ALP Encoding

ALP (Adaptive Lossless floating-Point) Encoding

This file contains the detailed specification of the ALP encoding (ALP = 10).

Overview

For each data page, ALP encoding consists of a header followed by an offset array and one or more encoded vectors (batches of values). Each vector contains up to vector_size elements (default 1024).

+-------------+-----------------------------+--------------------------------------+
|   Header    |        Offset Array         |            Vector Data               |
|  (7 bytes)  |   (num_vectors * 4 bytes)   |            (variable)                |
+-------------+------+------+-----+---------+----------+----------+-----+----------+
| Page Header | off0 | off1 | ... | off N-1 | Vector 0 | Vector 1 | ... | Vec N-1  |
|  (7 bytes)  | (4B) | (4B) |     |  (4B)   |(variable)|(variable)|     |(variable)|
+-------------+------+------+-----+---------+----------+----------+-----+----------+

The compression pipeline below describes one way to produce a conforming vector. It is informative, not normative: an encoder may use any strategy as long as it emits the byte layout defined in Page Layout. Only that byte layout and the Decoding procedure are normative.

                    Input: float/double array
                              |
                              v
    +----------------------------------------------------------+
    |  1. CHOOSE PARAMETERS                                    |
    |     Select (exponent, factor) pair for this array        |
    +----------------------------------------------------------+
                              |
                              v
    +----------------------------------------------------------+
    |  2. DECIMAL ENCODING                                     |
    |     encoded[i] = fast_round(value[i] * 10^e * 10^(-f))  |
    |     Detect exceptions where decode(encode(v)) != v       |
    +----------------------------------------------------------+
                              |
                              v
    +----------------------------------------------------------+
    |  3. FRAME OF REFERENCE (FOR)                             |
    |     min_val = min(encoded[:])                            |
    |     delta[i] = encoded[i] - min_val                      |
    +----------------------------------------------------------+
                              |
                              v
    +----------------------------------------------------------+
    |  4. BIT PACKING                                          |
    |     bit_width = ceil(log2(max_delta + 1))                |
    |     Pack each delta into bit_width bits                  |
    +----------------------------------------------------------+
                              |
                              v
                   Output: Serialized vector bytes

The fast_round used in step 2 is one recommended rounding technique, described in Fast Rounding below; it is informative, not normative.

Page Layout

Header (7 bytes)

All multi-byte values are stored in little-endian order.

 Byte:         0                1              2         3    4    5    6
       +----------------+---------------+--------------+----+----+----+----+
       | compression    | integer       | log_vector   |     num_elements  |
       | _mode          | _encoding     | _size        |     (int32 LE)    |
       +----------------+---------------+--------------+----+----+----+----+

The number of vectors is ceil(num_elements / vector_size). The last vector may contain fewer than vector_size elements.

Note: The number of elements per vector is NOT stored in the header — it is derived: vector_size for all vectors except the last, which may be smaller.

Offset Array

Immediately following the header is an array of num_vectors little-endian uint32 values. Each offset gives the byte position of the corresponding vector’s data, measured from the start of the offset array itself.

The first offset always equals num_vectors * 4 (pointing just past the offset array). Each subsequent offset equals the previous offset plus the stored size of the previous vector. No padding is inserted between vectors.

Offsets are relative to the start of the offset array. A vector’s absolute byte position is alp_data_start + 7 + offset, where alp_data_start is the first byte of the ALP header within the decoded page data — that is, after the page has been decompressed and after any repetition/definition levels — and 7 is the size of the ALP header. (When the page is uncompressed and carries no repetition or definition levels, alp_data_start coincides with the first byte after the page’s Thrift header.)

Vector Format

Each vector is self-describing and contains the encoding parameters, FOR metadata, bit-packed encoded values, and exception data. The layout described here applies when compression_mode = 0 (ALP) and integer_encoding = 0 (FOR + bit-packing); future modes may define different vector contents and need not include AlpInfo or ForInfo.

<----------- Vector Header -----------><----------------------- Data Section ----------------------->
+-------------------+-----------------+-------------------+---------------------+-------------------+
|      AlpInfo      |     ForInfo     |   PackedValues    | ExceptionPositions  | ExceptionValues   |
|     (4 bytes)     | (5B or 9B)      |    (variable)     |     (variable)      |    (variable)     |
+-------------------+-----------------+-------------------+---------------------+-------------------+

The first two components (AlpInfo and ForInfo) form the vector header; the remaining three (PackedValues, ExceptionPositions, ExceptionValues) form the data section.

Vector header sizes:

Data section sizes:

Here bit_width and num_exceptions are read from the vector header (ForInfo and AlpInfo respectively), described below.

AlpInfo (4 bytes, both types)

 Byte:    0           1          2       3
       +----------+----------+---------+---------+
       | exponent |  factor  |  num_exceptions   |
       |  (uint8) | (uint8)  |   (uint16 LE)     |
       +----------+----------+---------+---------+

ForInfo for FLOAT (5 bytes)

 Byte:    0    1    2    3       4
       +----+----+----+----+-----------+
       | frame_of_reference | bit_width |
       |    (int32 LE)      |  (uint8)  |
       +----+----+----+----+-----------+

ForInfo for DOUBLE (9 bytes)

 Byte:    0    1    2    3    4    5    6    7       8
       +----+----+----+----+----+----+----+----+-----------+
       |          frame_of_reference           | bit_width |
       |              (int64 LE)               |  (uint8)  |
       +----+----+----+----+----+----+----+----+-----------+

PackedValues

The FOR-encoded deltas, bit-packed into ceil(num_elements_in_vector * bit_width / 8) bytes. Values are bit-packed using the same LSB-first packing order as the RLE/Bit-Packing Hybrid encoding. When the total number of packed bits is not a multiple of 8, the final byte is padded with zero bits in its most significant positions.

Each delta is encoded[i] - frame_of_reference, computed in unsigned (wrapping) arithmetic and stored as an unsigned integer. Computing it as unsigned avoids signed-integer overflow when the vector’s range (max - min) exceeds the signed maximum of the encoded type, and it means no sign extension is applied when unpacking. Because frame_of_reference is the minimum encoded integer in the vector, every delta is non-negative.

If bit_width is 0, no bytes are stored (all deltas are zero, meaning all encoded integers are equal to frame_of_reference).

ExceptionPositions

An array of num_exceptions little-endian uint16 values, each giving the 0-based index within the vector of an exception value.

ExceptionValues

An array of num_exceptions values in the original floating-point type (4 bytes little-endian IEEE 754 for FLOAT, 8 bytes for DOUBLE), stored in the same order as the corresponding positions. Each value is stored as its exact IEEE 754 bit pattern; implementations MUST NOT canonicalize NaN or otherwise alter the bits, so that decoding reproduces the original value bit-for-bit.

Encoding

Encoding Formula

+-------------------------------------------------------------------+
|                                                                   |
|   encoded = fast_round( value  *  10^e  *  10^(-f) )             |
|                                                                   |
|   decoded = encoded  *  10^f  *  10^(-e)                          |
|                                                                   |
+-------------------------------------------------------------------+

The formula uses two separate multiplications (not a single multiplication by 10^(e-f), and not division). This is a requirement of the decode path, which is normative: to reconstruct a value every reader MUST compute decoded = encoded * 10^f * 10^(-e) using the same two-step multiplication and the same power-of-10 constants, so that all implementations reproduce the stored value bit-for-bit. The power-of-10 constants MUST be the correctly-rounded IEEE 754 values of the decimal literals 1e0, 1e1, …, 1e18 and 1e-1, 1e-2, …, 1e-18 as defined by the decimal-to-binary conversion in IEEE 754-2008 §5.12.2. Implementations MUST NOT compute these constants at runtime via pow() or equivalent functions, which are not guaranteed to be correctly rounded.

The encode direction — mapping each value to the integer it will be stored as, via fast_round(value * 10^e * 10^(-f)) — is informative, not normative. An encoder MAY choose that integer by any means, because every value is checked against the normative decode above and any value that does not round-trip exactly is stored as an exception. The rounding method therefore affects only compression ratio and exception count, never correctness or what a reader decodes. The fast_round technique below is one recommended implementation.

Fast Rounding (informative)

fast_round recovers the integer intended by value * 10^e * 10^(-f) — which carries floating-point rounding noise — by rounding it to the nearest integer (ties to even), without a division or a call to a library rounding function. It is not normative: an encoder MAY use any rounding method, since values that do not round-trip under the normative decode are stored as exceptions.

The technique adds then subtracts a “magic number” (a power of two large enough to discard the fractional bits), leaving the nearest integer. Implementations vary: some apply it in a single branch-free form, others add a sign test.

The value ± magic operations must be evaluated in the value’s own precision (FLOAT in binary32, DOUBLE in binary64); only the final cast converts to an integer. The two forms round some large-magnitude inputs differently, but since any value that fails to round-trip is stored as an exception, the choice affects only compression ratio, never correctness.

Parameter Selection

Any valid (exponent, factor) pair produces a correct encoding — the decoder is agnostic to the selection strategy, and the exception mechanism guarantees round-trip fidelity regardless of which pair is chosen. The choice only affects compression ratio.

The encoder SHOULD select the (exponent, factor) pair that produces the smallest encoded output. A simple heuristic is to minimize exception count; a more precise approach accounts for both bit-width and exception overhead.

Valid combinations satisfy 0 ≤ factor ≤ exponent:

To avoid the cost of exhaustive search on every vector, implementations can use a sampling approach. One such approach, described in the paper, is to select up to 5 candidate (exponent, factor) combinations (the “encoding preset”) at the start of each column chunk, and when encoding each vector, evaluate each candidate for the best compression.

Suggested sampling parameters (from the paper):

Exception Detection

A value becomes an exception if any of the following is true:

Exception values at positions in the vector are replaced with a placeholder (the encoded integer of the first non-exception value, or 0 if all values are exceptions) before FOR encoding. This keeps the FOR range tight.

Example: Frame of Reference and Bit-Packing

Given the following data after decimal encoding and exception substitution:

+---------------------------------------------------------------------+
|  Encoded:   [ 123,  456,  789,   12 ]                               |
|                                                                     |
|  min_val = 12  (stored as frame_of_reference)                       |
|                                                                     |
|  Deltas:    [ 111,  444,  777,    0 ]   <-- all non-negative        |
+---------------------------------------------------------------------+

Special case: If all values are identical, bit_width = 0 and no packed data is stored.

Decoding

                    Input: Serialized vector bytes
                              |
                              v
    +----------------------------------------------------------+
    |  1. BIT UNPACKING                                        |
    |     Unpack num_elements values at bit_width bits each    |
    +----------------------------------------------------------+
                              |
                              v
    +----------------------------------------------------------+
    |  2. REVERSE FOR                                          |
    |     encoded[i] = delta[i] + frame_of_reference           |
    +----------------------------------------------------------+
                              |
                              v
    +----------------------------------------------------------+
    |  3. DECIMAL DECODING                                     |
    |     value[i] = encoded[i] * 10^factor * 10^(-exponent)   |
    +----------------------------------------------------------+
                              |
                              v
    +----------------------------------------------------------+
    |  4. PATCH EXCEPTIONS                                     |
    |     value[pos[j]] = exception_values[j]                  |
    +----------------------------------------------------------+
                              |
                              v
                  Output: Original float/double array

For each vector:

1. Read AlpInfo and ForInfo from the vector header.
2. Unpack bit_width-bit integers from PackedValues.
3. Add frame_of_reference to each unpacked integer.
4. Decode: multiply each integer by 10^factor then by 10^(-exponent).
5. Patch exceptions: for each (position, value) in the exception arrays, overwrite the decoded output at that position with the stored value.

Worked Example: Exceptions and Non-Zero Factor

Input: double values[4] = { 1500.0, NaN, 2500.0, 333.5 }

Best encoding found: (exponent=4, factor=3). This means: encoded = fast_round(value * 10^4 * 10^(-3)) = fast_round(value * 10)

Step 1: Decimal Encoding

Step 2: Handle Exceptions

Exception positions: [1] Exception values: [NaN] Placeholder: 15000 (first non-exception encoded value) Encoded with placeholders: [15000, 15000, 25000, 3335]

Step 3: Frame of Reference

Step 4: Bit Packing

max_delta = 21665, bit_width = ceil(log2(21666)) = 15 bits, packed_size = ceil(4 * 15 / 8) = 8 bytes

Serialized Vector:

Compared to PLAIN encoding (4 * 8 = 32 bytes). With 1024 values, the 13-byte vector header becomes negligible and compression ratios of 2-8x are typical.