SICP Exercise 2.70 rock song lyrics

Exercise 2.70.  The following eight-symbol alphabet with associated relative frequencies was designed to efficiently encode the lyrics of 1950s rock songs. (Note that the ``symbols'' of an ``alphabet'' need not be individual letters.)

A2NA16
BOOM1SHA3
GET2YIP9
JOB2WAH1

Use generate-huffman-tree (exercise 2.69) to generate a corresponding Huffman tree, and use encode (exercise 2.68) to encode the following message:

Get a job
Sha na na na na na na na na
Get a job
Sha na na na na na na na na
Wah yip yip yip yip yip yip yip yip yip
Sha boom

How many bits are required for the encoding? What is the smallest number of bits that would be needed to encode this song if we used a fixed-length code for the eight-symbol alphabet?

 SOLUTION

 The code and tests are here.

 Encoding tree is: 
((leaf NA 16) ((leaf YIP 9) (((leaf A 2) ((leaf WAH 1) (leaf BOOM 1) (WAH BOOM) 2) (A WAH BOOM) 4) ((leaf SHA 3) ((leaf JOB 2) (leaf GET 2) (JOB GET) 4) (SHA JOB GET) 7) (A WAH BOOM SHA JOB GET) 11) (YIP A WAH BOOM SHA JOB GET) 20) (NA YIP A WAH BOOM SHA JOB GET) 36)

 Test message is: (GET A JOB SHA NA NA NA NA NA NA NA NA GET A JOB SHA NA NA NA NA NA NA NA NA WAH YIP YIP YIP YIP YIP YIP YIP YIP YIP SHA BOOM)

 Encoded message is: (1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1)

 Decoded bits is: (GET A JOB SHA NA NA NA NA NA NA NA NA GET A JOB SHA NA NA NA NA NA NA NA NA WAH YIP YIP YIP YIP YIP YIP YIP YIP YIP SHA BOOM)

 Number of bits in the encoded message is: 84

 Number of symbols in the test message is: 36

 Minimum Number of bits per symbol in fixed-length representation: 3 (Since there are eight symbols and 2 raised to 3 = 8)

 The smallest number of bits that would be needed to encode this song if we used a fixed-length code for the eight-symbol alphabet: 108

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