Universal compression of memoryless sources over unknown alphabets

Alon Orlitsky, Narayana P. Santhanam, Junan Zhang

Research output: Contribution to journalArticlepeer-review

107 Scopus citations


It has long been known that the compression redundancy of independent and identically distributed (i.i.d.) strings increases to infinity as the alphabet size grows. It is also apparent that any string can be described by separately conveying its symbols, and its pattern - the order in which the symbols appear. Concentrating on the latter, we show that the patterns of i.i.d. strings over all, including infinite and even unknown, alphabets, can be compressed with diminishing redundancy, both in block and sequentially, and that the compression can be performed in linear time. To establish these results, we show that the number of patterns is the Bell number, that the number of patterns with a given number of symbols is the Stirling number of the second kind, and that the redundancy of patterns can be bounded using results of Hardy and Ramanujan on the number of integer partitions. The results also imply an asymptotically optimal solution for the Good-Turing probability-estimation problem.

Original languageEnglish (US)
Pages (from-to)1469-1481
Number of pages13
JournalIEEE Transactions on Information Theory
Issue number7
StatePublished - Jul 2004
Externally publishedYes


  • Large and unknown alphabets
  • Patterns
  • Set and integer partitions
  • Universal compression

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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