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. (Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput., 23(4):738–761, 1994). The space usage is similar to that of binary search trees, i.e., three words per key on average. 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. Key Words: data structures, dictionaries, information retrieval, searching, hashing, experiments * Partially supported by the Future and Emerging Technologies programme of the EU

Bloom filtering is an important technique for space efficient storage of a conservative approximation of a set S. The set stored may have up to some specified number of false positive members, but all elements of S are included. In this paper we consider lossy dictionaries that are also allowed to have false negatives, i.e., leave out elements of S. The aim is to maximize the weight of included keys within a given space constraint. This relaxation allows a very fast and simple data structure making almost optimal use of memory. Being more time efficient than Bloom filters, we believe our data structure to be well suited for replacing Bloom filters in some applications. Also, the fact that our data structure supports information associated to keys paves the way for new uses, as illustrated by an application in lossy image compression.

Suppose each of k ≤ no(1) players holds an n-bit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a public-coin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (public-coin) Ω(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are one-way, like ours. We give a protocol to determine if ∑ xi> s with error 1 % and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg 2 n) bound. A similar improvement holds for computing degree-(k − 1) polynomial-threshold functions in the number-on-forehead model. We give a (public-coin, 2-player, tight) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).