Linear probing

Abstract: We present a simple dictionary with worst case constant lookup time, equaling the theoretical performance of the classic dynamic perfect hashing scheme of Dietzfelbinger et al. [SIAM J. Comput. 23 (4) (1994) 738-761]. The space usage is similar to that of binary search trees. Besides being conceptually much simpler than previous dynamic dictionaries with worst case constant lookup time, our data structure is interesting in that it does not use perfect hashing, but rather a variant of open addressing where keys can be moved back in their probe sequences. An implementation inspired by our algorithm, but using weaker hash functions, is found to be quite practical. It is competitive with the best known dictionaries having an average case (but no nontrivial worst case) guarantee on lookup time.

Abstract: Hashing is critical for high performance computer architecture. Hashing is used extensively in hardware applications, such as page tables, for address translation. Bit extraction and exclusive ORing hashing "methods" are two commonly used hashing functions for hardware applications. There is no study of the performance of these functions and no mention anywhere of the practical performance of the hashing functions in comparison with the theoretical performance prediction of hashing schemes. In this paper, we show that, by choosing hashing functions at random from a particular class, called H/sub 3/, of hashing functions, the analytical performance of hashing can be achieved in practice on real-life data. Our results about the expected worst case performance of hashing are of special significance, as they provide evidence for earlier theoretical predictions.

Abstract: We demonstrate an efficient data-parallel algorithm for building large hash tables of millions of elements in real-time. We consider two parallel algorithms for the construction: a classical sparse perfect hashing approach, and cuckoo hashing, which packs elements densely by allowing an element to be stored in one of multiple possible locations. Our construction is a hybrid approach that uses both algorithms. We measure the construction time, access time, and memory usage of our implementations and demonstrate real-time performance on large datasets: for 5 million key-value pairs, we construct a hash table in 35.7 ms using 1.42 times as much memory as the input data itself, and we can access all the elements in that hash table in 15.3 ms. For comparison, sorting the same data requires 36.6 ms, but accessing all the elements via binary search requires 79.5 ms. Furthermore, we show how our hashing methods can be applied to two graphics applications: 3D surface intersection for moving data and geometric hashing for image matching.

Abstract: This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n 3/2 ) , the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices $ -\frac{1}{3},\frac{2}{3} $ .) For sparse tables, the construction cost has expectation O(n) , standard deviation O ( $ \sqrt{n} $ ), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.