Compressing Integer Sequences

Years and Authors of Summarized Original Work

1966; Golomb

1974; Elias

2000; Moffat, Stuiver

Problem Definition

An n-symbol message $M =\langle s_{0},s_{1},\ldots ,s_{n-1}\rangle$ is given, where each symbol s _i is an integer in the range 0 ≤ s _i  < U. If the s _i s are strictly increasing, then M identifies an n-subset of {0, 1, …, U − 1}.

Objective To economically encode M as a binary string over {0, 1}.

Constraints

• 1. Short messages. The message length n may be small relative to U.

• 2. Monotonic equivalence. Message M is converted to a strictly increasing message M ^′ over the alphabet U ^′ ≤ Un by taking prefix sums, $s_{i}^{{\prime}} = i +\sum _{ j=0}^{i}s_{j}$ and $U^{{\prime}} = s_{n-1}^{{\prime}} + 1$. The inverse is to “take gaps,” $g_{i} = s_{i} - s_{i-1} - 1$, with g ₀ = s ₀.

• 3. Combinatorial Limit. If M is monotonic then $\left \lceil \log _{2}\left (\begin{array}{@{}c@{}} U\\ n\end{array} \right )\right \rceil \leq U$ bits are required in the worst case. When n ≪ U, \...