A Constant-Time Colored Choice Dictionary with Almost Robust Iteration

Abstract

A (colored) choice dictionary is a data structure that is initialized with positive integers n and c and subsequently maintains a sequence of n elements of {0,...,c-1}, called colors, under operations to inspect and to update the color in a given position and to return the position of an occurrence of a given color. Choice dictionaries are fundamental in space-efficient computing. Some applications call for the additional operation of dynamic iteration, i.e., enumeration of the positions containing a given color while the sequence of colors may change. An iteration is robust if it enumerates every position that contains the relevant color throughout the iteration but never enumerates a position more than once or when it does not contain the color in question. We describe the first choice dictionary that executes every operation in constant amortized time and almost robust iteration in constant amortized time per element enumerated. The iteration is robust, except that it may enumerate some elements a second time. The data structure occupies n log_2 c+O((log n)^2) bits. The time and space bounds assume that c=O((log n)^{1/2}(log log n)^{-{3/2}}).