Could you give me some explain of LZ77 Algorithm?

Question

I'm trying to learn LZ77 algorithm with my friend, and some case give us a confusion.

for example)

init

• search buffer size: 7
• look-ahead buffer size: 8
• original string: abcabbcabbcabca

• current window: abcabbc view: abbcabca

What I thought the LLD tuple is : Literal: 'a' Length: 4 Distance: 4

What my friend thought the LLD tuple is : Literal: 'c' Length: 6 Distance: 4

I thought the algorithm to find longest match string only checks in search buffer, but he says there is no limit to find match string.

Who's answer is correct?

Answer 1

I can't give you a definitive answer, but I can show there's no reason LZ77 couldn't support "overly long" tuples.

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I believe you're asking if the length component of a tuple can be larger than its distance component.

In other words, I believe you're asking which of the following streams of tuples would be produced:

1. ( Length: 0, Distance: 0, Char: 'a' )
2. (0,0,'b')
3. (0,0,'c')
4. (2,3,'b')
5. (4,4,'c')^[1] Length capped to distance.
6. (2,4,'c')
7. (0,0,'a')

or

1. (0,0,'a')
2. (0,0,'b')
3. (0,0,'c')
4. (2,3,'b')
5. (7,4,'c')^[2] Length larger than distance.
6. (0,0,'a')

Obviously, the latter would produce better compression.

I can't give you a definitive answer, but I can show that the decoder can handle "overly long" tuples just as easily as the encoder can produce them. As such, there's no reason LZ77 couldn't support them.

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Reconstructing the input from the first stream

1. (0,0,'a'): "" + "a" ⇒ "a"
2. (0,0,'b'): "" + "b" ⇒ "ab"
3. (0,0,'c'): "" + "c" ⇒ "abc"
4. (2,3,'b'): "ab" + "b" ⇒ "abcabb"
5. (4,4,'c'): "cabb" + "c" ⇒ "abcabbcabbc"
6. (2,4,'c'): "ab" + "c" ⇒ "abcabbcabbcabc"
7. (0,0,'a'): "" + "a" ⇒ "abcabbcabbcabca"

Simple.

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Reconstructing the input from the second stream

1. (0,0,'a'): "" + "a" ⇒ "a"

2. (0,0,'b'): "" + "b" ⇒ "ab"

3. (0,0,'c'): "" + "c" ⇒ "abc"

4. (2,3,'b'): "ab" + "b" ⇒ "abcabb"

5. (7,4,'c')

   So far, we've produced abcabb. The tuple says the next 7 characters start 4 characters from the end of abcabb.

   a b c c a b b _ _ _ _ _ _ _ c
         | | | | ^ ^ ^ ^ ^ ^ ^
         | | | | | | | | | | |
         +-)-)-)-+ | | | ? ? ?
           +-)-)---+ | |
             +-)-----+ |
               +-------+
   

   We're missing three characters! Are are we? If it's all one buffer and we copy from left to right, we just have to keep going.

                     +-------+
                   +-)-----+ |
                 +-)-)---+ | |
                 | | |   | | |
                 | | |   v v v
   a b c c a b b _ _ _ _ _ _ _ c
         | | | | ^ ^ ^ ^
         | | | | | | | |
         +-)-)-)-+ | | |
           +-)-)---+ | |
             +-)-----+ |
               +-------+
   

   So this works with no extra effort!

   (7,4,'c'): "cabbcab" + "c" ⇒ "abcabbcabbcabc"

6. (0,0,'a'): "" + "a" ⇒ "abcabbcabbcabca"

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1. You incorrectly said (4,4,'a').
2. You incorrectly said (6,4,'c').

Answer 2

Your friend is right. A simpler example is that LZ77 accomplishes run-length coding with a distance 1 match, with length n-1 for a run of n copies of the same byte. (The first occurrence of the byte is a literal.)