It is shown that a static dictionary that offers constant-time access to n elements with w-bit keys and occupies O(n) words of memory can be constructed deterministically in O(n log n) time on a unit-cost RAM with word length w and a standard instruction set including multiplication. Whereas a randomized construction working in linear expected time was known, the running time of the best previous deterministic algorithm was Ω(n²). Using a standard dynamization technique, the first deterministic dynamic dictionary with constant lookup time and sublinear update time is derived. The new algorithms are weakly nonuniform; i.e., they require access to a fixed number of precomputed constants dependent on w. The main technical tools employed are unit-cost error-correcting codes, word parallelism, and derandomization using conditional expectations.

We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully-dynamic linear space data structures. This leads to an optimal bound of O ( � log n / log log n) for searching and updating a dynamic set X of n integer keys in linear space. Searching X for an integer y means finding the maximum key in X which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set. The best previous deterministic linear space bound was O(log n / log log n) due to Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space. We also get the following worst-case linear space trade-offs between the number n, the word length W, and the maximal key U < 2W: O(min{log log n + log log U log n / log W, log log n · log log log U}). These trade-offs are, however, not likely to be optimal. Our results are generalized to finger searching and string searching, providing optimal results for both in terms of n.

We obtain subquadratic algorithms for 3SUM on integers and rationals in several models. On a standard word RAM with w-bit words, we obtain a running time of O(n 2 / max { w lg 2 w, lg 2 n (lg lg n) 2}). In the circuit RAM with one nonstandard AC0 operation, we obtain O(n2 / w2 lg2). In external w memory, we achieve O(n2 /(MB)), even under the standard assumption of data indivisibility. Cache-obliviously, we obtain a running time of O(n2 / MB lg2). In all cases, our speedup is almost M quadratic in the parallelism the model can afford, which may be the best possible. Our algorithms are Las Vegas randomized; time bounds hold in expectation, and in most cases, with high probability. 1

We describe new techniques for proving lower bounds on data-structure problems, with the following broad consequences: • the first Ω(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989; • for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show Ω(lg n/ lg lg n) bounds when the space is O(n · polylog n). Using these techniques, we analyze a variety of central data-structure problems, and obtain improved lower bounds for the following: • the partial-sums problem (a fundamental application of augmented binary search trees); • the predecessor problem (which is equivalent to IP lookup in Internet routers); • dynamic trees and dynamic connectivity; • orthogonal range stabbing; • orthogonal range counting, and orthogonal range reporting; • the partial match problem (searching with wild-cards); • (1 + ε)-approximate near neighbor on the hypercube; • approximate nearest neighbor in the l∞ metric. Our new techniques lead to surprisingly non-technical proofs. For several problems, we obtain simpler proofs for bounds that were already known.

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