Deterministic oblivious distribution (and tight compaction) in linear time

Abstract

In an array of N elements, M positions and M elements are "marked". We show how to permute the elements in the array so that all marked elements end in marked positions, in time O(N) (in the standard word-RAM model), deterministically, and obliviously - i.e. with a sequence of memory accesses that depends only on N and not on which elements or positions are marked. As a corollary, we answer affirmatively to an open question about the existence of a deterministic oblivious algorithm with O(N) running time for tight compaction (move the M marked elements to the first M positions of the array), a building block for several cryptographic constructions. Our O(N) result improves the running-time upper bounds for deterministic tight compaction, for randomized tight compaction, and for the simpler problem of randomized loose compaction (move the M marked elements to the first O(M) positions) - until now respectively O(N lg N), O(N lg lg N), and O(N lg*N).