Integer Representation and Counting in the Bit Probe Model

We examine the problem of integer representation in a nearly minimal number of bits so that the increment and the decrement (and indeed the addition and the subtraction) operations can be performed using few bit inspections and fewer bit changes. The model of computation we considered is the bit probe model, where the complexity measure counts only the bitwise accesses to the data structure. We present several efficient data structures to represent integer that use a logarithmic number of bit inspections and a constant number of bit changes per operation. The most space-efficient data structure uses only one extra bit. We also present an extension to our data structure to support efficient addition and subtraction, where the larger value is replaced by the result, while retaining the same asymptotic bounds for the increment and the decrement operations.