Compressing Integer Sequences and Sets

Problem Definition

Suppose that a message ${ M=\langle s_1, s_2, \dots, s_n\rangle }$ of length ${ n=|M| }$ symbols is to be represented, where each symbol s _i is an integer in the range ${ 1 \leq s_i \leq U }$, for some upper limit U that may or may not be known, and may or may not be finite. Messages in this form are commonly the output of some kind of modeling step in a data compression system. The objective is to represent the message over a binary output alphabet ${ \{ \texttt{0},\texttt{1}\} }$ using as few as possible output bits. A special case of the problem arises when the elements of the message are strictly increasing, ${ s_i < s_{i+1} }$. In this case the message M can be thought of as identifying a subset of ${ \{1, 2, \dots, U\} }$. Examples include storing sets of IP addresses or product codes, and recording the destinations of hyperlinks in the graph representation of the world wide web....