A linearly computable measure of string complexity

Abstract

We present a measure of string complexity, called I-complexity, computable in linear time and space. It counts the number of different substrings in a given string. The least complex strings are the runs of a single symbol, the most complex are the de Bruijn strings. Although the I-complexity of a string is not the length of any minimal description of the string, it satisfies many basic properties of classical description complexity. In particular, the number of strings with I-complexity up to a given value is bounded, and most strings of each length have high I-complexity.