Partitioning multi-dimensional sets in a small number of "uniform" parts

Noga Alon; Ilan Newman; Alexander Shen; Gábor Tardos; Nikolai Vereshchagin

doi:10.1016/j.ejc.2005.08.002

Partitioning multi-dimensional sets in a small number of "uniform" parts

Noga Alon, Ilan Newman, Alexander Shen, Gábor Tardos, Nikolai Vereshchagin

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log | E |) subsets so that all the resulting bipartite graphs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of the left nodes is bounded by a constant and the same condition holds for the right nodes. Stated differently, every finite 2-dimensional set S ⊂ N² can be partitioned into poly (log | S |) parts so that in every part the ratio between the maximal size and the minimal size of non-empty horizontal section is bounded by a constant and the same condition holds for vertical sections. We prove a similar statement for n-dimensional sets for any n and show how it can be used to relate information inequalities for Shannon entropy of random variables to inequalities between sizes of sections and their projections of multi-dimensional finite sets.

Original language	English
Pages (from-to)	134-144
Number of pages	11
Journal	European Journal of Combinatorics
Volume	28
Issue number	1
DOIs	https://doi.org/10.1016/j.ejc.2005.08.002
State	Published - Jan 2007

Bibliographical note

Funding Information:
Alon’s research was supported in part by the Israel Science Foundation, by the Hermann Minkowski Minerva Center for Geometry, and by the Von Neumann Fund. Shen’s research was supported in part by CNRS and RFBR. Tardos was partially supported by the Hungarian National Research Grants OTKA-T-037846 and OTKA-T-048826. Vereshchagin’s research was supported in part by RFBR grants 03-01-00475, 358.2003.1.

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1016/j.ejc.2005.08.002

Cite this

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abstract = "Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log | E |) subsets so that all the resulting bipartite graphs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of the left nodes is bounded by a constant and the same condition holds for the right nodes. Stated differently, every finite 2-dimensional set S ⊂ N2 can be partitioned into poly (log | S |) parts so that in every part the ratio between the maximal size and the minimal size of non-empty horizontal section is bounded by a constant and the same condition holds for vertical sections. We prove a similar statement for n-dimensional sets for any n and show how it can be used to relate information inequalities for Shannon entropy of random variables to inequalities between sizes of sections and their projections of multi-dimensional finite sets.",

author = "Noga Alon and Ilan Newman and Alexander Shen and G{\'a}bor Tardos and Nikolai Vereshchagin",

note = "Funding Information: Alon{\textquoteright}s research was supported in part by the Israel Science Foundation, by the Hermann Minkowski Minerva Center for Geometry, and by the Von Neumann Fund. Shen{\textquoteright}s research was supported in part by CNRS and RFBR. Tardos was partially supported by the Hungarian National Research Grants OTKA-T-037846 and OTKA-T-048826. Vereshchagin{\textquoteright}s research was supported in part by RFBR grants 03-01-00475, 358.2003.1.",

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AU - Tardos, Gábor

AU - Vereshchagin, Nikolai

N1 - Funding Information: Alon’s research was supported in part by the Israel Science Foundation, by the Hermann Minkowski Minerva Center for Geometry, and by the Von Neumann Fund. Shen’s research was supported in part by CNRS and RFBR. Tardos was partially supported by the Hungarian National Research Grants OTKA-T-037846 and OTKA-T-048826. Vereshchagin’s research was supported in part by RFBR grants 03-01-00475, 358.2003.1.

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Partitioning multi-dimensional sets in a small number of "uniform" parts

Abstract

Bibliographical note

ASJC Scopus subject areas

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