# Run-Length Encoding in a Finite Universe

Text compression schemes and compact data structures usually combine sophisticated probability models with basic coding methods whose average codeword length closely match the entropy of known distributions. In the frequent case where basic coding represents run-lengths of outcomes that have probability p, i.e. the geometric distribution (i)=p^i(1-p), a Golomb code is an optimal instantaneous code, which has the additional advantage that codewords can be computed using only an integer parameter calculated from p, without need for a large or sophisticated data structure. Golomb coding does not, however, gracefully handle the case where run-lengths are bounded by a known integer n. In this case, codewords allocated for the case i>n are wasted. While negligible for large n, this makes Golomb coding unattractive in situations where n is recurrently small, e.g., when representing many short lists of integers drawn from limited ranges, or when the range of n is narrowed down by a recursive algorithm. We address the problem of choosing a code for this case, considering efficiency from both information-theoretic and computational perspectives, and arrive at a simple code that allows computing a codeword using only O(1) simple computer operations and O(1) machine words. We demonstrate experimentally that the resulting representation length is very close (equal in a majority of tested cases) to the optimal Huffman code, to the extent that the expected difference is practically negligible. We describe efficient branch-free implementation of encoding and decoding.

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