4 Search Results for "Sarma M. N., Jayalal"

Document
DOI: 10.4230/LIPIcs.CCC.2024.4
Derandomizing Logspace with a Small Shared Hard Drive

Authors: Edward Pyne

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S²] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S] ⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α ∈ [1,1.5], BPSPACE[S] ⊆ CSPACE[S^α,S^(3-α)]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S], due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S²]. As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
Cite as

Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{pyne:LIPIcs.CCC.2024.4,
  author =	{Pyne, Edward},
  title =	{{Derandomizing Logspace with a Small Shared Hard Drive}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.4},
  URN =		{urn:nbn:de:0030-drops-204006},
  doi =		{10.4230/LIPIcs.CCC.2024.4},
  annote =	{Keywords: Catalytic computation, space-bounded computation, derandomization}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.622
Pebbling, Entropy and Branching Program Size Lower Bounds

Authors: Balagopal Komarath and Jayalal M. N. Sarma

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced in (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012). Proving an exponential lower bound for the size of the non-deterministic thrifty branching programs would separate NL from P under the thrifty hypothesis. In this context, we consider a restriction of non-deterministic thrifty branching programs called bitwise-independence. We show that any bitwise-independent non-deterministic thrifty branching program solving BT_2(h,k) must have at least 1/2 k^{h/2} states. Prior to this work, lower bounds were known for general branching programs only for fixed heights h=2,3,4 (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012). Our lower bounds are also tight (up to a factor of k), since the known (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012) non-deterministic thrifty branching programs for this problem of size O(k^{h/2+1}) are bitwise-independent. We prove our results by associating a fractional pebbling strategy with any bitwise-independent non-deterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Zak (S. Jukna and S. Žák, 2003) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.
Cite as

Balagopal Komarath and Jayalal M. N. Sarma. Pebbling, Entropy and Branching Program Size Lower Bounds. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 622-633, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{komarath_et_al:LIPIcs.STACS.2013.622,
  author =	{Komarath, Balagopal and Sarma, Jayalal M. N.},
  title =	{{Pebbling, Entropy and Branching Program Size Lower Bounds}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{622--633},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.622},
  URN =		{urn:nbn:de:0030-drops-39709},
  doi =		{10.4230/LIPIcs.STACS.2013.622},
  annote =	{Keywords: Pebbling, Entropy Method, Branching Programs, Size Lower Bounds.}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2011.115
Isomorphism testing of read-once functions and polynomials

Authors: Raghavendra Rao B .V. and Jayalal Sarma M. N.

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract
In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of Boolean formulas in which a variable is read at most once (known as read-once formulas) is complete for log-space. In contrast, we observe that the problem becomes polynomial time equivalent to the graph isomorphism problem, when the input formulas can be represented as OR of two or more monotone read-once formulas. This classifies the complexity of the problem in terms of the number of reads, as read-3 formula isomorphism problem is hard for \co\NP. We address the polynomial isomorphism problem, a special case of polynomial equivalence problem which in turn is important from a cryptographic perspective[Patarin EUROCRYPT'96, and Kayal SODA'11]. As our main result, we propose a deterministic polynomial time canonization scheme for polynomials computed by constant-free read-once arithmetic formulas. In contrast, we show that when the arithmetic formula is allowed to read a variable twice, this problem is as hard as the graph isomorphism problem.
Cite as

Raghavendra Rao B .V. and Jayalal Sarma M. N.. Isomorphism testing of read-once functions and polynomials. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 115-126, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{raob.v._et_al:LIPIcs.FSTTCS.2011.115,
  author =	{Rao B .V., Raghavendra and Sarma M. N., Jayalal},
  title =	{{Isomorphism testing of read-once functions and polynomials}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{115--126},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.115},
  URN =		{urn:nbn:de:0030-drops-33202},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.115},
  annote =	{Keywords: Isomorphism Problems, Computational Complexity, Read-once formulas, Read-once Polynomials}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2009.2327
Using Elimination Theory to construct Rigid Matrices

Authors: Abhinav Kumar, Satyanarayana V. Lokam, Vijay M. Patankar, and Jayalal Sarma M. N.

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract
The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that must be changed to ensure that the rank of the altered matrix is at most $r$. Since its introduction by Valiant \cite{Val77}, rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all $\nbyn$ matrices over an infinite field have a rigidity of $(n-r)^2$. It is a long-standing open question to construct infinite families of \emph{explicit} matrices even with superlinear rigidity when $r=\Omega(n)$. In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., $(n-r)^2$, rigidity. The entries of an $\nbyn$ matrix in this family are distinct primitive roots of unity of orders roughly \SL{$\exp(n^4 \log n)$}. To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most $k$ is exactly $n^2 - (n-r)^2 +k$. Finally, we use elimination theory to examine whether the rigidity function is semicontinuous.
Cite as

Abhinav Kumar, Satyanarayana V. Lokam, Vijay M. Patankar, and Jayalal Sarma M. N.. Using Elimination Theory to construct Rigid Matrices. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 299-310, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2009.2327,
  author =	{Kumar, Abhinav and Lokam, Satyanarayana V. and Patankar, Vijay M. and Sarma M. N., Jayalal},
  title =	{{Using Elimination Theory to construct Rigid Matrices}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{299--310},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2327},
  URN =		{urn:nbn:de:0030-drops-23278},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2327},
  annote =	{Keywords: Matrix Rigidity, Lower Bounds, Circuit Complexity}
}
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