In this paper we introduce the notion of approximate data structures, in which a small amount of error is tolerated in the output. Approximate data structures trade error of approximation for faster operation, leading to theoretical and practical speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure, except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is 1/polylog(n), and in O(log log n) time provided the error is 1/polynomial(n), for n elements in the data structure. We consider the tolerance of prototypical algorithms to approximate data structures. We study in particular Prim's minimum spanning tree algorithm, Dijkstra's single-source shortest paths algorithm, and an on-line variant of Graham's convex hull algorithm. To obtain output which approximates the desired output with the error of approximation tending to zero, Prim's algorithm requires only linear time, Dijkstra's algorithm requires O(m log log n) time, and the on-line variant of Graham's algorithm requires constant amortized time per operation.

A generalization of the Seidel-Entringer-Arnold method for calculating the alternating permutation numbers (or secant-tangent numbers) leads to a new operation on sequences, the boustrophedon transform. © 1996 Academic Press, Inc. * Present address: Math. and Computer Science Department, Dartmouth College, Hanover, New Hampshire.