PARQUET-2249: Introduce IEEE 754 total order & NaN-counts

This commit is a combination of the following PRs:

* Introduce IEEE 754 total order
  apache#221
* Add nan_count to handle NaNs in statistics
  apache#196

Both these PRs try to solve the same problems; read
the description of the respective PRs for explanation.

This PR is the result of an extended discussion in
which it was repeatedly brought up that another possible
solution to the problem could be the combination of
the two approaches. Please refer to this discussion
on the mailing list and in the two PRs for details.
the mailing list discussion can be found here:
https://lists.apache.org/thread/lzh0dvrvnsy8kvflvl61nfbn6f9js81s

The contents of this PR are basically a straightforward
combination of the two approaches:
* IEEE total order is introduced as a new order for floating
  point types
* nan_count and nan_counts fields are added

Legacy writers may not write nan_count(s) fields,
so readers have to handle them being absent. Also, legacy
writers may have included NaNs into min/max bounds, so readers
also have to handle that.

As there are no legacy writers writing IEEE total order,
nan_count(s) are defined to be mandatory if this order is used,
so readers can assume their presense when this order is used.

This commit removes `nan_pages` from the ColumnIndex,
which the nan_counts PR mandated. We don't need them anymore,
as IEEE total order solves this: As this is a new order and
there are thus no legacy writers and readers, we have the
freedom to define that for only-NaN pages using this order,
we can actually write NaNs into min & max bounds in this case,
and readers can assume that isNaN(min) signals an only-NaN
page.

Author: JFinis

LogicalTypes.md

@@ -254,7 +254,10 @@ Used in contexts where precision is traded off for smaller footprint and potenti
 
 The primitive type is a 2-byte `FIXED_LEN_BYTE_ARRAY`.
 
-The sort order for `FLOAT16` is signed (with special handling of NANs and signed zeros); it uses the same [logic](https://github.com/apache/parquet-format#sort-order) as `FLOAT` and `DOUBLE`.
+The type-defined sort order for `FLOAT16` is signed (with special handling of NaNs and signed zeros),
+as for `FLOAT` and `DOUBLE`. It is recommended that writers use IEEE754TotalOrder when writing columns
+of this type for a well-defined handling of NaNs and signed zeros. See the `ColumnOrder` union in the
+[Thrift definition](src/main/thrift/parquet.thrift) for details.
 
 ## Temporal Types
 

README.md

@@ -158,7 +158,9 @@ documented in [LogicalTypes.md][logical-types].
 Parquet stores min/max statistics at several levels (such as Column Chunk,
 Column Index, and Data Page). These statistics are according to a sort order,
 which is defined for each column in the file footer. Parquet supports common
-sort orders for logical and primitve types. The details are documented in the
+sort orders for logical and primitve types and also special orders for types
+where the common sort order is not unambiguously defined (e.g., NaN ordering
+for floating point types). The details are documented in the
 [Thrift definition](src/main/thrift/parquet.thrift) in the `ColumnOrder` union.
 
 ## Nested Encoding

src/main/thrift/parquet.thrift

@@ -309,6 +309,13 @@ struct Statistics {
    7: optional bool is_max_value_exact;
    /** If true, min_value is the actual minimum value for a column */
    8: optional bool is_min_value_exact;
+   /** 
+    * count of NaN values in the column; only present if physical type is FLOAT
+    * or DOUBLE, or logical type is FLOAT16. 
+    * Readers MUST distinguish between nan_count not being present and nan_count == 0.
+    * If nan_count is not present, readers MUST NOT assume nan_count == 0.
+    */
+   9: optional i64 nan_count;
 }
 
 /** Empty structs to use as logical type annotations */
@@ -670,7 +677,7 @@ enum BoundaryOrder {
 /** Data page header */
 struct DataPageHeader {
   /**
-   * Number of values, including NULLs, in this data page.
+   * Number of values, including nulls, in this data page.
    *
    * If a OffsetIndex is present, a page must begin at a row
    * boundary (repetition_level = 0). Otherwise, pages may begin
@@ -717,9 +724,9 @@ struct DictionaryPageHeader {
  * The remaining section containing the data is compressed if is_compressed is true
  **/
 struct DataPageHeaderV2 {
-  /** Number of values, including NULLs, in this data page. **/
+  /** Number of values, including nulls, in this data page. **/
   1: required i32 num_values
-  /** Number of NULL values, in this data page.
+  /** Number of null values, in this data page.
       Number of non-null = num_values - num_nulls which is also the number of values in the data section **/
   2: required i32 num_nulls
   /**
@@ -1030,6 +1037,9 @@ struct RowGroup {
 /** Empty struct to signal the order defined by the physical or logical type */
 struct TypeDefinedOrder {}
 
+/** Empty struct to signal IEEE 754 total order for floating point types */
+struct IEEE754TotalOrder {}
+
 /**
  * Union to specify the order used for the min_value and max_value fields for a
  * column. This union takes the role of an enhanced enum that allows rich
@@ -1038,6 +1048,7 @@ struct TypeDefinedOrder {}
  * Possible values are:
  * * TypeDefinedOrder - the column uses the order defined by its logical or
  *                      physical type (if there is no logical type).
+ * * IEEE754TotalOrder - the floating point column uses IEEE 754 total order.
  *
  * If the reader does not support the value of this union, min and max stats
  * for this column should be ignored.
@@ -1082,23 +1093,105 @@ union ColumnOrder {
    *   BYTE_ARRAY - unsigned byte-wise comparison
    *   FIXED_LEN_BYTE_ARRAY - unsigned byte-wise comparison
    *
-   * (*) Because the sorting order is not specified properly for floating
-   *     point values (relations vs. total ordering) the following
+   * (*) Because the precise sorting order is ambiguous for floating
+   *     point types due to underspecified handling of NaN and -0/+0,
+   *     it is recommended that writers use IEEE_754_TOTAL_ORDER
+   *     for these types.
+   *
+   *     If TYPE_ORDER is used for floating point types, then the following
    *     compatibility rules should be applied when reading statistics:
    *     - If the min is a NaN, it should be ignored.
    *     - If the max is a NaN, it should be ignored.
+   *     - If the nan_count field is set, a reader can compute
+   *       nan_count + null_count == num_values to deduce whether all non-null
+   *       values are NaN.
+   *     - When looking for NaN values, min and max should be ignored.
+   *       If the nan_count field is set, it can be used to check whether
+   *       NaNs are present.
    *     - If the min is +0, the row group may contain -0 values as well.
    *     - If the max is -0, the row group may contain +0 values as well.
    *     - When looking for NaN values, min and max should be ignored.
    * 
    *     When writing statistics the following rules should be followed:
-   *     - NaNs should not be written to min or max statistics fields.
+   *     - It is suggested to always set the nan_count field for floating
+   *       point types, especially also if it is zero.
+   *     - NaNs should not be written to min or max statistics fields except
+   *       in the column index, where min_values and max_values are not optional
+   *       so a NaN value must be written if all non-null values in a page
+   *       are NaN.
    *     - If the computed max value is zero (whether negative or positive),
    *       `+0.0` should be written into the max statistics field.
    *     - If the computed min value is zero (whether negative or positive),
    *       `-0.0` should be written into the min statistics field.
    */
   1: TypeDefinedOrder TYPE_ORDER;
+
+  /*
+   * The floating point type is ordered according to the totalOrder predicate,
+   * as defined in section 5.10 of IEEE-754 (2008 revision). Only columns of
+   * physical type FLOAT or DOUBLE, or logical type FLOAT16 may use this ordering.
+   *
+   * Intuitively, this orders floats mathematically, but defines -0 to be less
+   * than +0, -NaN to be less than anything else, and +NaN to be greater than
+   * anything else. It also defines an order between different bit representations
+   * of the same value.
+   *
+   * The formal definition is as follows:
+   *   a) If x<y, totalOrder(x, y) is true.
+   *   b) If x>y, totalOrder(x, y) is false.
+   *   c) If x=y:
+   *     1) totalOrder(−0, +0) is true.
+   *     2) totalOrder(+0, −0) is false.
+   *     3) If x and y represent the same floating-point datum:
+   *        i) If x and y have negative sign, totalOrder(x, y) is true if and
+   *           only if the exponent of x ≥ the exponent of y
+   *       ii) otherwise totalOrder(x, y) is true if and only if the exponent
+   *           of x ≤ the exponent of y.
+   *   d) If x and y are unordered numerically because x or y is NaN:
+   *     1) totalOrder(−NaN, y) is true where −NaN represents a NaN with
+   *        negative sign bit and y is a non-NaN floating-point number.
+   *     2) totalOrder(x, +NaN) is true where +NaN represents a NaN with
+   *        positive sign bit and x is a non-NaN floating-point number.
+   *     3) If x and y are both NaNs, then totalOrder reflects a total ordering
+   *        based on:
+   *         i) negative sign orders below positive sign
+   *        ii) signaling orders below quiet for +NaN, reverse for −NaN
+   *       iii) lesser payload, when regarded as an integer, orders below
+   *            greater payload for +NaN, reverse for −NaN.
+   *
+   * Note that this ordering can be implemented efficiently in software by bit-wise
+   * operations on the integer representation of the floating point values.
+   * E.g., this is a possible implementation for DOUBLE in Rust:
+   *
+   *   pub fn totalOrder(x: f64, y: f64) -> bool {
+   *     let mut x_int = x.to_bits() as i64;
+   *     let mut y_int = y.to_bits() as i64;
+   *     x_int ^= (((x_int >> 63) as u64) >> 1) as i64;
+   *     y_int ^= (((y_int >> 63) as u64) >> 1) as i64;
+   *     return x_int <= y_int;
+   *   }
+   *
+   * When writing statistics for columns with this order, the following rules
+   * must be followed:
+   * - Writing the nan_count field is mandatory when using this ordering,
+   *   especialy also if it is zero.
+   * - NaNs should not be written to min or max statistics fields except
+   *   in the column index, where min_values and max_values are not optional
+   *   so a NaN value must be written if all non-null values in a page
+   *   are NaN. In this case, the min_values[i] and max_values[i] fields
+   *   should be set to the smallest and largest NaN values contained
+   *   in the page, as defined by the IEEE 754 total order.
+   *
+   * When reading statistics for columns with this order, the following rules
+   * should be followed:
+   * - Readers should consult the nan_count field to determine whether NaNs
+   *   are present.
+   * - A reader can compute nan_count + null_count == num_values to deduce
+   *   whether all non-null values are NaN. In the page index, which does not
+   *   have a num_values field, the presence of a NaN value in min_values
+   *   or max_values indicates that all non-null values are NaN.
+   */
+  2: IEEE754TotalOrder IEEE_754_TOTAL_ORDER;
 }
 
 struct PageLocation {
@@ -1170,6 +1263,19 @@ struct ColumnIndex {
    * Such more compact values must still be valid values within the column's
    * logical type. Readers must make sure that list entries are populated before
    * using them by inspecting null_pages.
+   * For columns of physical type FLOAT or DOUBLE, or logical type FLOAT16,
+   * NaN values are not to be included in these bounds. If all non-null values
+   * of a page are NaN, then a writer must do the following:
+   * - If the order of this column is TypeDefinedOrder, then no column index
+   *   must be written for this column chunk. While this is unfortunate for
+   *   performance, it is necessary to avoid conflict with legacy files that
+   *   still included NaN in min_values and max_values even if the page had
+   *   non-NaN values. To mitigate this, IEEE754_TOTAL_ORDER is recommended.
+   * - If the order of this column is IEEE754_TOTAL_ORDER, then min_values[i]
+   * * If IEEE754_TOTAL_ORDER is used for the column and all non-null values
+   *   of a page are NaN, then min_values[i] and max_values[i] must be set to
+   *   the smallest and largest NaN value contained in the page, as defined
+   *   by the IEEE 754 total order.
    */
   2: required list<binary> min_values
   3: required list<binary> max_values
@@ -1193,7 +1299,6 @@ struct ColumnIndex {
    * null counts are 0.
    */
   5: optional list<i64> null_counts
-
   /**
    * Contains repetition level histograms for each page
    * concatenated together.  The repetition_level_histogram field on
@@ -1211,6 +1316,16 @@ struct ColumnIndex {
     * Same as repetition_level_histograms except for definitions levels.
     **/
    7: optional list<i64> definition_level_histograms;
+
+   /** 
+    * A list containing the number of NaN values for each page. Only present
+    * for columns of physical type FLOAT or DOUBLE, or logical type FLOAT16.
+    * If this field is not present, readers MUST assume that there might or
+    * might not be NaN values in any page, as NaNs should not be included
+    * in min_values or max_values.
+    */
+   8: optional list<i64> nan_counts
+
 }
 
 struct AesGcmV1 {