Further Developments
Prof. Kurt Eisemann of San Diego State University in his letter to the Communications of the ACM further improves the latter seemingly unbeatable estimate. He begins with an observation that for practical applications the value of the "hidden constant" c in the Big Oh notation may be crucial in making the difference between the feasibility and unfeasibility: for example a constant value of 1080 would exceed the capacity of any known device. He further notices that a technique has already been known in Mediaeval Europe whereby textual content of an arbitrary tune can be recorded basing on the recurrence relation, where, yielding the value of the big-Oh constant c equal to 2. However it turns out that another culture achieved the absolute lower bound of O(0). As Prof. Eisemann puts it:
"When the Mayflower voyagers first descended on these shores, the native Americans proud of their achievement in the theory of information storage and retrieval, at first welcomed the strangers with the complete silence. This was meant to convey their peak achievement in the complexity of songs, namely the demonstration that a limit as low as c = 0 is indeed obtainable."
However the Europeans were unprepared to grasp this notion, and the Indian chiefs, in order to establish a common ground to convey their achievements later proceeded to demonstrate an approach described by the recurrent relation, where, with a suboptimal complexity given by c = 1.
The 0(1) space complexity result was also implemented by Guy L. Steele, Jr., perhaps challenged by Knuth's article. Dr. Steele's TELNET Song used a completely different algorithm based on exponential recursion, a parody on some implementations of TELNET.
It has been suggested that the complexity analysis of human songs can be a useful pedagogic device for teaching students complexity theory.
The article On Superpolylogarithmic Subexponential Functions by prof. Alan Sherman writes that Knuth's article was seminal for analysis of a special class of functions.
Read more about this topic: The Complexity Of Songs
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