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.
Bibliographical noteFunding 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