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Gathering rights

As we saw in section [*], we can write $ R=R_C \cup
R_F$ with $ R_C=U \times C$ and $ R_F=U \times A \times F$.

We saw in section [*] that we can reduce the size of the starting set, and thus the size of the function, using a subset of $ F$ and the file system structure. We can do the same here.

For each file $ f \in F$, we will adapt the set of rights. let $ H_f$ be a subset of $ F$ so that "/"$ \in H_f$. We have, as defined in section [*],

$\displaystyle \psi_{H_f}~:~F$ $\displaystyle \rightarrow$ $\displaystyle H_f$  
$\displaystyle x$ $\displaystyle \mapsto$ $\displaystyle \psi^{\alpha_{H_f}(x)}(x)$  

The binary relation $ \mathcal{R}_{H_f}$ defined by

$\displaystyle x \mathcal{R}_{H_f} y \Leftrightarrow \psi_{H_f}(x)=\psi_{H_f}(y)$

is an equivalence relation.

If $ x \in F$, we write $ \tilde{x}=\{y \in F ~\vert~ y \mathcal{R}_{H_f} x\} \in
\frac {F}{\mathcal{R}_{H_f}}$ the equivalence class of $ x$. $ \frac {F}{\mathcal{R}_{H_f}}$ is the quotient set, i.e. the set of all the equivalence classes.

Let's define the function

$\displaystyle \Psi_{H_f}~:~ U \times C \cup U \times A \times H_f$ $\displaystyle \rightarrow$ $\displaystyle \mathcal{P}(R)$  
$\displaystyle (u,c) \in U \times C$ $\displaystyle \mapsto$ $\displaystyle \{(u,c)\}$  
$\displaystyle (u,a,h) \in U \times A \times H_f$ $\displaystyle \mapsto$ $\displaystyle \{(u,a)\} \times \tilde{h}$  

With a given $ f \in F$, if we correctly choose $ H_f$, we can factorize $ \sigma'$ and $ \mu'$. Let's call $ \beta(f)$ the best set $ H_f$ that can be found for $ f$. Let's call $ \gamma:f \mapsto \Psi_{\beta(f)}$. And let's call $ \delta(f):G
\rightarrow \beta(f)$ so that $ \forall f \in F \quad\sigma(f)=\left(\gamma(f)\circ
\delta(f)\circ \psi_G\right)(f)$. Here is our factorization.

Thus we have replaced the data of the graph of $ \sigma ~:~ F \rightarrow R$ by the data of the graphes of $ \beta ~:~ F \rightarrow \mathcal{P}(F)$ and

$\displaystyle \delta ~:~ F$ $\displaystyle \rightarrow$ $\displaystyle \fonc(G,\beta(f))$  
$\displaystyle f$ $\displaystyle \mapsto$ $\displaystyle \delta(f) ~:~ G \rightarrow \beta(f)$  

which is smaller in practical cases.


next up previous contents
Next: LIDS 2.0 description Up: Enhancing rights mapping Previous: Gathering files   Contents
Biondi Philippe 2000-12-15