ABSTRACT: A standard technique from the hashing literature is to use two hash functions h1(x) and h2(x) to simulate additional hash functions of the form gi(x) = h1(x) + ih2(x). We demonstrate that this technique can be usefully applied to Bloom filters and related data structures. Specifically, only two hash functions are necessary to effectively implement a Bloom filter without any loss in the asymptotic false positive probability. This leads to less computation and potentially less need for

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Abstract. Behrmann et al. posed the question whether “To Store or Not To Store ” [1] states during reachability analysis, in order to counter the effects of the well-known state space explosion problem in explicitstate model checking. Their answer was to store not all but only some strategical states. They pay in run-time if the answer too often is “Not To Store”. We propose a different strategy to adaptively trade time for space: “To Store ” as many states as memory limits permit. If free memory becomes scarce, we gradually swap states out to secondary storage. We are careful to minimize revisits, and I/O overhead, and also stay sound, i.e. on termination it is guaranteed that the full state space has been explored. It is also available for counterexample reconstruction. In our experiments we tackled state spaces of industrial-scale models with more than 10 9 explicit states with still modest storage requirements. 1

In a multiple-choice hashing scheme, each item is stored in one of d> = 2 possible hash tablebuckets. The availability of these multiple choices allows for a substantial reduction in the maximum load of the buckets. However, a lookup may now require examining each of the d locations. Forapplications where this cost is undesirable, Song et al. propose keeping a summary that allows one to determine which of the d locations is appropriate for each item, where the summary may allowfalse positives for items not in hash table. We propose alternative, simple constructions of such summaries that use less space for both the summary and the underlying hash table. Moreover, ourconstructions are easily analyzable and tunable.

A technique from the hashing literature is to use two hash functions h1(x) and h2(x) to simulate additional hash functions of the form gi(x) = h1(x) + ih2(x). We demonstrate that this technique can be usefully applied to Bloom filters and related data structures. Specifically, only two hash functions are necessary to effectively implement a Bloom filter without any loss in the asymptotic false positive probability. This leads to less computation and potentially less need for randomness in practice. 1

Abstract. In order to apply formal methods in practice, the practitioner has to comprehend a vast amount of research literature and realistically evaluate practical merits of different approaches. In this paper we focus on explicit finite state model checking and study this area from practitioner’s point of view. We provide a systematic overview of techniques for fighting state space explosion and we analyse trends in the research. We also report on our own experience with practical performance of techniques. Our main conclusion and recommendation for practitioner is the following: be critical to claims of dramatic improvement brought by a single sophisticated technique, rather use many different simple techniques and combine them. 1

In a multiple-choice hashing scheme, each item is stored in one of P possible hash table buckets. The availability of these multiple choices allows for a substantial reduction in the maximum load of the buckets. However, a lookup may now require examining each of the locations. For applications where this cost is undesirable, Song et al. propose keeping a summary that allows one to determine which of the locations is appropriate for each item, where the summary may allow false positives for items not in hash table. We propose alternative, simple constructions of such summaries that use less space for both the summary and the underlying hash table. Moreover, our constructions are easily analyzable and tunable.

Abstract — In recent work, the authors introduced a data structure with the same functionality as a counting Bloom filter (CBF) based on fingerprints and the d-left hashing technique. This paper describes dynamic bit reassignment, an approach that allows the size of the fingerprint to flexibly change with the load in each hash bucket, thereby reducing the probability of a false positive. This technique allows us to not only improve our d-left counting Bloom filter, but also to construct a data structure with the same functionality as a Bloom filter, including the ability to handle insertions online, that yields fewer false positives for sufficiently large filters. Our results show that our d-left Bloom filter data structure begins achieving smaller false positive rates than the standard construction at 16 bits per element. We explain the technique, describe why it is amenable to hardware implementation, and provide experimental results. I.

Abstract. A Bloom filter is a very compact data structure that supports approximate membership queries on a set, allowing false positives. We propose several new variants of Bloom filters and replacements with similar functionality. All of them have a better cache-efficiency and need less hash bits than regular Bloom filters. Some use SIMD functionality, while the others provide an even better space efficiency. As a consequence, we get a more flexible trade-off between false positive rate, spaceefficiency, cache-efficiency, hash-efficiency, and computational effort. We analyze the efficiency of Bloom filters and the proposed replacements in detail, in terms of the false positive rate, the number of expected cachemisses, and the number of required hash bits. We also describe and experimentally evaluate the performance of highly-tuned implementations. For many settings, our alternatives perform better than the methods proposed so far. 1

Abstract. This paper presents the ComBack method for explicit state space exploration. The ComBack method extends the well-known hash compaction method such that full coverage of the state space is guaranteed. Each encountered state is mapped into a compressed state descriptor (hash value) as in hash compaction. The method additionally stores for each state an integer representing the identity of the state and a backedge to a predecessor state. This allows hash collisions to be resolved on-the-fly during state space exploration using backtracking to reconstruct the full state descriptors when required for comparison with newly encountered states. A prototype implementation of the ComBack method is used to evaluate the method on several example systems and compare its performance to related methods. The results show a reduction in memory usage at an acceptable cost in exploration time. 1