10th GAMM - IMACS International Symposium on Scientific by Alt R., Vignes J.

By Alt R., Vignes J.

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Thus for each H− , there is a minimal hitting set Hi with respect to OS i , such that |Hi | ≤ m, H− = Hi \ HA and Hi ⊆ H An extensive class H that complies to these conditions is called minimal, hence an extensive hypothesis set consists of only minimal classes. This defines a surjective 6 The set OS i results from removing all Alice’s peers from those observations in OS i that do not contain any of Alice’s peers of H. A Combinatorial Approach for an Anonymity Metric 35 mapping of the extensive hypothesis set Li to the minimal hitting sets with respect to OS i .

Mean Number of Minimal Hitting Sets. In this section, we derive closed formulas for the mean number of hypotheses after t observations for distinct classes. Let Vi be a random variable, where Vi = 1 is the event that a particular hypothesis H of the class Hi remains valid after t observations, while Vi = 0 denotes the inverse event. The probability of Vi = 1 corresponds to t stochastically independent Bernoulli trials, where the outcome of each of the t trials shows that the minimal hitting set remains valid.

I For sets A, B and integer i we define Ai = j=0 Aj , H0 = {HA } 0 0 k m−1 where A is a neutral element, such that A ×B = B k . H1 ⊆ (R \ HA )1 × HA For example the class H2 might contain the (minimal m−2 2 hitting) sets H2 = {r21 , r22 , a23 . . , a2m } and H2 = H2 ⊆ (R \ HA ) × HA .. {r21 , a23 . . , a2m }, where each rij represents a non. peer and each aik represents an Alice’s peer. All peers (1) Hm ⊆ (R \ HA )m . within a set must be disjoint. Bounds of Minimal Hitting Set Classes. The last section derives the bound of bm for the number of minimal hitting sets.

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