Lempel–Ziv–Markov chain algorithm

LZMA data is at the lowest level decoded one bit at a time by the range decoder, at the direction of the LZMA decoder.

Context-based range decoding is invoked by the LZMA algorithm passing it a reference to the "context", which consists of the unsigned 11-bit variable prob (typically implemented using a 16-bit data type) representing the predicted probability of the bit being 0, which is read and updated by the range decoder (and should be initialized to ${\displaystyle 2^{10}}$, representing 0.5 probability).

Fixed probability range decoding instead assumes a 0.5 probability, but operates slightly differently from context-based range decoding.

The range decoder state consists of two unsigned 32-bit variables, range (representing the range size), and code (representing the encoded point within the range).

Initialization of the range decoder consists of setting range to 2³² − 1, and code to the 32-bit value starting at the second byte in the stream interpreted as big-endian; the first byte in the stream is completely ignored.

Normalization proceeds in this way:

1. Shift both range and code left by 8 bits
2. Read a byte from the compressed stream
3. Set the least significant 8 bits of code to the byte value read

Context-based range decoding of a bit using the prob probability variable proceeds in this way:

1. If range is less than ${\displaystyle 2^{24}}$, perform normalization
2. Set bound to ${\displaystyle \lfloor range/2^{11}\rfloor \times prob}$
3. If code is less than bound:
   1. Set range to bound
   2. Set prob to prob + ${\displaystyle \lfloor 2^{11}-prob\rfloor /2^{5}}$
   3. Return bit 0
4. Otherwise (if code is greater than or equal to the bound):
   1. Set range to range − bound
   2. Set code to code − bound
   3. Set prob to ${\displaystyle prob-\lfloor prob/2^{5}\rfloor }$
   4. Return bit 1

Fixed-probability range decoding of a bit proceeds in this way:

1. If range is less than ${\displaystyle 2^{24}}$, perform normalization
2. Set range to ${\displaystyle \lfloor range/2\rfloor }$
3. If code is less than range:
   1. Return bit 0
4. Otherwise (if code is greater or equal than range):
   1. Set code to code − range
   2. Return bit 1

The Linux kernel implementation of fixed-probability decoding in rc_direct(), for performance reasons, does not include a conditional branch, but instead subtracts range from code unconditionally. The resulting sign bit is used to both decide the bit to return and to generate a mask that is combined with code and added to range.

Note that:

1. The division by ${\displaystyle 2^{11}}$ when computing bound and floor operation is done before the multiplication, not after (apparently to avoid requiring fast hardware support for 32-bit multiplication with a 64-bit result)
2. Fixed probability decoding is not strictly equivalent to context-based range decoding with any prob value, due to the fact that context-based range decoding discards the lower 11 bits of range before multiplying by prob as just described, while fixed probability decoding only discards the last bit

The range decoder also provides the bit-tree, reverse bit-tree and fixed probability integer decoding facilities, which are used to decode integers, and generalize the single-bit decoding described above. To decode unsigned integers less than limit, an array of (limit − 1) 11-bit probability variables is provided, which are conceptually arranged as the internal nodes of a complete binary tree with limit leaves.

Non-reverse bit-tree decoding works by keeping a pointer to the tree of variables, which starts at the root. As long as the pointer does not point to a leaf, a bit is decoded using the variable indicated by the pointer, and the pointer is moved to either the left or right children depending on whether the bit is 0 or 1; when the pointer points to a leaf, the number associated with the leaf is returned.

Non-reverse bit-tree decoding thus happens from most significant to least significant bit, stopping when only one value in the valid range is possible (this conceptually allows to have range sizes that are not powers of two, even though LZMA does not make use of this).

Reverse bit-tree decoding instead decodes from least significant bit to most significant bits, and thus only supports ranges that are powers of two, and always decodes the same number of bits. It is equivalent to performing non-reverse bittree decoding with a power of two limit, and reversing the last log₂(limit) bits of the result.

In the rc_bittree function in the Linux kernel, integers are actually returned in the [limit, 2 × limit) range (with limit added to the conceptual value), and the variable at index 0 in the array is unused, while the one at index 1 is the root, and the left and right children indices are computed as 2i and 2i + 1. The rc_bittree_reverse function instead adds integers in the [0, limit) range to a caller-provided variable, where limit is implicitly represented by its logarithm, and has its own independent implementation for efficiency reasons.

Fixed probability integer decoding simply performs fixed probability bit decoding repeatedly, reading bits from the most to the least significant.

The LZMA decoder is configured by an lclppb "properties" byte and a dictionary size. The value of the lclppb byte is lc + lp × 9 + pb × 9 × 5, where:

• lc is the number of high bits of the previous byte to use as a context for literal encoding (the default value used by the LZMA SDK is 3)
• lp is the number of low bits of the dictionary position to include in literal_pos_state (the default value used by the LZMA SDK is 0)
• pb is the number of low bits of the dictionary position to include in pos_state (the default value used by the LZMA SDK is 2)

In non-LZMA2 streams, lc must not be greater than 8, and lp and pb must not be greater than 4. In LZMA2 streams, (lc + lp) and pb must not be greater than 4.

In the 7-zip LZMA file format, configuration is performed by a header containing the "properties" byte followed by the 32-bit little-endian dictionary size in bytes. In LZMA2, the properties byte can optionally be changed at the start of LZMA2 LZMA packets, while the dictionary size is specified in the LZMA2 header as later described.

The LZMA packet format has already been described, and this section specifies how LZMA statistically models the LZ-encoded streams, or in other words which probability variables are passed to the range decoder to decode each bit.

Those probability variables are implemented as multi-dimensional arrays; before introducing them, a few values that are used as indices in these multidimensional arrays are defined.

The state value is conceptually based on which of the patterns in the following table match the latest 2–4 packet types seen, and is implemented as a state machine state updated according to the transition table listed in the table every time a packet is output.

The initial state is 0, and thus packets before the beginning are assumed to be LIT packets.

The pos_state and literal_pos_state values consist of respectively the pb and lp (up to 4, from the LZMA header or LZMA2 properties packet) least significant bits of the dictionary position (the number of bytes coded since the last dictionary reset modulo the dictionary size). Note that the dictionary size is normally the multiple of a large power of 2, so these values are equivalently described as the least significant bits of the number of uncompressed bytes seen since the last dictionary reset.

The prev_byte_lc_msbs value is set to the lc (up to 4, from the LZMA header or LZMA2 properties packet) most significant bits of the previous uncompressed byte.

The is_REP value denotes whether a packet that includes a length is a LONGREP rather than a MATCH.

The match_byte value is the byte that would have been decoded if a SHORTREP packet had been used (in other words, the byte found at the dictionary at the last used distance); it is only used just after a *MATCH packet.

literal_bit_mode is an array of 8 values in the 0–2 range, one for each bit position in a byte, which are 1 or 2 if the previous packet was a *MATCH and it is either the most significant bit position or all the more significant bits in the literal to encode/decode are equal to the bits in the corresponding positions in match_byte, while otherwise it is 0; the choice between the 1 or 2 values depends on the value of the bit at the same position in match_byte.

The literal/Literal set of variables can be seen as a "pseudo-bit-tree" similar to a bit-tree but with 3 variables instead of 1 in every node, chosen depending on the literal_bit_mode value at the bit position of the next bit to decode after the bit-tree context denoted by the node.

The claim, found in some sources, that literals after a *MATCH are coded as the XOR of the byte value with match_byte is incorrect; they are instead coded simply as their byte value, but using the pseudo-bit-tree just described and the additional context listed in the table below.

The probability variable groups used in LZMA are those:

The LZMA2 container supports multiple runs of compressed LZMA data and uncompressed data. Each LZMA compressed run can have a different LZMA configuration and dictionary. This improves the compression of partially or completely incompressible files and allows multithreaded compression and multithreaded decompression by breaking the file into runs that can be compressed or decompressed independently in parallel. Criticism of LZMA2's changes over LZMA include header fields not being covered by CRCs, and parallel decompression not being possible in practice.[6]

The LZMA2 header consists of a byte indicating the dictionary size:

• 40 indicates a 4 GB − 1 dictionary size
• Even values less than 40 indicate a 2^{v/2 + 12} bytes dictionary size
• Odd values less than 40 indicate a 3×2^{(v − 1)/2 + 11} bytes dictionary size
• Values higher than 40 are invalid

LZMA2 data consists of packets starting with a control byte, with the following values:

• 0 denotes the end of the file
• 1 denotes a dictionary reset followed by an uncompressed chunk
• 2 denotes an uncompressed chunk without a dictionary reset
• 3–0x7f are invalid values
• 0x80–0xff denotes an LZMA chunk, where the lowest 5 bits are used as bit 16–20 of the uncompressed size minus one, and bit 5–6 indicates what should be reset

Bits 5–6 for LZMA chunks can be:

• 0: nothing reset
• 1: state reset
• 2: state reset, properties reset using properties byte
• 3: state reset, properties reset using properties byte, dictionary reset

LZMA state resets cause a reset of all LZMA state except the dictionary, and specifically:

• The range coder
• The state value
• The last distances for repeated matches
• All LZMA probabilities

Uncompressed chunks consist of:

• A 16-bit big-endian value encoding the data size minus one
• The data to be copied verbatim into the dictionary and the output

LZMA chunks consist of:

• A 16-bit big-endian value encoding the low 16 bits of the uncompressed size minus one
• A 16-bit big-endian value encoding the compressed size minus one
• A properties/lclppb byte if bit 6 in the control byte is set
• The LZMA compressed data, starting with the 5 bytes (of which the first is ignored) used to initialize the range coder (which are included in the compressed size)

The .xz format, which can contain LZMA2 data, is documented at tukaani.org,[10] while the .7z file format, which can contain either LZMA or LZMA2 data, is documented in the 7zformat.txt file contained in the LZMA SDK.[11]

The range encoder cannot make any interesting choices, and can be readily constructed based on the decoder description.

Initialization and termination are not fully determined; the xz encoder outputs 0 as the first byte which is ignored by the decompressor, and encodes the lower bound of the range (which matters for the final bytes).

The xz encoder uses an unsigned 33-bit variable called low (typically implemented as a 64-bit integer, initialized to 0), an unsigned 32-bit variable called range (initialized to 2³² − 1), an unsigned 8-bit variable called cache (initialized to 0), and an unsigned variable called cache_size which needs to be large enough to store the uncompressed size (initialized to 1, typically implemented as a 64-bit integer).

The cache/cache_size variables are used to properly handle carries, and represent a number defined by a big-endian sequence starting with the cache value, and followed by cache_size 0xff bytes, which has been shifted out of the low register, but has not been written yet, because it could be incremented by one due to a carry.

Note that the first byte output will always be 0 due to the fact that cache and low are initialized to 0, and the encoder implementation; the xz decoder ignores this byte.

Normalization proceeds in this way:

1. If low is less than (${\displaystyle 2^{32}-2^{24}}$):
   1. Output the byte stored in cache to the compressed stream
   2. Output cache_size − 1 bytes with 0xff value
   3. Set cache to bits 24–31 of low
   4. Set cache_size to 0
2. If low is greater or equal than ${\displaystyle 2^{32}}$:
   1. Output the byte stored in cache plus one to the compressed stream
   2. Output cache_size − 1 bytes with 0 value
   3. Set cache to bits 24–31 of low
   4. Set cache_size to 0
3. Increment cache_size
4. Set low to the lowest 24 bits of low shifted left by 8 bits
5. Set range to range shifted left by 8 bits

Context-based range encoding of a bit using the prob probability variable proceeds in this way:

1. If range is less than ${\displaystyle 2^{24}}$, perform normalization
2. Set bound to ${\displaystyle \lfloor range/2^{11}\rfloor \times prob}$
3. If encoding a 0 bit:
   1. Set range to bound
   2. Set prob to ${\displaystyle prob+\lfloor 2^{11}-prob\rfloor /2^{5}}$
4. Otherwise (if encoding a 1 bit):
   1. Set range to range − bound
   2. Set low to low + bound
   3. Set prob to ${\displaystyle prob-\lfloor prob/2^{5}\rfloor }$

Fixed-probability range encoding of a bit proceeds in this way:

1. If range is less than ${\displaystyle 2^{24}}$, perform normalization
2. Set range to ${\displaystyle \lfloor range/2\rfloor }$
3. If encoding a 1 bit:
   1. Set low to low + range

Termination proceeds this way:

1. Perform normalization 5 times

Bit-tree encoding is performed like decoding, except that bit values are taken from the input integer to be encoded rather than from the result of the bit decoding functions.

For algorithms that try to compute the encoding with the shortest post-range-encoding size, the encoder also needs to provide an estimate of that.

The encoder needs to be able to quickly locate matches in the dictionary. Since LZMA uses very large dictionaries (potentially on the order of gigabytes) to improve compression, simply scanning the whole dictionary would result in an encoder too slow to be practically usable, so sophisticated data structures are needed to support fast match searches.

The simplest approach, called "hash chains", is parameterized by a constant N which can be either 2, 3 or 4, which is typically chosen so that 2^8×N is greater than or equal to the dictionary size.

It consists of creating, for each k less than or equal to N, a hash table indexed by tuples of k bytes, where each of the buckets contains the last position where the first k bytes hashed to the hash value associated with that hash table bucket.

Chaining is achieved by an additional array which stores, for every dictionary position, the last seen previous position whose first N bytes hash to the same value of the first N bytes of the position in question.

To find matches of length N or higher, a search is started using the N-sized hash table, and continued using the hash chain array; the search stop after a pre-defined number of hash chain nodes has been traversed, or when the hash chains "wraps around", indicating that the portion of the input that has been overwritten in the dictionary has been reached.

Matches of size less than N are instead found by simply looking at the corresponding hash table, which either contains the latest such match, if any, or a string that hashes to the same value; in the latter case, the encoder will not be able to find the match. This issue is mitigated by the fact that for distant short matches using multiple literals might require less bits, and having hash conflicts in nearby strings is relatively unlikely; using larger hash tables or even direct lookup tables can reduce the problem at the cost of higher cache miss rate and thus lower performance.

Note that all matches need to be validated to check that the actual bytes match currently at that specific dictionary position match, since the hashing mechanism only guarantees that at some past time there were characters hashing to the hash table bucket index (some implementations may not even guarantee that, because they do not initialize the data structures).

LZMA uses Markov chains, as implied by "M" in its name.

The binary tree approach follows the hash chain approach, except that it logically uses a binary tree instead of a linked list for chaining.

The binary tree is maintained so that it is always both a search tree relative to the suffix lexicographic ordering, and a max-heap for the dictionary position[13] (in other words, the root is always the most recent string, and a child cannot have been added more recently than its parent): assuming all strings are lexicographically ordered, these conditions clearly uniquely determine the binary tree (this is trivially provable by induction on the size of the tree).

Since the string to search for and the string to insert are the same, it is possible to perform both dictionary search and insertion (which requires to rotate the tree) in a single tree traversal.

Some old LZMA encoders also supported a data structure based on Patricia tries, but such support has since been dropped since it was deemed inferior to the other options.[13]

LZMA encoders can freely decide which match to output, or whether to ignore the presence of matches and output literals anyway.

The ability to recall the 4 most recently used distances means that, in principle, using a match with a distance that will be needed again later may be globally optimal even if it is not locally optimal, and as a result of this, optimal LZMA compression probably requires knowledge of the whole input and might require algorithms too slow to be usable in practice.

Due to this, practical implementations tend to employ non-global heuristics.

The xz encoders use a value called nice_len (the default is 64): when any match of length at least nice_len is found, the encoder stops the search and outputs it, with the maximum matching length.

The XZ fast encoder[14] (derived from the 7-zip fast encoder) is the shortest LZMA encoder in the xz source tree.

It works like this:

1. Perform combined search and insertion in the dictionary data structure
2. If any repeated distance matches with length at least nice_len:
   • Output the most frequently used such distance with a REP packet
3. If a match was found of length at least nice_len:
   • Output it with a MATCH packet
4. Set the main match to the longest match
5. Look at the nearest match of every length in decreasing length order, and until no replacement can be made:
   • Replace the main match with a match which is one character shorter, but whose distance is less than 1/128 the current main match distance
6. Set the main match length to 1 if the current main match is of length 2 and distance at least 128
7. If a repeated match was found, and it is shorter by at most 1 character than the main match:
   • Output the repeated match with a REP packet
8. If a repeated match was found, and it is shorter by at most 2 characters than the main match, and the main match distance is at least 512:
   • Output the repeated match with a REP packet
9. If a repeated match was found, and it is shorter by at most 3 characters than the main match, and the main match distance is at least 32768:
   • Output the repeated match with a REP packet
10. If the main match size is less than 2 (or there is not any match):
    • Output a LIT packet
11. Perform a dictionary search for the next byte
12. If the next byte is shorter by at most 1 character than the main match, with distance less than 1/128 times the main match distance, and if the main match length is at least 3:
    • Output a LIT packet
13. If the next byte has a match at least as long as the main match, and with less distance than the main match:
    • Output a LIT packet
14. If the next byte has a match at least one character longer than the main match, and such that 1/128 of its distance is less or equal than the main match distance:
    • Output a LIT packet
15. If the next byte has a match more than one character longer than the main match:
    • Output a LIT packet
16. If any repeated match is shorter by at most 1 character than the main match:
    • Output the most frequently used such distance with a REP packet
17. Output the main match with a MATCH packet

The XZ normal encoder[15] (derived from the 7-zip normal encoder) is the other LZMA encoder in the xz source tree, which adopts a more sophisticated approach that tries to minimize the post-range-encoding size of the generated packets.

Specifically, it encodes portions of the input using the result of a dynamic programming algorithm, where the subproblems are finding the approximately optimal encoding (the one with minimal post-range-encoding size) of the substring of length L starting at the byte being compressed.

The size of the portion of the input processed in the dynamic programming algorithm is determined to be the maximum between the longest dictionary match and the longest repeated match found at the start position (which is capped by the maximum LZMA match length, 273); furthermore, if a match longer than nice_len is found at any point in the range just defined, the dynamic programming algorithm stops, the solution for the subproblem up to that point is output, the nice_len-sized match is output, and a new dynamic programming problem instance is started at the byte after the match is output.

Subproblem candidate solutions are incrementally updated with candidate encodings, constructed taking the solution for a shorter substring of length L', extended with all possible "tails", or sets of 1–3 packets with certain constraints that encode the input at the L' position. Once the final solution of a subproblem is found, the LZMA state and least used distances for it are computed, and are then used to appropriately compute post-range-encoding sizes of its extensions.

At the end of the dynamic programming optimization, the whole optimal encoding of the longest substring considered is output, and encoding continues at the first uncompressed byte not already encoded, after updating the LZMA state and least used distances.

Each subproblem is extended by a packet sequence which we call "tail", which must match one of the following patterns:

The reason for not only extending with single packets is that subproblems only have the substring length as the parameter for performance and algorithmic complexity reasons, while an optimal dynamic programming approach would also require to have the last used distances and LZMA state as parameter; thus, extending with multiple packets allows to better approximate the optimal solution, and specifically to make better use of LONGREP[0] packets.

The following data is stored for each subproblem (of course, the values stored are for the candidate solution with minimum price), where by "tail" we refer to the packets extending the solution of the smaller subproblem, which are described directly in the following structure:

Note that in the XZ for Java implementation, the optPrev and backPrev members are reused to store a forward single-linked list of packets as part of outputting the final solution.

The XZ LZMA2 encoder processes the input in chunks (of up to 2 MB uncompressed size or 64 KB compressed size, whichever is lower), handing each chunk to the LZMA encoder, and then deciding whether to output an LZMA2 LZMA chunk including the encoded data, or to output an LZMA2 uncompressed chunk, depending on which is shorter (LZMA, like any other compressor, will necessarily expand rather than compress some kinds of data).

The LZMA state is reset only in the first block, if the caller requests a change of properties and every time a compressed chunk is output. The LZMA properties are changed only in the first block, or if the caller requests a change of properties. The dictionary is only reset in the first block.

Before LZMA2 encoding, depending on the options provided, xz can apply the BCJ filter, which filters executable code to replace relative offsets with absolute ones that are more repetitive, or the delta filter, which replaces each byte with the difference between it and the byte N bytes before it.

Parallel encoding is performed by dividing the file in chunks which are distributed to threads, and ultimately each encoded (using, for instance, xz block encoding) separately, resulting in a dictionary reset between chunks in the output file.