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# Some Results Related to Dense Families of Database Relations

### Abstract (in LaTeX format)

The dense families of database relations were introduced by J\"{a}rvinen. The aim of this paper is to investigate some new properties of dense families of database relations, and their applications. That is, we characterize functional dependencies and minimal keys in terms of dense families. We give a necessary and sufficient condition for an abitrary family to be $R-$ dense family. We prove that with a given relation $R$ the equality set $E_R$ is an $R-$dense family whose size is at most $\frac{m(m-1)}{2}$, where $m$ is the number of tuples in $R$. We also prove that the set of all minimal keys of relation $R$ is the transversal hypergraph of the complement of the equality set $E_R$. We give an effective algorithm finding all minimal keys of a given relation $R$. We aslo give an algorithm which from a given relation $R$ finds a cover of functional dependencies that holds in $R$. The complexity of these algorithms is also esimated.

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### BibTeX entry

@article{DucThi:2005:ActaCybernetica,
author = {Vu Duc Thi and Nguyen Hoang Son},
title = {Some Results Related to Dense Families of Database Relations},
journal = {Acta Cybernetica},
year = {2005},
volume = {17},
number = {1},
pages = {173--182},
abstract = {The dense families of database relations were introduced by J\"{a}rvinen. The aim of this paper is to investigate some new properties of dense families of database relations, and their applications. That is, we characterize functional dependencies and minimal keys in terms of dense families. We give a necessary and sufficient condition for an abitrary family to be $R-$ dense family. We prove that with a given relation $R$ the equality set $E_R$ is an $R-$dense family whose size is at most $\frac{m(m-1)}{2}$, where $m$ is the number of tuples in $R$. We also prove that the set of all minimal keys of relation $R$ is the transversal hypergraph of the complement of the equality set $E_R$. We give an effective algorithm finding all minimal keys of a given relation $R$. We aslo give an algorithm which from a given relation $R$ finds a cover of functional dependencies that holds in $R$. The complexity of these algorithms is also esimated.}
}

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