# Entropy

Source : ACM ICPC Greater New York 2000 |
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Time limit : 1 sec |
Memory limit : 32 M |

**Submitted** : 442, **Accepted** : 199

**Background**

An entropy encoder is a data encoding method that achieves lossless data compression
by encoding a message with “wasted” or “extra” information removed. In other
words, entropy encoding removes information that was not necessary in the first
place to accurately encode the message. A high degree of entropy implies a message
with a great deal of wasted information; english text encoded in ASCII is an
example of a message type that has very high entropy. Already compressed messages,
such as JPEG graphics or ZIP archives, have very little entropy and do not benefit
from further attempts at entropy encoding.

English text encoded in ASCII has a high degree of entropy because all characters
are encoded using the same number of bits, eight. It is a known fact that the
letters E, L, N, R, S and T occur at a considerably higher frequency than do
most other letters in english text. If a way could be found to encode just these
letters with four bits, then the new encoding would be smaller, would contain
all the original information, and would have less entropy. ASCII uses a fixed
number of bits for a reason, however: it’s easy, since one is always dealing
with a fixed number of bits to represent each possible glyph or character. How
would an encoding scheme that used four bits for the above letters be able to
distinguish between the four-bit codes and eight-bit codes? This seemingly difficult
problem is solved using what is known as a “prefix-free variable-length” encoding.

In such an encoding, any number of bits can be used to represent any glyph,
and glyphs not present in the message are simply not encoded. However, in order
to be able to recover the information, no bit pattern that encodes a glyph is
allowed to be the prefix of any other encoding bit pattern. This allows the
encoded bitstream to be read bit by bit, and whenever a set of bits is encountered
that represents a glyph, that glyph can be decoded. If the prefix-free constraint
was not enforced, then such a decoding would be impossible.

Consider the text “AAAAABCD”. Using ASCII, encoding this would require 64 bits.
If, instead, we encode “A” with the bit pattern “00”, “B” with “01”, “C” with
“10”, and “D” with “11” then we can encode this text in only 16 bits; the resulting
bit pattern would be “0000000000011011”. This is still a fixed-length encoding,
however; we’re using two bits per glyph instead of eight. Since the glyph “A”
occurs with greater frequency, could we do better by encoding it with fewer
bits? In fact we can, but in order to maintain a prefix-free encoding, some
of the other bit patterns will become longer than two bits. An optimal encoding
is to encode “A” with “0”, “B” with “10”, “C” with “110”, and “D” with “111”.
(This is clearly not the only optimal encoding, as it is obvious that the encodings
for B, C and D could be interchanged freely for any given encoding without increasing
the size of the final encoded message.) Using this encoding, the message encodes
in only 13 bits to “0000010110111”, a compression ratio of 4.9 to 1 (that is,
each bit in the final encoded message represents as much information as did
4.9 bits in the original encoding). Read through this bit pattern from left
to right and you’ll see that the prefix-free encoding makes it simple to decode
this into the original text even though the codes have varying bit lengths.

As a second example, consider the text “THE CAT IN THE HAT”. In this text, the
letter “T” and the space character both occur with the highest frequency, so
they will clearly have the shortest encoding bit patterns in an optimal encoding.
The letters “C”, “I’ and “N” only occur once, however, so they will have the
longest codes.

There are many possible sets of prefix-free variable-length bit patterns that
would yield the optimal encoding, that is, that would allow the text to be encoded
in the fewest number of bits. One such optimal encoding is to encode spaces
with “00”, “A” with “100”, “C” with “1110”, “E” with “1111”, “H” with “110”,
“I” with “1010”, “N” with “1011” and “T” with “01”. The optimal encoding therefore
requires only 51 bits compared to the 144 that would be necessary to encode
the message with 8-bit ASCII encoding, a compression ratio of 2.8 to 1.

**Input**

The input file will contain a list of text strings, one per line. The text strings
will consist only of uppercase alphanumeric characters and underscores (which
are used in place of spaces). The end of the input will be signalled by a line
containing only the word “END” as the text string. This line should not be processed.

**Output
**

For each text string in the input, output the length in bits of the 8-bit ASCII encoding, the length in bits of an optimal prefix-free variable-length encoding, and the compression ratio accurate to one decimal point.

**Example**

**Input**

AAAAABCD THE_CAT_IN_THE_HAT END

**Output**

64 13 4.9 144 51 2.8