Article Summary
Knuth writes, with a grain of truth, that "our ancient ancestors invented the concept of refrain" to reduce the space complexity of songs, which becomes crucial when a large number of songs is to be committed to one's memory. Knuth's Lemma 1 states that if N is the length of a song, then the refrain decreases the song complexity to cN, where the factor c < 1.
Knuth further demonstrates a way of producing songs with O complexity, an approach "further improved by a Scottish farmer named O. MacDonald".
More ingenious approaches yield songs of complexity O, a class known as "m bottles of beer on the wall".
Finally, the progress during the 20th century—stimulated by the fact that "the advent of modern drugs has led to demands for still less memory"—leads to the ultimate improvement: arbitrarily long songs with space complexity O(1), e.g. for a song to be defined by the recurrence relation
- 'That's the way,' 'I like it,', for all
- 'uh huh,' 'uh huh'
Read more about this topic: The Complexity Of Songs
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