nach oben

Theory of Computing Systems

Erschienen in:

14.12.2016

Partition Expanders

verfasst von: Dmitry Gavinsky, Pavel Pudlák

Erschienen in: Theory of Computing Systems | Ausgabe 3/2017

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any pair of sufficiently large sets is close to the expected number, we consider partitions and require this condition only for most of the pairs of blocks. As a result, the blocks can be substantially smaller. We show that for some range of parameters, to be a partition expander a random graph needs exponentially smaller degree than any expander would require in order to achieve similar expanding properties. We apply the concept of partition expanders in communication complexity. First, we construct an optimal pseudo-random generator (PRG) for the Simultaneous Message Passing (SMP) model: it needs n + log k random bits against protocols of cost Ω(k). Second, we compare the SMP model to that of Simultaneous Two-Way Communication, and give a new separation that is stronger both qualitatively and quantitatively than the previously known ones.

Vorheriger Artikel Editor’s Note

Nächster Artikel Genericity of Weakly Computable Objects

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

We use “N” to denote the number of graph vertices and “n” for the input length throughout the paper.

We get quadratic improvement in terms of the partition size, and show that it is, essentially, the best possible general bound in terms of the spectral gap alone.

All previously known PRGs in communication complexity were given against stronger models, thus requiring exponentially larger “overhead” over n in terms of seed length—for details, see Section 5.

In those cases when we explicitly allow multiple edges, the edges of a graph will be viewed as a collection with repetitions.

Note that if v ₁, v ₂ ∈ S ₁ ∩ S ₂, then the edge (v ₁, v ₂) appears in E(S ₁, S ₂) twice: as ordered pairs (v ₁, v ₂) and (v ₂, v ₁).

Note also that the distribution \(\mathcal {G}_{N,d}^{\prime }\) is not uniform over its support—e.g., \(\mathcal {G}_{2,2}^{\prime }\) produces the graph with two parallel edges with probability 2/3.

Note that if e = (v _2i−1, v _2i) ∈ E then (v _2i, v _2i−1) ∈ E as well.

For example, the models \(\mathcal {R}^{1}\) and \(\mathcal {R}^{\leftrightarrow }\) (and more generally, any two-party model where a k-bit message from one player reaches the other player, who also receives his own n bits of input) require seed length at least n + k − O(1) even with a non-uniform PRG, as witnessed by the protocol where the sender sends the first k bits of his input and the computationally-unlimited recipient outputs “1” only if the message together with his own n bits of input have Kolmogorov complexity n + k−O(1).

The same applies to many other fields of complexity, where also most of the following discussion remains valid—e.g., in the field of circuit complexity.

Note that we required both \(\mathcal {U}_{f^{-1}(0)}\) and \(\mathcal {U}_{f^{-1}(1)}\) to be k-PRGs for \(\mathcal {M}\) when f is k-pseudo-random in order not to require f to be balanced; if it is balanced, either condition implies the other.

The power of the model \(\mathcal {R}^{\leftrightarrow }\) is not affected by allowing public randomness.

Alon, N., Roichman, Y.: Random cayley graphs and expanders. Random Struct. Algorithms 5, 271–284 (1994)MathSciNetCrossRefMATH

Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (2008)CrossRefMATH

Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 68, 357–367 (1967)MathSciNetCrossRefMATH

Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: Information theory methods in communication complexity. In: Proceedings of 17th IEEE Conference on Computational Complexity, pp 93–102 (2002)

Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb. 1, 311–316 (1980)MathSciNetCrossRefMATH

Capalbo, M., Reingold, O., Vadhan, S., Wigderson, A.: Randomness conductors and Constant-Degree lossless expanders. In: Proceedings of the 34th Symposium on Theory of Computing, pp 659–668 (2002)

Dvir, Z., Wigderson, A.: Monotone expanders: constructions and applications. Theory Comput. 6(1), 291–308 (2010)MathSciNetCrossRefMATH

Gavinsky, D., Regev, O., de Wolf, R.: Simultaneous communication protocols with quantum and classical messages. Chic. J. Theor. Comput. Sci. 7 (2008)

Hoffman, A.J., Wielandt, H.W.: The variation of the spectrum of a normal matrix. Duke Math. J. 20, 37–39 (1953)MathSciNetCrossRefMATH

10.

Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for network algorithms. In: Proceedings of the 26th Symposium on Theory of Computing, pp 356–364 (1994)

11.

Landau, Z., Russell, A.: Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem. Electron. J. Comb. 11 (2004)

12.

McKay, B.D., Wormald, N.C.: Asymptotic enumeration by degree sequence of graphs with degrees o(n ^1/2). Combinatorica 11(4), 369–382 (1991)MathSciNetCrossRefMATH

13.

Mendel, M., Naor, A.: Nonlinear spectral calculus and super-expanders. Publications mathématiques de l’IHÉ (2013)

14.

Wormald, N.C.: Models of random regular graphs. Surveys in combinatorics. Lecture Note Series 276, pp. 239–298 (1999)

Titel: Partition Expanders
verfasst von: Dmitry Gavinsky
Pavel Pudlák
Publikationsdatum: 14.12.2016
Verlag: Springer US
Erschienen in: Theory of Computing Systems / Ausgabe 3/2017
Print ISSN: 1432-4350
Elektronische ISSN: 1433-0490
DOI: https://doi.org/10.1007/s00224-016-9738-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2017

A Solution to Wiehagen’s Thesis

Genericity of Weakly Computable Objects

On Boolean Closed Full Trios and Rational Kripke Frames

New Bounds and Extended Relations Between Prefix Arrays, Border Arrays, Undirected Graphs, and Indeterminate Strings

Editor’s Note

Higher Randomness and Forcing with Closed Sets