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Bloom Filters are a desirable data structure for distinguishing new values in sequences of data (i.e., messages), due to their space efficiency, their low false positive rates (incorrectly classifying a new value as a repeat), and never producing false negatives (classifying a repeat value as new). However, as the Bloom Filter's bits are filled, false positive rates creep upward. To keep false positive rates below a reasonable threshold, applications periodically "recycle" the Bloom Filter, clearing the memory and then resuming the tracking of data. After a recycle point, subsequent arrivals of recycled messages are likely to be misclassified as new; recycling induces false negatives. Despite numerous applications of recycling, the corresponding false negative rates have never been analyzed. In this paper, we derive approximations, upper bounds, and lower bounds of false negative rates for several variants of recycling Bloom Filters. These approximations and bounds are functions of the size of memory used to store the Bloom Filter and the distributions on new arrivals and repeat messages, and can be efficiently computed on conventional hardware. We show, via comparison to simulation, that our upper bounds and approximations are extremely tight, and can be efficiently computed for megabyte-sized Bloom Filters on conventional hardware. 

Bloom Filters are a space-efficient data structure used for the testing of membership in a set that errs only in the False Positive direction. However, the standard analysis that measures this False Positive rate provides a form of worst case bound that is both overly conservative for the majority of network applications that utilize Bloom Filters, and reduces accuracy by not taking into account the actual state (number of bits set) of the Bloom Filter after each arrival. In this paper, we more accurately characterize the False Positive dynamics of Bloom Filters as they are commonly used in networking applications. In particular, network applications often utilize a Bloom Filter that “recycles”: it repeatedly fills, and upon reaching a certain level of saturation, empties and fills again. In this context, it makes more sense to evaluate performance using the average False Positive rate instead of the worst case bound. We show how to efficiently compute the average False Positive rate of recycling Bloom Filter variants via renewal and Markov models. We apply our models to both the standard Bloom Filter and a “two-phase” variant, verify the accuracy of our model with simulations, and find that the previous analysis’ worst-case formulation leads to up to a 30% reduction in the efficiency of Bloom Filter when applied in network applications, while two-phase overhead diminishes as the needed False Positive rate is tightened.