon for n-bit sources having min-entropy {polylog}(n/epsilon). Unfortunately, the construction’s running-time is {poly}(n/epsilon), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a {poly}(n,log(1/epsilon))-time computable two-source condenser. For any k >= {polylog}(n/epsilon), our condenser transforms two independent (n,k)-sources to a distribution over m = k-O(log(1/epsilon)) bits that is epsilon-close to having min-entropy m - o(log(1/epsilon)). Hence, achieving entropy gap of o(log(1/epsilon)). The bottleneck for obtaining low error in recent constructions of two-source extractors lies in the use of resilient functions. Informally, this is a function that receives input bits from r players with the property that the function’s output has small bias even if a bounded number of corrupted players feed adversarial inputs after seeing the inputs of the other players. The drawback of using resilient functions is that the error cannot be smaller than ln r/r. This, in return, forces the running time of the construction to be polynomial in 1/epsilon. A key component in our construction is a variant of resilient functions which we call entropy-resilient functions. This variant can be seen as playing the above game for several rounds, each round outputting one bit. The goal of the corrupted players is to reduce, with as high probability as they can, the min-entropy accumulated throughout the rounds. We show that while the bias decreases only polynomially with the number of players in a one-round game, their success probability decreases exponentially in the entropy gap they are attempting to incur in a repeated game.

Cite as

Avraham Ben-Aroya, Gil Cohen, Dean Doron, and Amnon Ta-Shma. Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 43:1-43:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{benaroya_et_al:LIPIcs.APPROX-RANDOM.2019.43,
  author =	{Ben-Aroya, Avraham and Cohen, Gil and Doron, Dean and Ta-Shma, Amnon},
  title =	{{Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{43:1--43:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.43},
  URN =		{urn:nbn:de:0030-drops-112587},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.43},
  annote =	{Keywords: Condensers, Extractors, Resilient functions, Explicit constructions}
}
Document
Non-Malleable Extractors and Non-Malleable Codes: Partially Optimal Constructions

Authors: Xin Li

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
The recent line of study on randomness extractors has been a great success, resulting in exciting new techniques, new connections, and breakthroughs to long standing open problems in several seemingly different topics. These include seeded non-malleable extractors, privacy amplification protocols with an active adversary, independent source extractors (and explicit Ramsey graphs), and non-malleable codes in the split state model. Previously, the best constructions are given in [Xin Li, 2017]: seeded non-malleable extractors with seed length and entropy requirement O(log n+log(1/epsilon)log log (1/epsilon)) for error epsilon; two-round privacy amplification protocols with optimal entropy loss for security parameter up to Omega(k/log k), where k is the entropy of the shared weak source; two-source extractors for entropy O(log n log log n); and non-malleable codes in the 2-split state model with rate Omega(1/log n). However, in all cases there is still a gap to optimum and the motivation to close this gap remains strong. In this paper, we introduce a set of new techniques to further push the frontier in the above questions. Our techniques lead to improvements in all of the above questions, and in several cases partially optimal constructions. This is in contrast to all previous work, which only obtain close to optimal constructions. Specifically, we obtain: 1) A seeded non-malleable extractor with seed length O(log n)+log^{1+o(1)}(1/epsilon) and entropy requirement O(log log n+log(1/epsilon)), where the entropy requirement is asymptotically optimal by a recent result of Gur and Shinkar [Tom Gur and Igor Shinkar, 2018]; 2) A two-round privacy amplification protocol with optimal entropy loss for security parameter up to Omega(k), which solves the privacy amplification problem completely; 3) A two-source extractor for entropy O((log n log log n)/(log log log n)), which also gives an explicit Ramsey graph on N vertices with no clique or independent set of size (log N)^{O((log log log N)/(log log log log N))}; and 4) The first explicit non-malleable code in the 2-split state model with constant rate, which has been a major goal in the study of non-malleable codes for quite some time. One small caveat is that the error of this code is only (an arbitrarily small) constant, but we can also achieve negligible error with rate Omega(log log log n/log log n), which already improves the rate in [Xin Li, 2017] exponentially. We believe our new techniques can help to eventually obtain completely optimal constructions in the above questions, and may have applications in other settings.

Cite as

Xin Li. Non-Malleable Extractors and Non-Malleable Codes: Partially Optimal Constructions. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 28:1-28:49, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{li:LIPIcs.CCC.2019.28,
  author =	{Li, Xin},
  title =	{{Non-Malleable Extractors and Non-Malleable Codes: Partially Optimal Constructions}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{28:1--28:49},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.28},
  URN =		{urn:nbn:de:0030-drops-108507},
  doi =		{10.4230/LIPIcs.CCC.2019.28},
  annote =	{Keywords: extractor, non-malleable, privacy, codes}
}
Document
Algorithms for the Discrete Fréchet Distance Under Translation

Authors: Omrit Filtser and Matthew J. Katz

Published in: LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)


Abstract
The (discrete) Fréchet distance (DFD) is a popular similarity measure for curves. Often the input curves are not aligned, so one of them must undergo some transformation for the distance computation to be meaningful. Ben Avraham et al. [Rinat Ben Avraham et al., 2015] presented an O(m^3n^2(1+log(n/m))log(m+n))-time algorithm for DFD between two sequences of points of sizes m and n in the plane under translation. In this paper we consider two variants of DFD, both under translation. For DFD with shortcuts in the plane, we present an O(m^2n^2 log^2(m+n))-time algorithm, by presenting a dynamic data structure for reachability queries in the underlying directed graph. In 1D, we show how to avoid the use of parametric search and remove a logarithmic factor from the running time of (the 1D versions of) these algorithms and of an algorithm for the weak discrete Fréchet distance; the resulting running times are thus O(m^2n(1+log(n/m))), for the discrete Fréchet distance, and O(mn log(m+n)), for its two variants. Our 1D algorithms follow a general scheme introduced by Martello et al. [Martello et al., 1984] for the Balanced Optimization Problem (BOP), which is especially useful when an efficient dynamic version of the feasibility decider is available. We present an alternative scheme for BOP, whose advantage is that it yields efficient algorithms quite easily, without having to devise a specially tailored dynamic version of the feasibility decider. We demonstrate our scheme on the most uniform path problem (significantly improving the known bound), and observe that the weak DFD under translation in 1D is a special case of it.

Cite as

Omrit Filtser and Matthew J. Katz. Algorithms for the Discrete Fréchet Distance Under Translation. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 20:1-20:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{filtser_et_al:LIPIcs.SWAT.2018.20,
  author =	{Filtser, Omrit and Katz, Matthew J.},
  title =	{{Algorithms for the Discrete Fr\'{e}chet Distance Under Translation}},
  booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-068-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{101},
  editor =	{Eppstein, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.20},
  URN =		{urn:nbn:de:0030-drops-88466},
  doi =		{10.4230/LIPIcs.SWAT.2018.20},
  annote =	{Keywords: curve similarity, discrete Fr\'{e}chet distance, translation, algorithms, BOP}
}
Document
A New Approach for Constructing Low-Error, Two-Source Extractors

Authors: Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, and Amnon Ta-Shma

Published in: LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)


Abstract
Our main contribution in this paper is a new reduction from explicit two-source extractors for polynomially-small entropy rate and negligible error to explicit t-non-malleable extractors with seed-length that has a good dependence on t. Our reduction is based on the Chattopadhyay and Zuckerman framework (STOC 2016), and surprisingly we dispense with the use of resilient functions which appeared to be a major ingredient there and in follow-up works. The use of resilient functions posed a fundamental barrier towards achieving negligible error, and our new reduction circumvents this bottleneck. The parameters we require from t-non-malleable extractors for our reduction to work hold in a non-explicit construction, but currently it is not known how to explicitly construct such extractors. As a result we do not give an unconditional construction of an explicit low-error two-source extractor. Nonetheless, we believe our work gives a viable approach for solving the important problem of low-error two-source extractors. Furthermore, our work highlights an existing barrier in constructing low-error two-source extractors, and draws attention to the dependence of the parameter t in the seed-length of the non-malleable extractor. We hope this work would lead to further developments in explicit constructions of both non-malleable and two-source extractors.

Cite as

Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, and Amnon Ta-Shma. A New Approach for Constructing Low-Error, Two-Source Extractors. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 3:1-3:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{benaroya_et_al:LIPIcs.CCC.2018.3,
  author =	{Ben-Aroya, Avraham and Chattopadhyay, Eshan and Doron, Dean and Li, Xin and Ta-Shma, Amnon},
  title =	{{A New Approach for Constructing Low-Error, Two-Source Extractors}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Servedio, Rocco A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.3},
  URN =		{urn:nbn:de:0030-drops-88877},
  doi =		{10.4230/LIPIcs.CCC.2018.3},
  annote =	{Keywords: Two-Source Extractors, Non-Malleable Extractors, Pseudorandomness, Explicit Constructions}
}
Document
Semidefinite Programs for Randomness Extractors

Authors: Mario Berta, Omar Fawzi, and Volkher B. Scholz

Published in: LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)


Abstract
Randomness extractors are an important building block for classical and quantum cryptography. However, for many applications it is crucial that the extractors are quantum-proof, i.e., that they work even in the presence of quantum adversaries. In general, quantum-proof extractors are poorly understood and we would like to argue that in the same way as Bell inequalities (multiprover games) and communication complexity, the setting of randomness extractors provides a operationally useful framework for studying the power and limitations of a quantum memory compared to a classical one. We start by recalling how to phrase the extractor property as a quadratic program with linear constraints. We then construct a semidefinite programming (SDP) relaxation for this program that is tight for some extractor constructions. Moreover, we show that this SDP relaxation is even sufficient to certify quantum-proof extractors. This gives a unifying approach to understand the stability properties of extractors against quantum adversaries. Finally, we analyze the limitations of this SDP relaxation.

Cite as

Mario Berta, Omar Fawzi, and Volkher B. Scholz. Semidefinite Programs for Randomness Extractors. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 73-91, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{berta_et_al:LIPIcs.TQC.2015.73,
  author =	{Berta, Mario and Fawzi, Omar and Scholz, Volkher B.},
  title =	{{Semidefinite Programs for Randomness Extractors}},
  booktitle =	{10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
  pages =	{73--91},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-96-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{44},
  editor =	{Beigi, Salman and K\"{o}nig, Robert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.73},
  URN =		{urn:nbn:de:0030-drops-55507},
  doi =		{10.4230/LIPIcs.TQC.2015.73},
  annote =	{Keywords: Randomness Extractors, Quantum adversaries, Semidefinite programs}
}
Document
Approximability of the Discrete Fréchet Distance

Authors: Karl Bringmann and Wolfgang Mulzer

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds. In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399. This raises the question of how well we can approximate the Fréchet distance (of two given d-dimensional point sequences of length n) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be 2^Theta(n). Moreover, we design an alpha-approximation algorithm that runs in time O(n log n + n^2 / alpha), for any alpha in [1, n]. Hence, an n^epsilon-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any epsilon > 0.

Cite as

Karl Bringmann and Wolfgang Mulzer. Approximability of the Discrete Fréchet Distance. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 739-753, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bringmann_et_al:LIPIcs.SOCG.2015.739,
  author =	{Bringmann, Karl and Mulzer, Wolfgang},
  title =	{{Approximability of the Discrete Fr\'{e}chet Distance}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{739--753},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.739},
  URN =		{urn:nbn:de:0030-drops-51072},
  doi =		{10.4230/LIPIcs.SOCG.2015.739},
  annote =	{Keywords: Fr\'{e}chet distance, approximation, lower bounds, Strong Exponential Time Hypothesis}
}
Document
Generalized Quantum Arthur-Merlin Games

Authors: Hirotada Kobayashi, Francois Le Gall, and Harumichi Nishimura

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having $m$-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

Cite as

Hirotada Kobayashi, Francois Le Gall, and Harumichi Nishimura. Generalized Quantum Arthur-Merlin Games. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 488-511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Copy BibTex To Clipboard

@InProceedings{kobayashi_et_al:LIPIcs.CCC.2015.488,
  author =	{Kobayashi, Hirotada and Le Gall, Francois and Nishimura, Harumichi},
  title =	{{Generalized Quantum Arthur-Merlin Games}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{488--511},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.488},
  URN =		{urn:nbn:de:0030-drops-50697},
  doi =		{10.4230/LIPIcs.CCC.2015.488},
  annote =	{Keywords: interactive proof systems, Arthur-Merlin games, quantum computing, complete problems, entanglement}
}
Document
Optimal Bounds for Parity-Oblivious Random Access Codes with Applications

Authors: André Chailloux, Iordanis Kerenidis, Srijita Kundu, and Jamie Sikora

Published in: LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)


Abstract
Random Access Codes is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an n-bit string x, and wishes to encode x into a quantum state rho_x, such that Bob, when receiving the state rho_x, can choose any bit i in [n] and recover the input bit x_i with high probability. Here we study a variant called parity-oblivious random acres codes, where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input, apart form the single bits x_i. We provide the optimal quantum parity-oblivious random access codes and show that they are asymptotically better than the optimal classical ones. For this, we relate such encodings to a non-local game and provide tight bounds for the success probability of the non-local game via semi-definite programming. Our results provide a large non-contextuality inequality violation and resolve the main open question in [Spekkens et al., Phys. Review Letters, 2009].

Cite as

André Chailloux, Iordanis Kerenidis, Srijita Kundu, and Jamie Sikora. Optimal Bounds for Parity-Oblivious Random Access Codes with Applications. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 76-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Copy BibTex To Clipboard

@InProceedings{chailloux_et_al:LIPIcs.TQC.2014.76,
  author =	{Chailloux, Andr\'{e} and Kerenidis, Iordanis and Kundu, Srijita and Sikora, Jamie},
  title =	{{Optimal Bounds for Parity-Oblivious Random Access Codes with Applications}},
  booktitle =	{9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)},
  pages =	{76--87},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-73-6},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{27},
  editor =	{Flammia, Steven T. and Harrow, Aram W.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.76},
  URN =		{urn:nbn:de:0030-drops-48084},
  doi =		{10.4230/LIPIcs.TQC.2014.76},
  annote =	{Keywords: quantum information theory, contextuality, semidefinite programming}
}
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