Abstract
Compression of files is an important practical question. Memory is always a limiting resource so if our files can be stored in a more economic fashion, this has to be done. Some files, like pictures, contain a lot of redundancy and can be compressed significantly even without loss of quality of pictures. Firstly, we assume that the frequency of each symbol is known and introduce optimal Huffman’s code which encodes information using variable-length strings to represent symbols depending on how frequently they appear. Secondly, we give a glimpse of the combinatorial approach to the problem of compressing data whose source is unknown. These codes are called universal. We describe Fitingof’s compression codes that have an elegant polynomial-time decoding procedure using the Pascal triangle.
Good things, when short, are twice as good.
Baltasar Gracián y Morales (1601–1658)
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Notes
- 1.
The triangle is called after French mathematician Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui almost 500 years earlier, and the Persian astronomer Omar Khayyám, who is better known for his poetry.
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Slinko, A. (2020). Compression. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-44074-9_8
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DOI: https://doi.org/10.1007/978-3-030-44074-9_8
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