xref: /external/lzma/DOC/lzma-specification.txt

LZMA specification (DRAFT version)
----------------------------------

Author: Igor Pavlov
Date: 2013-07-28

This specification defines the format of LZMA compressed data and lzma file format.

Notation 
--------

We use the syntax of C++ programming language.
We use the following types in C++ code:
  unsigned - unsigned integer, at least 16 bits in size
  int      - signed integer, at least 16 bits in size
  UInt64   - 64-bit unsigned integer
  UInt32   - 32-bit unsigned integer
  UInt16   - 16-bit unsigned integer
  Byte     - 8-bit unsigned integer
  bool     - boolean type with two possible values: false, true


lzma file format
================

The lzma file contains the raw LZMA stream and the header with related properties.

The files in that format use ".lzma" extension.

The lzma file format layout:

Offset Size Description

  0     1   LZMA model properties (lc, lp, pb) in encoded form
  1     4   Dictionary size (32-bit unsigned integer, little-endian)
  5     8   Uncompressed size (64-bit unsigned integer, little-endian)
 13         Compressed data (LZMA stream)

LZMA properties:

    name  Range          Description

      lc  [0, 8]         the number of "literal context" bits
      lp  [0, 4]         the number of "literal pos" bits
      pb  [0, 4]         the number of "pos" bits
dictSize  [0, 2^32 - 1]  the dictionary size 

The following code encodes LZMA properties:

void EncodeProperties(Byte *properties)
{
  properties[0] = (Byte)((pb * 5 + lp) * 9 + lc);
  Set_UInt32_LittleEndian(properties + 1, dictSize);
}

If the value of dictionary size in properties is smaller than (1 << 12),
the LZMA decoder must set the dictionary size variable to (1 << 12).

#define LZMA_DIC_MIN (1 << 12)

  unsigned lc, pb, lp;
  UInt32 dictSize;
  UInt32 dictSizeInProperties;

  void DecodeProperties(const Byte *properties)
  {
    unsigned d = properties[0];
    if (d >= (9 * 5 * 5))
      throw "Incorrect LZMA properties";
    lc = d % 9;
    d /= 9;
    pb = d / 5;
    lp = d % 5;
    dictSizeInProperties = 0;
    for (int i = 0; i < 4; i++)
      dictSizeInProperties |= (UInt32)properties[i + 1] << (8 * i);
    dictSize = dictSizeInProperties;
    if (dictSize < LZMA_DIC_MIN)
      dictSize = LZMA_DIC_MIN;
  }

If "Uncompressed size" field contains ones in all 64 bits, it means that
uncompressed size is unknown and there is the "end marker" in stream,
that indicates the end of decoding point.
In opposite case, if the value from "Uncompressed size" field is not
equal to ((2^64) - 1), the LZMA stream decoding must be finished after
specified number of bytes (Uncompressed size) is decoded. And if there 
is the "end marker", the LZMA decoder must read that marker also.


The new scheme to encode LZMA properties
----------------------------------------

If LZMA compression is used for some another format, it's recommended to
use a new improved scheme to encode LZMA properties. That new scheme was
used in xz format that uses the LZMA2 compression algorithm.
The LZMA2 is a new compression algorithm that is based on the LZMA algorithm.

The dictionary size in LZMA2 is encoded with just one byte and LZMA2 supports
only reduced set of dictionary sizes:
  (2 << 11), (3 << 11),
  (2 << 12), (3 << 12),
  ...
  (2 << 30), (3 << 30),
  (2 << 31) - 1

The dictionary size can be extracted from encoded value with the following code:

  dictSize = (p == 40) ? 0xFFFFFFFF : (((UInt32)2 | ((p) & 1)) << ((p) / 2 + 11));

Also there is additional limitation (lc + lp <= 4) in LZMA2 for values of 
"lc" and "lp" properties:

  if (lc + lp > 4)
    throw "Unsupported properties: (lc + lp) > 4";

There are some advantages for LZMA decoder with such (lc + lp) value
limitation. It reduces the maximum size of tables allocated by decoder.
And it reduces the complexity of initialization procedure, that can be 
important to keep high speed of decoding of big number of small LZMA streams.

It's recommended to use that limitation (lc + lp <= 4) for any new format
that uses LZMA compression. Note that the combinations of "lc" and "lp" 
parameters, where (lc + lp > 4), can provide significant improvement in 
compression ratio only in some rare cases.

The LZMA properties can be encoded into two bytes in new scheme:

Offset Size Description

  0     1   The dictionary size encoded with LZMA2 scheme
  1     1   LZMA model properties (lc, lp, pb) in encoded form


The RAM usage 
=============

The RAM usage for LZMA decoder is determined by the following parts:

1) The Sliding Window (from 4 KiB to 4 GiB).
2) The probability model counter arrays (arrays of 16-bit variables).
3) Some additional state variables (about 10 variables of 32-bit integers).


The RAM usage for Sliding Window
--------------------------------

There are two main scenarios of decoding:

1) The decoding of full stream to one RAM buffer.

  If we decode full LZMA stream to one output buffer in RAM, the decoder 
  can use that output buffer as sliding window. So the decoder doesn't 
  need additional buffer allocated for sliding window.

2) The decoding to some external storage.

  If we decode LZMA stream to external storage, the decoder must allocate
  the buffer for sliding window. The size of that buffer must be equal 
  or larger than the value of dictionary size from properties of LZMA stream.

In this specification we describe the code for decoding to some external
storage. The optimized version of code for decoding of full stream to one
output RAM buffer can require some minor changes in code.


The RAM usage for the probability model counters
------------------------------------------------

The size of the probability model counter arrays is calculated with the 
following formula:

size_of_prob_arrays = 1846 + 768 * (1 << (lp + lc))

Each probability model counter is 11-bit unsigned integer.
If we use 16-bit integer variables (2-byte integers) for these probability 
model counters, the RAM usage required by probability model counter arrays 
can be estimated with the following formula:

  RAM = 4 KiB + 1.5 KiB * (1 << (lp + lc))

For example, for default LZMA parameters (lp = 0 and lc = 3), the RAM usage is

  RAM_lc3_lp0 = 4 KiB + 1.5 KiB * 8 = 16 KiB

The maximum RAM state usage is required for decoding the stream with lp = 4 
and lc = 8:

  RAM_lc8_lp4 = 4 KiB + 1.5 KiB * 4096 = 6148 KiB

If the decoder uses LZMA2's limited property condition 
(lc + lp <= 4), the RAM usage will be not larger than

  RAM_lc_lp_4 = 4 KiB + 1.5 KiB * 16 = 28 KiB


The RAM usage for encoder
-------------------------

There are many variants for LZMA encoding code.
These variants have different values for memory consumption.
Note that memory consumption for LZMA Encoder can not be 
smaller than memory consumption of LZMA Decoder for same stream.

The RAM usage required by modern effective implementation of 
LZMA Encoder can be estimated with the following formula:

  Encoder_RAM_Usage = 4 MiB + 11 * dictionarySize.

But there are some modes of the encoder that require less memory.


LZMA Decoding
=============

The LZMA compression algorithm uses LZ-based compression with Sliding Window
and Range Encoding as entropy coding method.


Sliding Window
--------------

LZMA uses Sliding Window compression similar to LZ77 algorithm.

LZMA stream must be decoded to the sequence that consists
of MATCHES and LITERALS:
  
  - a LITERAL is a 8-bit character (one byte).
    The decoder just puts that LITERAL to the uncompressed stream.
  
  - a MATCH is a pair of two numbers (DISTANCE-LENGTH pair).
    The decoder takes one byte exactly "DISTANCE" characters behind
    current position in the uncompressed stream and puts it to 
    uncompressed stream. The decoder must repeat it "LENGTH" times.

The "DISTANCE" can not be larger than dictionary size.
And the "DISTANCE" can not be larger than the number of bytes in
the uncompressed stream that were decoded before that match.

In this specification we use cyclic buffer to implement Sliding Window
for LZMA decoder:

class COutWindow
{
  Byte *Buf;
  UInt32 Pos;
  UInt32 Size;
  bool IsFull;

public:
  unsigned TotalPos;
  COutStream OutStream;

  COutWindow(): Buf(NULL) {}
  ~COutWindow() { delete []Buf; }
 
  void Create(UInt32 dictSize)
  {
    Buf = new Byte[dictSize];
    Pos = 0;
    Size = dictSize;
    IsFull = false;
    TotalPos = 0;
  }

  void PutByte(Byte b)
  {
    TotalPos++;
    Buf[Pos++] = b;
    if (Pos == Size)
    {
      Pos = 0;
      IsFull = true;
    }
    OutStream.WriteByte(b);
  }

  Byte GetByte(UInt32 dist) const
  {
    return Buf[dist <= Pos ? Pos - dist : Size - dist + Pos];
  }

  void CopyMatch(UInt32 dist, unsigned len)
  {
    for (; len > 0; len--)
      PutByte(GetByte(dist));
  }

  bool CheckDistance(UInt32 dist) const
  {
    return dist <= Pos || IsFull;
  }

  bool IsEmpty() const
  {
    return Pos == 0 && !IsFull;
  }
};


In another implementation it's possible to use one buffer that contains 
Sliding Window and the whole data stream after uncompressing.


Range Decoder
-------------

LZMA algorithm uses Range Encoding (1) as entropy coding method.

LZMA stream contains just one very big number in big-endian encoding.
LZMA decoder uses the Range Decoder to extract a sequence of binary
symbols from that big number.

The state of the Range Decoder:

struct CRangeDecoder
{
  UInt32 Range; 
  UInt32 Code;
  InputStream *InStream;

  bool Corrupted;
}

The notes about UInt32 type for the "Range" and "Code" variables:

  It's possible to use 64-bit (unsigned or signed) integer type
  for the "Range" and the "Code" variables instead of 32-bit unsigned,
  but some additional code must be used to truncate the values to 
  low 32-bits after some operations.

  If the programming language does not support 32-bit unsigned integer type 
  (like in case of JAVA language), it's possible to use 32-bit signed integer, 
  but some code must be changed. For example, it's required to change the code
  that uses comparison operations for UInt32 variables in this specification.

The Range Decoder can be in some states that can be treated as 
"Corruption" in LZMA stream. The Range Decoder uses the variable "Corrupted":

  (Corrupted == false), if the Range Decoder has not detected any corruption.
  (Corrupted == true), if the Range Decoder has detected some corruption.

The reference LZMA Decoder ignores the value of the "Corrupted" variable.
So it continues to decode the stream, even if the corruption can be detected
in the Range Decoder. To provide the full compatibility with output of the 
reference LZMA Decoder, another LZMA Decoder implementations must also 
ignore the value of the "Corrupted" variable.

The LZMA Encoder is required to create only such LZMA streams, that will not 
lead the Range Decoder to states, where the "Corrupted" variable is set to true.

The Range Decoder reads first 5 bytes from input stream to initialize
the state:

void CRangeDecoder::Init()
{
  Corrupted = false;
  
  if (InStream->ReadByte() != 0)
    Corrupted = true;
  
  Range = 0xFFFFFFFF;
  Code = 0;
  for (int i = 0; i < 4; i++)
    Code = (Code << 8) | InStream->ReadByte();
  
  if (Code == Range)
    Corrupted = true;
}

The LZMA Encoder always writes ZERO in initial byte of compressed stream.
That scheme allows to simplify the code of the Range Encoder in the 
LZMA Encoder.

After the last bit of data was decoded by Range Decoder, the value of the
"Code" variable must be equal to 0. The LZMA Decoder must check it by 
calling the IsFinishedOK() function:

  bool IsFinishedOK() const { return Code == 0; }

If there is corruption in data stream, there is big probability that
the "Code" value will be not equal to 0 in the Finish() function. So that
check in the IsFinishedOK() function provides very good feature for 
corruption detection.

The value of the "Range" variable before each bit decoding can not be smaller 
than ((UInt32)1 << 24). The Normalize() function keeps the "Range" value in 
described range.

#define kTopValue ((UInt32)1 << 24)

void CRangeDecoder::Normalize()
{
  if (Range < kTopValue)
  {
    Range <<= 8;
    Code = (Code << 8) | InStream->ReadByte();
  }
}

Notes: if the size of the "Code" variable is larger than 32 bits, it's
required to keep only low 32 bits of the "Code" variable after the change
in Normalize() function.

If the LZMA Stream is not corrupted, the value of the "Code" variable is
always smaller than value of the "Range" variable.
But the Range Decoder ignores some types of corruptions, so the value of
the "Code" variable can be equal or larger than value of the "Range" variable
for some "Corrupted" archives.


LZMA uses Range Encoding only with binary symbols of two types:
  1) binary symbols with fixed and equal probabilities (direct bits)
  2) binary symbols with predicted probabilities

The DecodeDirectBits() function decodes the sequence of direct bits:

UInt32 CRangeDecoder::DecodeDirectBits(unsigned numBits)
{
  UInt32 res = 0;
  do
  {
    Range >>= 1;
    Code -= Range;
    UInt32 t = 0 - ((UInt32)Code >> 31);
    Code += Range & t;
    
    if (Code == Range)
      Corrupted = true;
    
    Normalize();
    res <<= 1;
    res += t + 1;
  }
  while (--numBits);
  return res;
}


The Bit Decoding with Probability Model
---------------------------------------

The task of Bit Probability Model is to estimate probabilities of binary
symbols. And then it provides the Range Decoder with that information.
The better prediction provides better compression ratio.
The Bit Probability Model uses statistical data of previous decoded
symbols.

That estimated probability is presented as 11-bit unsigned integer value
that represents the probability of symbol "0".

#define kNumBitModelTotalBits 11

Mathematical probabilities can be presented with the following formulas:
     probability(symbol_0) = prob / 2048.
     probability(symbol_1) =  1 - Probability(symbol_0) =  
                           =  1 - prob / 2048 =  
                           =  (2048 - prob) / 2048
where the "prob" variable contains 11-bit integer probability counter.

It's recommended to use 16-bit unsigned integer type, to store these 11-bit
probability values:

typedef UInt16 CProb;

Each probability value must be initialized with value ((1 << 11) / 2),
that represents the state, where probabilities of symbols 0 and 1 
are equal to 0.5:

#define PROB_INIT_VAL ((1 << kNumBitModelTotalBits) / 2)

The INIT_PROBS macro is used to initialize the array of CProb variables:

#define INIT_PROBS(p) \
 { for (unsigned i = 0; i < sizeof(p) / sizeof(p[0]); i++) p[i] = PROB_INIT_VAL; }


The DecodeBit() function decodes one bit.
The LZMA decoder provides the pointer to CProb variable that contains 
information about estimated probability for symbol 0 and the Range Decoder 
updates that CProb variable after decoding. The Range Decoder increases 
estimated probability of the symbol that was decoded:

#define kNumMoveBits 5

unsigned CRangeDecoder::DecodeBit(CProb *prob)
{
  unsigned v = *prob;
  UInt32 bound = (Range >> kNumBitModelTotalBits) * v;
  unsigned symbol;
  if (Code < bound)
  {
    v += ((1 << kNumBitModelTotalBits) - v) >> kNumMoveBits;
    Range = bound;
    symbol = 0;
  }
  else
  {
    v -= v >> kNumMoveBits;
    Code -= bound;
    Range -= bound;
    symbol = 1;
  }
  *prob = (CProb)v;
  Normalize();
  return symbol;
}


The Binary Tree of bit model counters
-------------------------------------

LZMA uses a tree of Bit model variables to decode symbol that needs
several bits for storing. There are two versions of such trees in LZMA:
  1) the tree that decodes bits from high bit to low bit (the normal scheme).
  2) the tree that decodes bits from low bit to high bit (the reverse scheme).

Each binary tree structure supports different size of decoded symbol
(the size of binary sequence that contains value of symbol).
If that size of decoded symbol is "NumBits" bits, the tree structure 
uses the array of (2 << NumBits) counters of CProb type. 
But only ((2 << NumBits) - 1) items are used by encoder and decoder.
The first item (the item with index equal to 0) in array is unused.
That scheme with unused array's item allows to simplify the code.

unsigned BitTreeReverseDecode(CProb *probs, unsigned numBits, CRangeDecoder *rc)
{
  unsigned m = 1;
  unsigned symbol = 0;
  for (unsigned i = 0; i < numBits; i++)
  {
    unsigned bit = rc->DecodeBit(&probs[m]);
    m <<= 1;
    m += bit;
    symbol |= (bit << i);
  }
  return symbol;
}

template <unsigned NumBits>
class CBitTreeDecoder
{
  CProb Probs[(unsigned)1 << NumBits];

public:

  void Init()
  {
    INIT_PROBS(Probs);
  }

  unsigned Decode(CRangeDecoder *rc)
  {
    unsigned m = 1;
    for (unsigned i = 0; i < NumBits; i++)
      m = (m << 1) + rc->DecodeBit(&Probs[m]);
    return m - ((unsigned)1 << NumBits);
  }

  unsigned ReverseDecode(CRangeDecoder *rc)
  {
    return BitTreeReverseDecode(Probs, NumBits, rc);
  }
};


LZ part of LZMA 
---------------

LZ part of LZMA describes details about the decoding of MATCHES and LITERALS.


The Literal Decoding
--------------------

The LZMA Decoder uses (1 << (lc + lp)) tables with CProb values, where 
each table contains 0x300 CProb values:

  CProb *LitProbs;

  void CreateLiterals()
  {
    LitProbs = new CProb[(UInt32)0x300 << (lc + lp)];
  }
  
  void InitLiterals()
  {
    UInt32 num = (UInt32)0x300 << (lc + lp);
    for (UInt32 i = 0; i < num; i++)
      LitProbs[i] = PROB_INIT_VAL;
  }

To select the table for decoding it uses the context that consists of
(lc) high bits from previous literal and (lp) low bits from value that
represents current position in outputStream.

If (State > 7), the Literal Decoder also uses "matchByte" that represents 
the byte in OutputStream at position the is the DISTANCE bytes before 
current position, where the DISTANCE is the distance in DISTANCE-LENGTH pair
of latest decoded match.

The following code decodes one literal and puts it to Sliding Window buffer:

  void DecodeLiteral(unsigned state, UInt32 rep0)
  {
    unsigned prevByte = 0;
    if (!OutWindow.IsEmpty())
      prevByte = OutWindow.GetByte(1);
    
    unsigned symbol = 1;
    unsigned litState = ((OutWindow.TotalPos & ((1 << lp) - 1)) << lc) + (prevByte >> (8 - lc));
    CProb *probs = &LitProbs[(UInt32)0x300 * litState];
    
    if (state >= 7)
    {
      unsigned matchByte = OutWindow.GetByte(rep0 + 1);
      do
      {
        unsigned matchBit = (matchByte >> 7) & 1;
        matchByte <<= 1;
        unsigned bit = RangeDec.DecodeBit(&probs[((1 + matchBit) << 8) + symbol]);
        symbol = (symbol << 1) | bit;
        if (matchBit != bit)
          break;
      }
      while (symbol < 0x100);
    }
    while (symbol < 0x100)
      symbol = (symbol << 1) | RangeDec.DecodeBit(&probs[symbol]);
    OutWindow.PutByte((Byte)(symbol - 0x100));
  }


The match length decoding
-------------------------

The match length decoder returns normalized (zero-based value) 
length of match. That value can be converted to real length of the match 
with the following code:

#define kMatchMinLen 2

    matchLen = len + kMatchMinLen;

The match length decoder can return the values from 0 to 271.
And the corresponded real match length values can be in the range 
from 2 to 273.

The following scheme is used for the match length encoding:

  Binary encoding    Binary Tree structure    Zero-based match length 
  sequence                                    (binary + decimal):

  0 xxx              LowCoder[posState]       xxx
  1 0 yyy            MidCoder[posState]       yyy + 8
  1 1 zzzzzzzz       HighCoder                zzzzzzzz + 16

LZMA uses bit model variable "Choice" to decode the first selection bit.

If the first selection bit is equal to 0, the decoder uses binary tree 
  LowCoder[posState] to decode 3-bit zero-based match length (xxx).

If the first selection bit is equal to 1, the decoder uses bit model 
  variable "Choice2" to decode the second selection bit.

  If the second selection bit is equal to 0, the decoder uses binary tree 
    MidCoder[posState] to decode 3-bit "yyy" value, and zero-based match
    length is equal to (yyy + 8).

  If the second selection bit is equal to 1, the decoder uses binary tree 
    HighCoder to decode 8-bit "zzzzzzzz" value, and zero-based 
    match length is equal to (zzzzzzzz + 16).

LZMA uses "posState" value as context to select the binary tree 
from LowCoder and MidCoder binary tree arrays:

    unsigned posState = OutWindow.TotalPos & ((1 << pb) - 1);

The full code of the length decoder:

class CLenDecoder
{
  CProb Choice;
  CProb Choice2;
  CBitTreeDecoder<3> LowCoder[1 << kNumPosBitsMax];
  CBitTreeDecoder<3> MidCoder[1 << kNumPosBitsMax];
  CBitTreeDecoder<8> HighCoder;

public:

  void Init()
  {
    Choice = PROB_INIT_VAL;
    Choice2 = PROB_INIT_VAL;
    HighCoder.Init();
    for (unsigned i = 0; i < (1 << kNumPosBitsMax); i++)
    {
      LowCoder[i].Init();
      MidCoder[i].Init();
    }
  }

  unsigned Decode(CRangeDecoder *rc, unsigned posState)
  {
    if (rc->DecodeBit(&Choice) == 0)
      return LowCoder[posState].Decode(rc);
    if (rc->DecodeBit(&Choice2) == 0)
      return 8 + MidCoder[posState].Decode(rc);
    return 16 + HighCoder.Decode(rc);
  }
};

The LZMA decoder uses two instances of CLenDecoder class.
The first instance is for the matches of "Simple Match" type,
and the second instance is for the matches of "Rep Match" type:

  CLenDecoder LenDecoder;
  CLenDecoder RepLenDecoder;


The match distance decoding
---------------------------

LZMA supports dictionary sizes up to 4 GiB minus 1.
The value of match distance (decoded by distance decoder) can be 
from 1 to 2^32. But the distance value that is equal to 2^32 is used to
indicate the "End of stream" marker. So real largest match distance 
that is used for LZ-window match is (2^32 - 1).

LZMA uses normalized match length (zero-based length) 
to calculate the context state "lenState" do decode the distance value:

#define kNumLenToPosStates 4

    unsigned lenState = len;
    if (lenState > kNumLenToPosStates - 1)
      lenState = kNumLenToPosStates - 1;

The distance decoder returns the "dist" value that is zero-based value 
of match distance. The real match distance can be calculated with the
following code:
  
  matchDistance = dist + 1; 

The state of the distance decoder and the initialization code: 

  #define kEndPosModelIndex 14
  #define kNumFullDistances (1 << (kEndPosModelIndex >> 1))
  #define kNumAlignBits 4

  CBitTreeDecoder<6> PosSlotDecoder[kNumLenToPosStates];
  CProb PosDecoders[1 + kNumFullDistances - kEndPosModelIndex];
  CBitTreeDecoder<kNumAlignBits> AlignDecoder;

  void InitDist()
  {
    for (unsigned i = 0; i < kNumLenToPosStates; i++)
      PosSlotDecoder[i].Init();
    AlignDecoder.Init();
    INIT_PROBS(PosDecoders);
  }

At first stage the distance decoder decodes 6-bit "posSlot" value with bit
tree decoder from PosSlotDecoder array. It's possible to get 2^6=64 different 
"posSlot" values.

    unsigned posSlot = PosSlotDecoder[lenState].Decode(&RangeDec);

The encoding scheme for distance value is shown in the following table:

posSlot (decimal) /
      zero-based distance (binary)
 0    0
 1    1
 2    10
 3    11

 4    10 x
 5    11 x
 6    10 xx
 7    11 xx
 8    10 xxx
 9    11 xxx
10    10 xxxx
11    11 xxxx
12    10 xxxxx
13    11 xxxxx

14    10 yy zzzz
15    11 yy zzzz
16    10 yyy zzzz
17    11 yyy zzzz
...
62    10 yyyyyyyyyyyyyyyyyyyyyyyyyy zzzz
63    11 yyyyyyyyyyyyyyyyyyyyyyyyyy zzzz

where 
  "x ... x" means the sequence of binary symbols encoded with binary tree and 
      "Reverse" scheme. It uses separated binary tree for each posSlot from 4 to 13.
  "y" means direct bit encoded with range coder.
  "zzzz" means the sequence of four binary symbols encoded with binary
      tree with "Reverse" scheme, where one common binary tree "AlignDecoder"
      is used for all posSlot values.

If (posSlot < 4), the "dist" value is equal to posSlot value.

If (posSlot >= 4), the decoder uses "posSlot" value to calculate the value of
  the high bits of "dist" value and the number of the low bits.

  If (4 <= posSlot < kEndPosModelIndex), the decoder uses bit tree decoders.
    (one separated bit tree decoder per one posSlot value) and "Reverse" scheme.
    In this implementation we use one CProb array "PosDecoders" that contains 
    all CProb variables for all these bit decoders.
  
  if (posSlot >= kEndPosModelIndex), the middle bits are decoded as direct 
    bits from RangeDecoder and the low 4 bits are decoded with a bit tree 
    decoder "AlignDecoder" with "Reverse" scheme.

The code to decode zero-based match distance:
  
  unsigned DecodeDistance(unsigned len)
  {
    unsigned lenState = len;
    if (lenState > kNumLenToPosStates - 1)
      lenState = kNumLenToPosStates - 1;
    
    unsigned posSlot = PosSlotDecoder[lenState].Decode(&RangeDec);
    if (posSlot < 4)
      return posSlot;
    
    unsigned numDirectBits = (unsigned)((posSlot >> 1) - 1);
    UInt32 dist = ((2 | (posSlot & 1)) << numDirectBits);
    if (posSlot < kEndPosModelIndex)
      dist += BitTreeReverseDecode(PosDecoders + dist - posSlot, numDirectBits, &RangeDec);
    else
    {
      dist += RangeDec.DecodeDirectBits(numDirectBits - kNumAlignBits) << kNumAlignBits;
      dist += AlignDecoder.ReverseDecode(&RangeDec);
    }
    return dist;
  }



LZMA Decoding modes
-------------------

There are 2 types of LZMA streams:

1) The stream with "End of stream" marker.
2) The stream without "End of stream" marker.

And the LZMA Decoder supports 3 modes of decoding:

1) The unpack size is undefined. The LZMA decoder stops decoding after 
   getting "End of stream" marker. 
   The input variables for that case:
    
      markerIsMandatory = true
      unpackSizeDefined = false
      unpackSize contains any value

2) The unpack size is defined and LZMA decoder supports both variants, 
   where the stream can contain "End of stream" marker or the stream is
   finished without "End of stream" marker. The LZMA decoder must detect 
   any of these situations.
   The input variables for that case:
    
      markerIsMandatory = false
      unpackSizeDefined = true
      unpackSize contains unpack size

3) The unpack size is defined and the LZMA stream must contain 
   "End of stream" marker
   The input variables for that case:
    
      markerIsMandatory = true
      unpackSizeDefined = true
      unpackSize contains unpack size


The main loop of decoder
------------------------

The main loop of LZMA decoder:

Initialize the LZMA state.
loop
{
  // begin of loop
  Check "end of stream" conditions.
  Decode Type of MATCH / LITERAL. 
    If it's LITERAL, decode LITERAL value and put the LITERAL to Window.
    If it's MATCH, decode the length of match and the match distance. 
        Check error conditions, check end of stream conditions and copy
        the sequence of match bytes from sliding window to current position
        in window.
  Go to begin of loop
}

The reference implementation of LZMA decoder uses "unpackSize" variable
to keep the number of remaining bytes in output stream. So it reduces 
"unpackSize" value after each decoded LITERAL or MATCH.

The following code contains the "end of stream" condition check at the start
of the loop:

    if (unpackSizeDefined && unpackSize == 0 && !markerIsMandatory)
      if (RangeDec.IsFinishedOK())
        return LZMA_RES_FINISHED_WITHOUT_MARKER;

LZMA uses three types of matches:

1) "Simple Match" -     the match with distance value encoded with bit models.

2) "Rep Match" -        the match that uses the distance from distance
                        history table.

3) "Short Rep Match" -  the match of single byte length, that uses the latest 
                        distance from distance history table.

The LZMA decoder keeps the history of latest 4 match distances that were used 
by decoder. That set of 4 variables contains zero-based match distances and 
these variables are initialized with zero values:

  UInt32 rep0 = 0, rep1 = 0, rep2 = 0, rep3 = 0;

The LZMA decoder uses binary model variables to select type of MATCH or LITERAL:

#define kNumStates 12
#define kNumPosBitsMax 4

  CProb IsMatch[kNumStates << kNumPosBitsMax];
  CProb IsRep[kNumStates];
  CProb IsRepG0[kNumStates];
  CProb IsRepG1[kNumStates];
  CProb IsRepG2[kNumStates];
  CProb IsRep0Long[kNumStates << kNumPosBitsMax];

The decoder uses "state" variable value to select exact variable 
from "IsRep", "IsRepG0", "IsRepG1" and "IsRepG2" arrays.
The "state" variable can get the value from 0 to 11.
Initial value for "state" variable is zero:

  unsigned state = 0;

The "state" variable is updated after each LITERAL or MATCH with one of the
following functions:

unsigned UpdateState_Literal(unsigned state)
{
  if (state < 4) return 0;
  else if (state < 10) return state - 3;
  else return state - 6;
}
unsigned UpdateState_Match   (unsigned state) { return state < 7 ? 7 : 10; }
unsigned UpdateState_Rep     (unsigned state) { return state < 7 ? 8 : 11; }
unsigned UpdateState_ShortRep(unsigned state) { return state < 7 ? 9 : 11; }

The decoder calculates "state2" variable value to select exact variable from 
"IsMatch" and "IsRep0Long" arrays:

unsigned posState = OutWindow.TotalPos & ((1 << pb) - 1);
unsigned state2 = (state << kNumPosBitsMax) + posState;

The decoder uses the following code flow scheme to select exact 
type of LITERAL or MATCH:

IsMatch[state2] decode
  0 - the Literal
  1 - the Match
    IsRep[state] decode
      0 - Simple Match
      1 - Rep Match
        IsRepG0[state] decode
          0 - the distance is rep0
            IsRep0Long[state2] decode
              0 - Short Rep Match
              1 - Rep Match 0
          1 - 
            IsRepG1[state] decode
              0 - Rep Match 1
              1 - 
                IsRepG2[state] decode
                  0 - Rep Match 2
                  1 - Rep Match 3


LITERAL symbol
--------------
If the value "0" was decoded with IsMatch[state2] decoding, we have "LITERAL" type.

At first the LZMA decoder must check that it doesn't exceed 
specified uncompressed size:

      if (unpackSizeDefined && unpackSize == 0)
        return LZMA_RES_ERROR;

Then it decodes literal value and puts it to sliding window:

      DecodeLiteral(state, rep0);

Then the decoder must update the "state" value and "unpackSize" value;

      state = UpdateState_Literal(state);
      unpackSize--;

Then the decoder must go to the begin of main loop to decode next Match or Literal.


Simple Match
------------

If the value "1" was decoded with IsMatch[state2] decoding,
we have the "Simple Match" type.

The distance history table is updated with the following scheme:
    
      rep3 = rep2;
      rep2 = rep1;
      rep1 = rep0;

The zero-based length is decoded with "LenDecoder":

      len = LenDecoder.Decode(&RangeDec, posState);

The state is update with UpdateState_Match function:

      state = UpdateState_Match(state);

and the new "rep0" value is decoded with DecodeDistance:

      rep0 = DecodeDistance(len);

That "rep0" will be used as zero-based distance for current match.

If the value of "rep0" is equal to 0xFFFFFFFF, it means that we have 
"End of stream" marker, so we can stop decoding and check finishing 
condition in Range Decoder:

      if (rep0 == 0xFFFFFFFF)
        return RangeDec.IsFinishedOK() ?
            LZMA_RES_FINISHED_WITH_MARKER :
            LZMA_RES_ERROR;

If uncompressed size is defined, LZMA decoder must check that it doesn't 
exceed that specified uncompressed size:

      if (unpackSizeDefined && unpackSize == 0)
        return LZMA_RES_ERROR;

Also the decoder must check that "rep0" value is not larger than dictionary size
and is not larger than the number of already decoded bytes:

      if (rep0 >= dictSize || !OutWindow.CheckDistance(rep0))
        return LZMA_RES_ERROR;

Then the decoder must copy match bytes as described in 
"The match symbols copying" section.


Rep Match
---------

If the LZMA decoder has decoded the value "1" with IsRep[state] variable,
we have "Rep Match" type.

At first the LZMA decoder must check that it doesn't exceed 
specified uncompressed size:

      if (unpackSizeDefined && unpackSize == 0)
        return LZMA_RES_ERROR;

Also the decoder must return error, if the LZ window is empty:

      if (OutWindow.IsEmpty())
        return LZMA_RES_ERROR;

If the match type is "Rep Match", the decoder uses one of the 4 variables of
distance history table to get the value of distance for current match.
And there are 4 corresponding ways of decoding flow. 

The decoder updates the distance history with the following scheme 
depending from type of match:

- "Rep Match 0" or "Short Rep Match":
      ; LZMA doesn't update the distance history    

- "Rep Match 1":
      UInt32 dist = rep1;
      rep1 = rep0;
      rep0 = dist;

- "Rep Match 2":
      UInt32 dist = rep2;
      rep2 = rep1;
      rep1 = rep0;
      rep0 = dist;

- "Rep Match 3":
      UInt32 dist = rep3;
      rep3 = rep2;
      rep2 = rep1;
      rep1 = rep0;
      rep0 = dist;

Then the decoder decodes exact subtype of "Rep Match" using "IsRepG0", "IsRep0Long",
"IsRepG1", "IsRepG2".

If the subtype is "Short Rep Match", the decoder updates the state, puts 
the one byte from window to current position in window and goes to next 
MATCH/LITERAL symbol (the begin of main loop):

          state = UpdateState_ShortRep(state);
          OutWindow.PutByte(OutWindow.GetByte(rep0 + 1));
          unpackSize--;
          continue;

In other cases (Rep Match 0/1/2/3), it decodes the zero-based 
length of match with "RepLenDecoder" decoder:

      len = RepLenDecoder.Decode(&RangeDec, posState);

Then it updates the state:

      state = UpdateState_Rep(state);

Then the decoder must copy match bytes as described in 
"The Match symbols copying" section.


The match symbols copying
-------------------------

If we have the match (Simple Match or Rep Match 0/1/2/3), the decoder must
copy the sequence of bytes with calculated match distance and match length.
If uncompressed size is defined, LZMA decoder must check that it doesn't 
exceed that specified uncompressed size:

    len += kMatchMinLen;
    bool isError = false;
    if (unpackSizeDefined && unpackSize < len)
    {
      len = (unsigned)unpackSize;
      isError = true;
    }
    OutWindow.CopyMatch(rep0 + 1, len);
    unpackSize -= len;
    if (isError)
      return LZMA_RES_ERROR;

Then the decoder must go to the begin of main loop to decode next MATCH or LITERAL.



NOTES
-----

This specification doesn't describe the variant of decoder implementation 
that supports partial decoding. Such partial decoding case can require some 
changes in "end of stream" condition checks code. Also such code 
can use additional status codes, returned by decoder.

This specification uses C++ code with templates to simplify describing.
The optimized version of LZMA decoder doesn't need templates.
Such optimized version can use just two arrays of CProb variables:
  1) The dynamic array of CProb variables allocated for the Literal Decoder.
  2) The one common array that contains all other CProb variables.


References:      

1. G. N. N. Martin, Range encoding: an algorithm for removing redundancy 
   from a digitized message, Video & Data Recording Conference, 
   Southampton, UK, July 24-27, 1979.