9 Search Results for "Gupta, Chetan"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.10
Locality in Online, Dynamic, Sequential, and Distributed Graph Algorithms

Authors: Amirreza Akbari, Navid Eslami, Henrik Lievonen, Darya Melnyk, Joona Särkijärvi, and Jukka Suomela

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL model: the adversary presents the nodes of the input graph one by one, in the same way as in classical online algorithms, but for each node we get to see its radius-T neighborhood before choosing the output. Instead of looking ahead in time, we have the power of looking around in space. We compare the online-LOCAL model with three other models: the LOCAL model of distributed computing, where each node produces its output based on its radius-T neighborhood, the SLOCAL model, which is the sequential counterpart of LOCAL, and the dynamic-LOCAL model, where changes in the dynamic input graph only influence the radius-T neighborhood of the point of change. The SLOCAL and dynamic-LOCAL models are sandwiched between the LOCAL and online-LOCAL models. In general, all four models are distinct, but we study in particular locally checkable labeling problems (LCLs), which is a family of graph problems extensively studied in the context of distributed graph algorithms. We prove that for LCL problems in paths, cycles, and rooted trees, all four models are roughly equivalent: the locality of any LCL problem falls in the same broad class - O(log* n), Θ(log n), or n^Θ(1) - in all four models. In particular, this result enables one to generalize prior lower-bound results from the LOCAL model to all four models, and it also allows one to simulate e.g. dynamic-LOCAL algorithms efficiently in the LOCAL model. We also show that this equivalence does not hold in two-dimensional grids or general bipartite graphs. We provide an online-LOCAL algorithm with locality O(log n) for the 3-coloring problem in bipartite graphs - this is a problem with locality Ω(n^{1/2}) in the LOCAL model and Ω(n^{1/10}) in the SLOCAL model.
Cite as

Amirreza Akbari, Navid Eslami, Henrik Lievonen, Darya Melnyk, Joona Särkijärvi, and Jukka Suomela. Locality in Online, Dynamic, Sequential, and Distributed Graph Algorithms. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{akbari_et_al:LIPIcs.ICALP.2023.10,
  author =	{Akbari, Amirreza and Eslami, Navid and Lievonen, Henrik and Melnyk, Darya and S\"{a}rkij\"{a}rvi, Joona and Suomela, Jukka},
  title =	{{Locality in Online, Dynamic, Sequential, and Distributed Graph Algorithms}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.10},
  URN =		{urn:nbn:de:0030-drops-180627},
  doi =		{10.4230/LIPIcs.ICALP.2023.10},
  annote =	{Keywords: Online computation, spatial advice, distributed algorithms, computational complexity}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.118
Dynamic Meta-Theorems for Distance and Matching

Authors: Samir Datta, Chetan Gupta, Rahul Jain, Anish Mukherjee, Vimal Raj Sharma, and Raghunath Tewari

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Reachability, distance, and matching are some of the most fundamental graph problems that have been of particular interest in dynamic complexity theory in recent years [Samir Datta et al., 2018; Samir Datta et al., 2018; Samir Datta et al., 2020]. Reachability can be maintained with first-order update formulas, or equivalently in DynFO in general graphs with n nodes [Samir Datta et al., 2018], even under O(log(n)/log log(n)) changes per step [Samir Datta et al., 2018]. In the context of how large the number of changes can be handled, it has recently been shown [Samir Datta et al., 2020] that under a polylogarithmic number of changes, reachability is in DynFOpar in planar, bounded treewidth, and related graph classes - in fact in any graph where small non-zero circulation weights can be computed in NC. We continue this line of investigation and extend the meta-theorem for reachability to distance and bipartite maximum matching with the same bounds. These are amongst the most general classes of graphs known where we can maintain these problems deterministically without using a majority quantifier and even maintain witnesses. For the bipartite matching result, modifying the approach from [Stephen A. Fenner et al., 2016], we convert the static non-zero circulation weights to dynamic matching-isolating weights. While reachability is in DynFOar under O(log(n)/log log(n)) changes, no such bound is known for either distance or matching in any non-trivial class of graphs under non-constant changes. We show that, in the same classes of graphs as before, bipartite maximum matching is in DynFOar under O(log(n)/log log(n)) changes per step. En route to showing this we prove that the rank of a matrix can be maintained in DynFOar, also under O(log(n)/log log(n)) entry changes, improving upon the previous O(1) bound [Samir Datta et al., 2018]. This implies a similar extension for the non-uniform DynFO bound for maximum matching in general graphs and an alternate algorithm for maintaining reachability under O(log(n)/log log(n)) changes [Samir Datta et al., 2018].
Cite as

Samir Datta, Chetan Gupta, Rahul Jain, Anish Mukherjee, Vimal Raj Sharma, and Raghunath Tewari. Dynamic Meta-Theorems for Distance and Matching. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 118:1-118:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{datta_et_al:LIPIcs.ICALP.2022.118,
  author =	{Datta, Samir and Gupta, Chetan and Jain, Rahul and Mukherjee, Anish and Sharma, Vimal Raj and Tewari, Raghunath},
  title =	{{Dynamic Meta-Theorems for Distance and Matching}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{118:1--118:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.118},
  URN =		{urn:nbn:de:0030-drops-164598},
  doi =		{10.4230/LIPIcs.ICALP.2022.118},
  annote =	{Keywords: Dynamic Complexity, Distance, Matching, Derandomization, Isolation, Matrix Rank}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.16
Reachability and Matching in Single Crossing Minor Free Graphs

Authors: Samir Datta, Chetan Gupta, Rahul Jain, Anish Mukherjee, Vimal Raj Sharma, and Raghunath Tewari

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
We show that for each single crossing graph H, a polynomially bounded weight function for all H-minor free graphs G can be constructed in logspace such that it gives nonzero weights to all the cycles in G. This class of graphs subsumes almost all classes of graphs for which such a weight function is known to be constructed in logspace. As a consequence, we obtain that for the class of H-minor free graphs where H is a single crossing graph, reachability can be solved in UL, and bipartite maximum matching can be solved in SPL, which are small subclasses of the parallel complexity class NC. In the restrictive case of bipartite graphs, our maximum matching result improves upon the recent result of Eppstein and Vazirani [David Eppstein and Vijay V. Vazirani, 2021], where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.
Cite as

Samir Datta, Chetan Gupta, Rahul Jain, Anish Mukherjee, Vimal Raj Sharma, and Raghunath Tewari. Reachability and Matching in Single Crossing Minor Free Graphs. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{datta_et_al:LIPIcs.FSTTCS.2021.16,
  author =	{Datta, Samir and Gupta, Chetan and Jain, Rahul and Mukherjee, Anish and Sharma, Vimal Raj and Tewari, Raghunath},
  title =	{{Reachability and Matching in Single Crossing Minor Free Graphs}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{16:1--16:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.16},
  URN =		{urn:nbn:de:0030-drops-155277},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.16},
  annote =	{Keywords: Reachability, Matching, Logspace, Single-crossing minor free graphs}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.23
Time Space Optimal Algorithm for Computing Separators in Bounded Genus Graphs

Authors: Chetan Gupta, Rahul Jain, and Raghunath Tewari

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
A graph separator is a subset of vertices of a graph whose removal divides the graph into small components. Computing small graph separators for various classes of graphs is an important computational task. In this paper, we present a polynomial-time algorithm that uses O(g^{1/2} n^{1/2} log n)-space to find an O(g^{1/2} n^{1/2})-sized separator of a graph having n vertices and embedded on an orientable surface of genus g.
Cite as

Chetan Gupta, Rahul Jain, and Raghunath Tewari. Time Space Optimal Algorithm for Computing Separators in Bounded Genus Graphs. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2021.23,
  author =	{Gupta, Chetan and Jain, Rahul and Tewari, Raghunath},
  title =	{{Time Space Optimal Algorithm for Computing Separators in Bounded Genus Graphs}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.23},
  URN =		{urn:nbn:de:0030-drops-155344},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.23},
  annote =	{Keywords: Graph algorithms, space-bounded algorithms, surface embedded graphs, reachability, Euler genus, algorithmic graph theory, computational complexity theory}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.43
Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs

Authors: Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari

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

Abstract
We show that given an embedding of an O(log n) genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists. As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for O(log n) genus bipartite graphs.
Cite as

Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{gupta_et_al:LIPIcs.MFCS.2020.43,
  author =	{Gupta, Chetan and Sharma, Vimal Raj and Tewari, Raghunath},
  title =	{{Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{43:1--43:13},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.43},
  URN =		{urn:nbn:de:0030-drops-127099},
  doi =		{10.4230/LIPIcs.MFCS.2020.43},
  annote =	{Keywords: Logspace computation, High genus, Matching isolation}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2020.122
Dynamic Complexity of Reachability: How Many Changes Can We Handle?

Authors: Samir Datta, Pankaj Kumar, Anish Mukherjee, Anuj Tawari, Nils Vortmeier, and Thomas Zeume

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges. Later, in 2018, this was extended for changes of size (log n)/(log log n), where n is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used. In this paper we extend these results by showing that, for changes of polylogarithmic size, first-order update formulas suffice for maintaining (1) undirected reachability, and (2) directed reachability under insertions. For classes of directed graphs for which efficient parallel algorithms can compute non-zero circulation weights, reachability can be maintained with update formulas that may use "modulo 2" quantifiers under changes of polylogarithmic size. Examples for these classes include the class of planar graphs and graphs with bounded treewidth. The latter is shown here. As the logics we consider cannot maintain reachability under changes of larger sizes, our results are optimal with respect to the size of the changes.
Cite as

Samir Datta, Pankaj Kumar, Anish Mukherjee, Anuj Tawari, Nils Vortmeier, and Thomas Zeume. Dynamic Complexity of Reachability: How Many Changes Can We Handle?. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 122:1-122:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{datta_et_al:LIPIcs.ICALP.2020.122,
  author =	{Datta, Samir and Kumar, Pankaj and Mukherjee, Anish and Tawari, Anuj and Vortmeier, Nils and Zeume, Thomas},
  title =	{{Dynamic Complexity of Reachability: How Many Changes Can We Handle?}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{122:1--122:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.122},
  URN =		{urn:nbn:de:0030-drops-125291},
  doi =		{10.4230/LIPIcs.ICALP.2020.122},
  annote =	{Keywords: Dynamic complexity, reachability, complex changes}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.16
Unambiguous Catalytic Computation

Authors: Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
The catalytic Turing machine is a model of computation defined by Buhrman, Cleve, Koucký, Loff, and Speelman (STOC 2014). Compared to the classical space-bounded Turing machine, this model has an extra space which is filled with arbitrary content in addition to the clean space. In such a model we study if this additional filled space can be used to increase the power of computation or not, with the condition that the initial content of this extra filled space must be restored at the end of the computation. In this paper, we define the notion of unambiguous catalytic Turing machine and prove that under a standard derandomization assumption, the class of problems solved by an unambiguous catalytic Turing machine is same as the class of problems solved by a general nondeterministic catalytic Turing machine in the logspace setting.
Cite as

Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Unambiguous Catalytic Computation. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2019.16,
  author =	{Gupta, Chetan and Jain, Rahul and Sharma, Vimal Raj and Tewari, Raghunath},
  title =	{{Unambiguous Catalytic Computation}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{16:1--16:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.16},
  URN =		{urn:nbn:de:0030-drops-115782},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.16},
  annote =	{Keywords: Catalytic computation, Logspace, Reinhardt-Allender}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.34
Reachability in O(log n) Genus Graphs is in Unambiguous Logspace

Authors: Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We show that given an embedding of an O(log n) genus graph G and two vertices s and t in G, deciding if there is a path from s to t in G is in unambiguous logarithmic space. Unambiguous computation is a restriction of nondeterministic computation where the nondeterministic machine has at most one accepting computation path on each input. An important fundamental question in computational complexity theory is whether this is an actual restriction or are unambiguous computations as powerful as general nondeterminism. We investigate this problem in the domain of logarithmic space bounded computations, where the corresponding unambiguous and general nondeterministic classes are UL and NL respectively. In 1997 Reinhardt and Allender showed that NL and UL are equal in a non-uniform model. More specifically they showed that if one can efficiently construct an O(log n)-bit min-unique weight function for a graph, then these classes are equal unconditionally as well. In other words, they gave a UL algorithm to solve reachability in graphs with a min-unique weight assignment. Using this approach reachability in various classes of graphs such as planar graphs, constant genus graphs, minor free graphs, etc., have been shown to be in UL by devising min-unique weight functions for those classes. In this paper we improve these results by constructing a min-unique weight function for O(log n) genus graphs. We define signature of a path in a graph as the parity of the number of crossings of that path with respect to each handle of the surface on which the graph is embedded. We construct our weight function in two steps. First we ensure that between any pair of vertices, amongst all paths having the same signature, the minimum weight path is unique. Now since in a genus g graph there are 2^{2g} many possible signatures, we use the hashing scheme of Fredman, Komlós and Szemerédi to isolate a unique minimum weight path among these 2^{2g} many paths isolated in the first step.
Cite as

Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Reachability in O(log n) Genus Graphs is in Unambiguous Logspace. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{gupta_et_al:LIPIcs.STACS.2019.34,
  author =	{Gupta, Chetan and Sharma, Vimal Raj and Tewari, Raghunath},
  title =	{{Reachability in O(log n) Genus Graphs is in Unambiguous Logspace}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{34:1--34:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.34},
  URN =		{urn:nbn:de:0030-drops-102730},
  doi =		{10.4230/LIPIcs.STACS.2019.34},
  annote =	{Keywords: logspace unambiguity, high genus, path isolation}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.23
On Higher-Order Fourier Analysis over Non-Prime Fields

Authors: Arnab Bhattacharyya, Abhishek Bhowmick, and Chetan Gupta

Published in: LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Abstract
The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character chi, either chi(P) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of chi(P) and its "structure"? Previously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field F_p. Our technical results are: 1. If P: K^n -> K is a polynomial of degree <= d, and E[chi(P(x))] > |K|^{-s} for some s > 0 and non-trivial additive character chi, then P is a function of O_{d, s}(1) many non-classical polynomials of weight degree < d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work. 2. Suppose K and F are of bounded order, and let H be an affine subspace of K^n. Then, if P: K^n -> K is a polynomial of degree d that is sufficiently regular, then (P(x): x in H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P. Such a theorem was previously known for H an affine subspace over a prime field. The tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas. (i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code. (ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P: K^n -> K can be decomposed as a particular composition of lesser degree polynomials. (iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f: K^n -> K are testable with one-sided error.
Cite as

Arnab Bhattacharyya, Abhishek Bhowmick, and Chetan Gupta. On Higher-Order Fourier Analysis over Non-Prime Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 23:1-23:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{bhattacharyya_et_al:LIPIcs.APPROX-RANDOM.2016.23,
  author =	{Bhattacharyya, Arnab and Bhowmick, Abhishek and Gupta, Chetan},
  title =	{{On Higher-Order Fourier Analysis over Non-Prime Fields}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{23:1--23:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.23},
  URN =		{urn:nbn:de:0030-drops-66463},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.23},
  annote =	{Keywords: finite fields, higher order fourier analysis, coding theory, property testing}
}
  • Refine by Author
  • 7 Gupta, Chetan
  • 6 Tewari, Raghunath
  • 5 Sharma, Vimal Raj
  • 4 Jain, Rahul
  • 3 Datta, Samir
  • Show More...

  • Refine by Classification
  • 3 Theory of computation → Complexity classes
  • 2 Theory of computation → Complexity theory and logic
  • 1 Computing methodologies → Distributed algorithms
  • 1 Mathematics of computing
  • 1 Mathematics of computing → Graph algorithms
  • Show More...

  • Refine by Keyword
  • 2 Logspace
  • 2 Matching
  • 2 reachability
  • 1 Catalytic computation
  • 1 Derandomization
  • Show More...

  • Refine by Type
  • 9 document

  • Refine by Publication Year
  • 2 2019
  • 2 2020
  • 2 2021
  • 1 2016
  • 1 2022
  • Show More...

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact