Towards the Efficient Generation of Gray Codes in the Bitprobe Model

Abstract

We examine the problem of representing integers modulo L so that both increment and decrement operations can be performed efficiently. This problem is studied in the bitprobe model, where the complexity of the underlying problem is measured by the number of bit operations performed on the data structure. In this thesis, we will primarily be interested in constructing space-optimal data structures. That is, we would like to use exactly n bits to represent integers modulo 2^n. Brodal et al. gave such a data structure, which requires n-1 bit reads and 3 bit writes, in the worst case, to perform increment and decrement operations We provide several improvements to their data structure. First, we give a data structure that requires n-1 bit reads and 2 bit writes, in the worst case, to perform increment and decrement operations. Then, we refine this result to obtain a data structure that requires n-1 bit reads and a single bit write to perform both operations. This disproves the conjecture that, when a space-optimal data structure uses only 1 bit write to perform these operations, then every bit in the data structure must be inspected in the worst case.