Fix behavior of ~> (cube, int) operator

~> (cube, int) operator was especially designed for knn-gist search.
However, it appears that knn-gist search can't work correctly with current
behavior of this operator when dataset contains cubes of variable
dimensionality. In this case, the same value of second operator argument
can point to different dimension depending on dimensionality of particular cube.
Such behavior is incompatible with gist indexing of cubes, and knn-gist doesn't
work correctly for it.

This patch changes behavior of ~> (cube, int) operator by introducing dimension
numbering where value of second argument unambiguously identifies number of
dimension. With new behavior, this operator can be correctly supported by
knn-gist. Relevant changes to cube operator class are also included.

Backpatch to v9.6 where operator was introduced.

Since behavior of ~> (cube, int) operator is changed, depending entities
must be refreshed after upgrade. Such as, expression indexes using this
operator must be reindexed, materialized views must be rebuilt, stored
procedures and client code must be revised to correctly use new behavior.
That should be mentioned in release notes.

Noticed by: Tomas Vondra
Author: Alexander Korotkov
Reviewed by: Tomas Vondra, Andrey Borodin
Discussion: https://www.postgresql.org/message-id/flat/a9657f6a-b497-36ff-e56-482a2c7e3292@2ndquadrant.com

5 files changed, 512 insertions, 286 deletions

diff --git a/contrib/cube/cube.c b/contrib/cube/cube.c
index 3e3d83323c8..dcc0850aa92 100644
--- a/contrib/cube/cube.c
+++ b/contrib/cube/cube.c
@@ -1337,15 +1337,55 @@ g_cube_distance(PG_FUNCTION_ARGS)
 
 	if (strategy == CubeKNNDistanceCoord)
 	{
+		/*
+		 * Handle ordering by ~> operator.  See comments of cube_coord_llur()
+		 * for details
+		 */
 		int			coord = PG_GETARG_INT32(1);
+		bool		isLeaf = GistPageIsLeaf(entry->page);
 
-		if (DIM(cube) == 0)
-			retval = 0.0;
-		else if (IS_POINT(cube))
-			retval = cube->x[(coord - 1) % DIM(cube)];
+		/* 0 is the only unsupported coordinate value */
+		if (coord <= 0)
+			ereport(ERROR,
+					(errcode(ERRCODE_ARRAY_ELEMENT_ERROR),
+					 errmsg("cube index %d is out of bounds", coord)));
+
+		if (coord <= 2 * DIM(cube))
+		{
+			/* dimension index */
+			int		index = (coord - 1) / 2;
+			/* whether this is upper bound (lower bound otherwise) */
+			bool	upper = ((coord - 1) % 2 == 1);
+
+			if (IS_POINT(cube))
+			{
+				retval = cube->x[index];
+			}
+			else
+			{
+				if (isLeaf)
+				{
+					/* For leaf just return required upper/lower bound */
+					if (upper)
+						retval = Max(cube->x[index], cube->x[index + DIM(cube)]);
+					else
+						retval = Min(cube->x[index], cube->x[index + DIM(cube)]);
+				}
+				else
+				{
+					/*
+					 * For non-leaf we should always return lower bound,
+					 * because even upper bound of a child in the subtree can
+					 * be as small as our lower bound.
+					 */
+					retval = Min(cube->x[index], cube->x[index + DIM(cube)]);
+				}
+			}
+		}
 		else
-			retval = Min(cube->x[(coord - 1) % DIM(cube)],
-						 cube->x[(coord - 1) % DIM(cube) + DIM(cube)]);
+		{
+			retval = 0.0;
+		}
 	}
 	else
 	{
@@ -1492,43 +1532,73 @@ cube_coord(PG_FUNCTION_ARGS)
 }
 
 
-/*
- * This function works like cube_coord(),
- * but rearranges coordinates of corners to get cube representation
- * in the form of (lower left, upper right).
- * For historical reasons that extension allows us to create cubes in form
- * ((2,1),(1,2)) and instead of normalizing such cube to ((1,1),(2,2)) it
- * stores cube in original way. But to get cubes ordered by one of dimensions
- * directly from the index without extra sort step we need some
- * representation-independent coordinate getter. This function implements it.
+/*----
+ * This function works like cube_coord(), but rearranges coordinates in the
+ * way suitable to support coordinate ordering using KNN-GiST.  For historical
+ * reasons this extension allows us to create cubes in form ((2,1),(1,2)) and
+ * instead of normalizing such cube to ((1,1),(2,2)) it stores cube in original
+ * way.  But in order to get cubes ordered by one of dimensions from the index
+ * without explicit sort step we need this representation-independent coordinate
+ * getter.  Moreover, indexed dataset may contain cubes of different dimensions
+ * number.  Accordingly, this coordinate getter should be able to return
+ * lower/upper bound for particular dimension independently on number of cube
+ * dimensions.
+ *
+ * Long story short, this function uses following meaning of coordinates:
+ * # (2 * N - 1) -- lower bound of Nth dimension,
+ * # (2 * N) -- upper bound of Nth dimension.
+ *
+ * When given coordinate exceeds number of cube dimensions, then 0 returned
+ * (reproducing logic of GiST indexing of variable-length cubes).
  */
 Datum
 cube_coord_llur(PG_FUNCTION_ARGS)
 {
 	NDBOX	   *cube = PG_GETARG_NDBOX_P(0);
 	int			coord = PG_GETARG_INT32(1);
+	bool		inverse = false;
+	float8		result;
 
-	if (coord <= 0 || coord > 2 * DIM(cube))
+	/* 0 is the only unsupported coordinate value */
+	if (coord <= 0)
 		ereport(ERROR,
 				(errcode(ERRCODE_ARRAY_ELEMENT_ERROR),
 				 errmsg("cube index %d is out of bounds", coord)));
 
-	if (coord <= DIM(cube))
+	if (coord <= 2 * DIM(cube))
 	{
+		/* dimension index */
+		int		index = (coord - 1) / 2;
+		/* whether this is upper bound (lower bound otherwise) */
+		bool	upper = ((coord - 1) % 2 == 1);
+
 		if (IS_POINT(cube))
-			PG_RETURN_FLOAT8(cube->x[coord - 1]);
+		{
+			result = cube->x[index];
+		}
 		else
-			PG_RETURN_FLOAT8(Min(cube->x[coord - 1],
-								 cube->x[coord - 1 + DIM(cube)]));
+		{
+			if (upper)
+				result = Max(cube->x[index], cube->x[index + DIM(cube)]);
+			else
+				result = Min(cube->x[index], cube->x[index + DIM(cube)]);
+		}
 	}
 	else
 	{
-		if (IS_POINT(cube))
-			PG_RETURN_FLOAT8(cube->x[(coord - 1) % DIM(cube)]);
-		else
-			PG_RETURN_FLOAT8(Max(cube->x[coord - 1],
-								 cube->x[coord - 1 - DIM(cube)]));
+		/*
+		 * Return zero if coordinate is out of bound.  That reproduces logic of
+		 * how cubes with low dimension number are expanded during GiST
+		 * indexing.
+		 */
+		result = 0.0;
 	}
+
+	/* Inverse value if needed */
+	if (inverse)
+		result = -result;
+
+	PG_RETURN_FLOAT8(result);
 }
 
 /* Increase or decrease box size by a radius in at least n dimensions. */
diff --git a/contrib/cube/expected/cube.out b/contrib/cube/expected/cube.out
index c430b4e1f0c..c586a73727d 100644
--- a/contrib/cube/expected/cube.out
+++ b/contrib/cube/expected/cube.out
@@ -1532,25 +1532,25 @@ SELECT cube(array[40,50,60], array[10,20,30])~>1;
 SELECT cube(array[10,20,30], array[40,50,60])~>2;
  ?column? 
 ----------
-       20
+       40
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>2;
  ?column? 
 ----------
-       20
+       40
 (1 row)
 
 SELECT cube(array[10,20,30], array[40,50,60])~>3;
  ?column? 
 ----------
-       30
+       20
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>3;
  ?column? 
 ----------
-       30
+       20
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>0;
@@ -1558,7 +1558,7 @@ ERROR:  cube index 0 is out of bounds
 SELECT cube(array[40,50,60], array[10,20,30])~>4;
  ?column? 
 ----------
-       40
+       50
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>(-1);
@@ -1611,25 +1611,28 @@ SELECT c FROM test_cube WHERE c <@ '(3000,1000),(0,0)' ORDER BY c;
 (4 rows)
 
 RESET enable_bitmapscan;
--- kNN with index
+-- Test kNN
+INSERT INTO test_cube VALUES ('(1,1)'), ('(100000)'), ('(0, 100000)'); -- Some corner cases
+SET enable_seqscan = false;
+-- Test different metrics
 SELECT *, c <-> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <-> '(100, 100),(500, 500)'::cube LIMIT 5;
             c            |       dist       
 -------------------------+------------------
  (337, 455),(240, 359)   |                0
+ (1, 1)                  | 140.007142674936
  (759, 187),(662, 163)   |              162
  (948, 1201),(907, 1156) | 772.000647668122
  (1444, 403),(1346, 344) |              846
- (369, 1457),(278, 1409) |              909
 (5 rows)
 
 SELECT *, c <=> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <=> '(100, 100),(500, 500)'::cube LIMIT 5;
             c            | dist 
 -------------------------+------
  (337, 455),(240, 359)   |    0
+ (1, 1)                  |   99
  (759, 187),(662, 163)   |  162
  (948, 1201),(907, 1156) |  656
  (1444, 403),(1346, 344) |  846
- (369, 1457),(278, 1409) |  909
 (5 rows)
 
 SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <#> '(100, 100),(500, 500)'::cube LIMIT 5;
@@ -1637,133 +1640,203 @@ SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c
 -------------------------+------
  (337, 455),(240, 359)   |    0
  (759, 187),(662, 163)   |  162
+ (1, 1)                  |  198
  (1444, 403),(1346, 344) |  846
  (369, 1457),(278, 1409) |  909
- (948, 1201),(907, 1156) | 1063
 (5 rows)
 
--- kNN-based sorting
-SELECT * FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by 1st coordinate of lower left corner
-             c             
----------------------------
- (54, 38679),(3, 38602)
- (83, 10271),(15, 10265)
- (122, 46832),(64, 46762)
- (167, 17214),(92, 17184)
- (161, 24465),(107, 24374)
- (162, 26040),(120, 25963)
- (154, 4019),(138, 3990)
- (259, 1850),(175, 1820)
- (207, 40886),(179, 40879)
- (288, 49588),(204, 49571)
- (270, 32616),(226, 32607)
- (318, 31489),(235, 31404)
- (337, 455),(240, 359)
- (270, 29508),(264, 29440)
- (369, 1457),(278, 1409)
+-- Test sorting by coordinates
+SELECT c~>1, c FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by left bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+        3 | (54, 38679),(3, 38602)
+       15 | (83, 10271),(15, 10265)
+       64 | (122, 46832),(64, 46762)
+       92 | (167, 17214),(92, 17184)
+      107 | (161, 24465),(107, 24374)
+      120 | (162, 26040),(120, 25963)
+      138 | (154, 4019),(138, 3990)
+      175 | (259, 1850),(175, 1820)
+      179 | (207, 40886),(179, 40879)
+      204 | (288, 49588),(204, 49571)
+      226 | (270, 32616),(226, 32607)
+      235 | (318, 31489),(235, 31404)
+      240 | (337, 455),(240, 359)
 (15 rows)
 
-SELECT * FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by 2nd coordinate or upper right corner
-             c             
----------------------------
- (30333, 50),(30273, 6)
- (43301, 75),(43227, 43)
- (19650, 142),(19630, 51)
- (2424, 160),(2424, 81)
- (3449, 171),(3354, 108)
- (18037, 155),(17941, 109)
- (28511, 208),(28479, 114)
- (19946, 217),(19941, 118)
- (16906, 191),(16816, 139)
- (759, 187),(662, 163)
- (22684, 266),(22656, 181)
- (24423, 255),(24360, 213)
- (45989, 249),(45910, 222)
- (11399, 377),(11360, 294)
- (12162, 389),(12103, 309)
+SELECT c~>2, c FROM test_cube ORDER BY c~>2 LIMIT 15; -- ascending by right bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+       54 | (54, 38679),(3, 38602)
+       83 | (83, 10271),(15, 10265)
+      122 | (122, 46832),(64, 46762)
+      154 | (154, 4019),(138, 3990)
+      161 | (161, 24465),(107, 24374)
+      162 | (162, 26040),(120, 25963)
+      167 | (167, 17214),(92, 17184)
+      207 | (207, 40886),(179, 40879)
+      259 | (259, 1850),(175, 1820)
+      270 | (270, 29508),(264, 29440)
+      270 | (270, 32616),(226, 32607)
+      288 | (288, 49588),(204, 49571)
+      318 | (318, 31489),(235, 31404)
 (15 rows)
 
-SELECT * FROM test_cube ORDER BY c~>1 DESC LIMIT 15; -- descending by 1st coordinate of lower left corner
-               c               
--------------------------------
- (50027, 49230),(49951, 49214)
- (49980, 35004),(49937, 34963)
- (49985, 6436),(49927, 6338)
- (49999, 27218),(49908, 27176)
- (49954, 1340),(49905, 1294)
- (49944, 25163),(49902, 25153)
- (49981, 34876),(49898, 34786)
- (49957, 43390),(49897, 43384)
- (49853, 18504),(49848, 18503)
- (49902, 41752),(49818, 41746)
- (49907, 30225),(49810, 30158)
- (49843, 5175),(49808, 5145)
- (49887, 24274),(49805, 24184)
- (49847, 7128),(49798, 7067)
- (49820, 7990),(49771, 7967)
+SELECT c~>3, c FROM test_cube ORDER BY c~>3 LIMIT 15; -- ascending by lower bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+        6 | (30333, 50),(30273, 6)
+       43 | (43301, 75),(43227, 43)
+       51 | (19650, 142),(19630, 51)
+       81 | (2424, 160),(2424, 81)
+      108 | (3449, 171),(3354, 108)
+      109 | (18037, 155),(17941, 109)
+      114 | (28511, 208),(28479, 114)
+      118 | (19946, 217),(19941, 118)
+      139 | (16906, 191),(16816, 139)
+      163 | (759, 187),(662, 163)
+      181 | (22684, 266),(22656, 181)
+      213 | (24423, 255),(24360, 213)
+      222 | (45989, 249),(45910, 222)
 (15 rows)
 
-SELECT * FROM test_cube ORDER BY c~>4 DESC LIMIT 15; -- descending by 2nd coordinate or upper right corner
-               c               
--------------------------------
- (36311, 50073),(36258, 49987)
- (30746, 50040),(30727, 49992)
- (2168, 50012),(2108, 49914)
- (21551, 49983),(21492, 49885)
- (17954, 49975),(17865, 49915)
- (3531, 49962),(3463, 49934)
- (19128, 49932),(19112, 49849)
- (31287, 49923),(31236, 49913)
- (43925, 49912),(43888, 49878)
- (29261, 49910),(29247, 49818)
- (14913, 49873),(14849, 49836)
- (20007, 49858),(19921, 49778)
- (38266, 49852),(38233, 49844)
- (37595, 49849),(37581, 49834)
- (46151, 49848),(46058, 49830)
+SELECT c~>4, c FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by upper bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+       50 | (30333, 50),(30273, 6)
+       75 | (43301, 75),(43227, 43)
+      142 | (19650, 142),(19630, 51)
+      155 | (18037, 155),(17941, 109)
+      160 | (2424, 160),(2424, 81)
+      171 | (3449, 171),(3354, 108)
+      187 | (759, 187),(662, 163)
+      191 | (16906, 191),(16816, 139)
+      208 | (28511, 208),(28479, 114)
+      217 | (19946, 217),(19941, 118)
+      249 | (45989, 249),(45910, 222)
+      255 | (24423, 255),(24360, 213)
+      266 | (22684, 266),(22656, 181)
 (15 rows)
 
--- same thing for index with points
-CREATE TABLE test_point(c cube);
-INSERT INTO test_point(SELECT cube(array[c->1,c->2,c->3,c->4]) FROM test_cube);
-CREATE INDEX ON test_point USING gist(c);
-SELECT * FROM test_point ORDER BY c~>1, c~>2 LIMIT 15; -- ascending by 1st then by 2nd coordinate
-            c             
---------------------------
- (54, 38679, 3, 38602)
- (83, 10271, 15, 10265)
- (122, 46832, 64, 46762)
- (154, 4019, 138, 3990)
- (161, 24465, 107, 24374)
- (162, 26040, 120, 25963)
- (167, 17214, 92, 17184)
- (207, 40886, 179, 40879)
- (259, 1850, 175, 1820)
- (270, 29508, 264, 29440)
- (270, 32616, 226, 32607)
- (288, 49588, 204, 49571)
- (318, 31489, 235, 31404)
- (326, 18837, 285, 18817)
- (337, 455, 240, 359)
+-- Same queries with sequential scan (should give the same results as above)
+RESET enable_seqscan;
+SET enable_indexscan = OFF;
+SELECT *, c <-> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <-> '(100, 100),(500, 500)'::cube LIMIT 5;
+            c            |       dist       
+-------------------------+------------------
+ (337, 455),(240, 359)   |                0
+ (1, 1)                  | 140.007142674936
+ (759, 187),(662, 163)   |              162
+ (948, 1201),(907, 1156) | 772.000647668122
+ (1444, 403),(1346, 344) |              846
+(5 rows)
+
+SELECT *, c <=> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <=> '(100, 100),(500, 500)'::cube LIMIT 5;
+            c            | dist 
+-------------------------+------
+ (337, 455),(240, 359)   |    0
+ (1, 1)                  |   99
+ (759, 187),(662, 163)   |  162
+ (948, 1201),(907, 1156) |  656
+ (1444, 403),(1346, 344) |  846
+(5 rows)
+
+SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <#> '(100, 100),(500, 500)'::cube LIMIT 5;
+            c            | dist 
+-------------------------+------
+ (337, 455),(240, 359)   |    0
+ (759, 187),(662, 163)   |  162
+ (1, 1)                  |  198
+ (1444, 403),(1346, 344) |  846
+ (369, 1457),(278, 1409) |  909
+(5 rows)
+
+SELECT c~>1, c FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by left bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+        3 | (54, 38679),(3, 38602)
+       15 | (83, 10271),(15, 10265)
+       64 | (122, 46832),(64, 46762)
+       92 | (167, 17214),(92, 17184)
+      107 | (161, 24465),(107, 24374)
+      120 | (162, 26040),(120, 25963)
+      138 | (154, 4019),(138, 3990)
+      175 | (259, 1850),(175, 1820)
+      179 | (207, 40886),(179, 40879)
+      204 | (288, 49588),(204, 49571)
+      226 | (270, 32616),(226, 32607)
+      235 | (318, 31489),(235, 31404)
+      240 | (337, 455),(240, 359)
+(15 rows)
+
+SELECT c~>2, c FROM test_cube ORDER BY c~>2 LIMIT 15; -- ascending by right bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+       54 | (54, 38679),(3, 38602)
+       83 | (83, 10271),(15, 10265)
+      122 | (122, 46832),(64, 46762)
+      154 | (154, 4019),(138, 3990)
+      161 | (161, 24465),(107, 24374)
+      162 | (162, 26040),(120, 25963)
+      167 | (167, 17214),(92, 17184)
+      207 | (207, 40886),(179, 40879)
+      259 | (259, 1850),(175, 1820)
+      270 | (270, 29508),(264, 29440)
+      270 | (270, 32616),(226, 32607)
+      288 | (288, 49588),(204, 49571)
+      318 | (318, 31489),(235, 31404)
+(15 rows)
+
+SELECT c~>3, c FROM test_cube ORDER BY c~>3 LIMIT 15; -- ascending by lower bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+        6 | (30333, 50),(30273, 6)
+       43 | (43301, 75),(43227, 43)
+       51 | (19650, 142),(19630, 51)
+       81 | (2424, 160),(2424, 81)
+      108 | (3449, 171),(3354, 108)
+      109 | (18037, 155),(17941, 109)
+      114 | (28511, 208),(28479, 114)
+      118 | (19946, 217),(19941, 118)
+      139 | (16906, 191),(16816, 139)
+      163 | (759, 187),(662, 163)
+      181 | (22684, 266),(22656, 181)
+      213 | (24423, 255),(24360, 213)
+      222 | (45989, 249),(45910, 222)
 (15 rows)
 
-SELECT * FROM test_point ORDER BY c~>4 DESC LIMIT 15; -- descending by 1st coordinate
-              c               
-------------------------------
- (30746, 50040, 30727, 49992)
- (36311, 50073, 36258, 49987)
- (3531, 49962, 3463, 49934)
- (17954, 49975, 17865, 49915)
- (2168, 50012, 2108, 49914)
- (31287, 49923, 31236, 49913)
- (21551, 49983, 21492, 49885)
- (43925, 49912, 43888, 49878)
- (19128, 49932, 19112, 49849)
- (38266, 49852, 38233, 49844)
- (14913, 49873, 14849, 49836)
- (37595, 49849, 37581, 49834)
- (46151, 49848, 46058, 49830)
- (29261, 49910, 29247, 49818)
- (19233, 49824, 19185, 49794)
+SELECT c~>4, c FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by upper bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+       50 | (30333, 50),(30273, 6)
+       75 | (43301, 75),(43227, 43)
+      142 | (19650, 142),(19630, 51)
+      155 | (18037, 155),(17941, 109)
+      160 | (2424, 160),(2424, 81)
+      171 | (3449, 171),(3354, 108)
+      187 | (759, 187),(662, 163)
+      191 | (16906, 191),(16816, 139)
+      208 | (28511, 208),(28479, 114)
+      217 | (19946, 217),(19941, 118)
+      249 | (45989, 249),(45910, 222)
+      255 | (24423, 255),(24360, 213)
+      266 | (22684, 266),(22656, 181)
 (15 rows)
 
+RESET enable_indexscan;
diff --git a/contrib/cube/expected/cube_2.out b/contrib/cube/expected/cube_2.out
index b979c4d6c88..8c75e27b465 100644
--- a/contrib/cube/expected/cube_2.out
+++ b/contrib/cube/expected/cube_2.out
@@ -1532,25 +1532,25 @@ SELECT cube(array[40,50,60], array[10,20,30])~>1;
 SELECT cube(array[10,20,30], array[40,50,60])~>2;
  ?column? 
 ----------
-       20
+       40
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>2;
  ?column? 
 ----------
-       20
+       40
 (1 row)
 
 SELECT cube(array[10,20,30], array[40,50,60])~>3;
  ?column? 
 ----------
-       30
+       20
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>3;
  ?column? 
 ----------
-       30
+       20
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>0;
@@ -1558,7 +1558,7 @@ ERROR:  cube index 0 is out of bounds
 SELECT cube(array[40,50,60], array[10,20,30])~>4;
  ?column? 
 ----------
-       40
+       50
 (1 row)
 
 SELECT cube(array[40,50,60], array[10,20,30])~>(-1);
@@ -1611,25 +1611,28 @@ SELECT c FROM test_cube WHERE c <@ '(3000,1000),(0,0)' ORDER BY c;
 (4 rows)
 
 RESET enable_bitmapscan;
--- kNN with index
+-- Test kNN
+INSERT INTO test_cube VALUES ('(1,1)'), ('(100000)'), ('(0, 100000)'); -- Some corner cases
+SET enable_seqscan = false;
+-- Test different metrics
 SELECT *, c <-> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <-> '(100, 100),(500, 500)'::cube LIMIT 5;
             c            |       dist       
 -------------------------+------------------
  (337, 455),(240, 359)   |                0
+ (1, 1)                  | 140.007142674936
  (759, 187),(662, 163)   |              162
  (948, 1201),(907, 1156) | 772.000647668122
  (1444, 403),(1346, 344) |              846
- (369, 1457),(278, 1409) |              909
 (5 rows)
 
 SELECT *, c <=> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <=> '(100, 100),(500, 500)'::cube LIMIT 5;
             c            | dist 
 -------------------------+------
  (337, 455),(240, 359)   |    0
+ (1, 1)                  |   99
  (759, 187),(662, 163)   |  162
  (948, 1201),(907, 1156) |  656
  (1444, 403),(1346, 344) |  846
- (369, 1457),(278, 1409) |  909
 (5 rows)
 
 SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <#> '(100, 100),(500, 500)'::cube LIMIT 5;
@@ -1637,133 +1640,203 @@ SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c
 -------------------------+------
  (337, 455),(240, 359)   |    0
  (759, 187),(662, 163)   |  162
+ (1, 1)                  |  198
  (1444, 403),(1346, 344) |  846
  (369, 1457),(278, 1409) |  909
- (948, 1201),(907, 1156) | 1063
 (5 rows)
 
--- kNN-based sorting
-SELECT * FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by 1st coordinate of lower left corner
-             c             
----------------------------
- (54, 38679),(3, 38602)
- (83, 10271),(15, 10265)
- (122, 46832),(64, 46762)
- (167, 17214),(92, 17184)
- (161, 24465),(107, 24374)
- (162, 26040),(120, 25963)
- (154, 4019),(138, 3990)
- (259, 1850),(175, 1820)
- (207, 40886),(179, 40879)
- (288, 49588),(204, 49571)
- (270, 32616),(226, 32607)
- (318, 31489),(235, 31404)
- (337, 455),(240, 359)
- (270, 29508),(264, 29440)
- (369, 1457),(278, 1409)
+-- Test sorting by coordinates
+SELECT c~>1, c FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by left bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+        3 | (54, 38679),(3, 38602)
+       15 | (83, 10271),(15, 10265)
+       64 | (122, 46832),(64, 46762)
+       92 | (167, 17214),(92, 17184)
+      107 | (161, 24465),(107, 24374)
+      120 | (162, 26040),(120, 25963)
+      138 | (154, 4019),(138, 3990)
+      175 | (259, 1850),(175, 1820)
+      179 | (207, 40886),(179, 40879)
+      204 | (288, 49588),(204, 49571)
+      226 | (270, 32616),(226, 32607)
+      235 | (318, 31489),(235, 31404)
+      240 | (337, 455),(240, 359)
 (15 rows)
 
-SELECT * FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by 2nd coordinate or upper right corner
-             c             
----------------------------
- (30333, 50),(30273, 6)
- (43301, 75),(43227, 43)
- (19650, 142),(19630, 51)
- (2424, 160),(2424, 81)
- (3449, 171),(3354, 108)
- (18037, 155),(17941, 109)
- (28511, 208),(28479, 114)
- (19946, 217),(19941, 118)
- (16906, 191),(16816, 139)
- (759, 187),(662, 163)
- (22684, 266),(22656, 181)
- (24423, 255),(24360, 213)
- (45989, 249),(45910, 222)
- (11399, 377),(11360, 294)
- (12162, 389),(12103, 309)
+SELECT c~>2, c FROM test_cube ORDER BY c~>2 LIMIT 15; -- ascending by right bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+       54 | (54, 38679),(3, 38602)
+       83 | (83, 10271),(15, 10265)
+      122 | (122, 46832),(64, 46762)
+      154 | (154, 4019),(138, 3990)
+      161 | (161, 24465),(107, 24374)
+      162 | (162, 26040),(120, 25963)
+      167 | (167, 17214),(92, 17184)
+      207 | (207, 40886),(179, 40879)
+      259 | (259, 1850),(175, 1820)
+      270 | (270, 29508),(264, 29440)
+      270 | (270, 32616),(226, 32607)
+      288 | (288, 49588),(204, 49571)
+      318 | (318, 31489),(235, 31404)
 (15 rows)
 
-SELECT * FROM test_cube ORDER BY c~>1 DESC LIMIT 15; -- descending by 1st coordinate of lower left corner
-               c               
--------------------------------
- (50027, 49230),(49951, 49214)
- (49980, 35004),(49937, 34963)
- (49985, 6436),(49927, 6338)
- (49999, 27218),(49908, 27176)
- (49954, 1340),(49905, 1294)
- (49944, 25163),(49902, 25153)
- (49981, 34876),(49898, 34786)
- (49957, 43390),(49897, 43384)
- (49853, 18504),(49848, 18503)
- (49902, 41752),(49818, 41746)
- (49907, 30225),(49810, 30158)
- (49843, 5175),(49808, 5145)
- (49887, 24274),(49805, 24184)
- (49847, 7128),(49798, 7067)
- (49820, 7990),(49771, 7967)
+SELECT c~>3, c FROM test_cube ORDER BY c~>3 LIMIT 15; -- ascending by lower bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+        6 | (30333, 50),(30273, 6)
+       43 | (43301, 75),(43227, 43)
+       51 | (19650, 142),(19630, 51)
+       81 | (2424, 160),(2424, 81)
+      108 | (3449, 171),(3354, 108)
+      109 | (18037, 155),(17941, 109)
+      114 | (28511, 208),(28479, 114)
+      118 | (19946, 217),(19941, 118)
+      139 | (16906, 191),(16816, 139)
+      163 | (759, 187),(662, 163)
+      181 | (22684, 266),(22656, 181)
+      213 | (24423, 255),(24360, 213)
+      222 | (45989, 249),(45910, 222)
 (15 rows)
 
-SELECT * FROM test_cube ORDER BY c~>4 DESC LIMIT 15; -- descending by 2nd coordinate or upper right corner
-               c               
--------------------------------
- (36311, 50073),(36258, 49987)
- (30746, 50040),(30727, 49992)
- (2168, 50012),(2108, 49914)
- (21551, 49983),(21492, 49885)
- (17954, 49975),(17865, 49915)
- (3531, 49962),(3463, 49934)
- (19128, 49932),(19112, 49849)
- (31287, 49923),(31236, 49913)
- (43925, 49912),(43888, 49878)
- (29261, 49910),(29247, 49818)
- (14913, 49873),(14849, 49836)
- (20007, 49858),(19921, 49778)
- (38266, 49852),(38233, 49844)
- (37595, 49849),(37581, 49834)
- (46151, 49848),(46058, 49830)
+SELECT c~>4, c FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by upper bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+       50 | (30333, 50),(30273, 6)
+       75 | (43301, 75),(43227, 43)
+      142 | (19650, 142),(19630, 51)
+      155 | (18037, 155),(17941, 109)
+      160 | (2424, 160),(2424, 81)
+      171 | (3449, 171),(3354, 108)
+      187 | (759, 187),(662, 163)
+      191 | (16906, 191),(16816, 139)
+      208 | (28511, 208),(28479, 114)
+      217 | (19946, 217),(19941, 118)
+      249 | (45989, 249),(45910, 222)
+      255 | (24423, 255),(24360, 213)
+      266 | (22684, 266),(22656, 181)
 (15 rows)
 
--- same thing for index with points
-CREATE TABLE test_point(c cube);
-INSERT INTO test_point(SELECT cube(array[c->1,c->2,c->3,c->4]) FROM test_cube);
-CREATE INDEX ON test_point USING gist(c);
-SELECT * FROM test_point ORDER BY c~>1, c~>2 LIMIT 15; -- ascending by 1st then by 2nd coordinate
-            c             
---------------------------
- (54, 38679, 3, 38602)
- (83, 10271, 15, 10265)
- (122, 46832, 64, 46762)
- (154, 4019, 138, 3990)
- (161, 24465, 107, 24374)
- (162, 26040, 120, 25963)
- (167, 17214, 92, 17184)
- (207, 40886, 179, 40879)
- (259, 1850, 175, 1820)
- (270, 29508, 264, 29440)
- (270, 32616, 226, 32607)
- (288, 49588, 204, 49571)
- (318, 31489, 235, 31404)
- (326, 18837, 285, 18817)
- (337, 455, 240, 359)
+-- Same queries with sequential scan (should give the same results as above)
+RESET enable_seqscan;
+SET enable_indexscan = OFF;
+SELECT *, c <-> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <-> '(100, 100),(500, 500)'::cube LIMIT 5;
+            c            |       dist       
+-------------------------+------------------
+ (337, 455),(240, 359)   |                0
+ (1, 1)                  | 140.007142674936
+ (759, 187),(662, 163)   |              162
+ (948, 1201),(907, 1156) | 772.000647668122
+ (1444, 403),(1346, 344) |              846
+(5 rows)
+
+SELECT *, c <=> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <=> '(100, 100),(500, 500)'::cube LIMIT 5;
+            c            | dist 
+-------------------------+------
+ (337, 455),(240, 359)   |    0
+ (1, 1)                  |   99
+ (759, 187),(662, 163)   |  162
+ (948, 1201),(907, 1156) |  656
+ (1444, 403),(1346, 344) |  846
+(5 rows)
+
+SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <#> '(100, 100),(500, 500)'::cube LIMIT 5;
+            c            | dist 
+-------------------------+------
+ (337, 455),(240, 359)   |    0
+ (759, 187),(662, 163)   |  162
+ (1, 1)                  |  198
+ (1444, 403),(1346, 344) |  846
+ (369, 1457),(278, 1409) |  909
+(5 rows)
+
+SELECT c~>1, c FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by left bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+        3 | (54, 38679),(3, 38602)
+       15 | (83, 10271),(15, 10265)
+       64 | (122, 46832),(64, 46762)
+       92 | (167, 17214),(92, 17184)
+      107 | (161, 24465),(107, 24374)
+      120 | (162, 26040),(120, 25963)
+      138 | (154, 4019),(138, 3990)
+      175 | (259, 1850),(175, 1820)
+      179 | (207, 40886),(179, 40879)
+      204 | (288, 49588),(204, 49571)
+      226 | (270, 32616),(226, 32607)
+      235 | (318, 31489),(235, 31404)
+      240 | (337, 455),(240, 359)
+(15 rows)
+
+SELECT c~>2, c FROM test_cube ORDER BY c~>2 LIMIT 15; -- ascending by right bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (0, 100000)
+        1 | (1, 1)
+       54 | (54, 38679),(3, 38602)
+       83 | (83, 10271),(15, 10265)
+      122 | (122, 46832),(64, 46762)
+      154 | (154, 4019),(138, 3990)
+      161 | (161, 24465),(107, 24374)
+      162 | (162, 26040),(120, 25963)
+      167 | (167, 17214),(92, 17184)
+      207 | (207, 40886),(179, 40879)
+      259 | (259, 1850),(175, 1820)
+      270 | (270, 29508),(264, 29440)
+      270 | (270, 32616),(226, 32607)
+      288 | (288, 49588),(204, 49571)
+      318 | (318, 31489),(235, 31404)
+(15 rows)
+
+SELECT c~>3, c FROM test_cube ORDER BY c~>3 LIMIT 15; -- ascending by lower bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+        6 | (30333, 50),(30273, 6)
+       43 | (43301, 75),(43227, 43)
+       51 | (19650, 142),(19630, 51)
+       81 | (2424, 160),(2424, 81)
+      108 | (3449, 171),(3354, 108)
+      109 | (18037, 155),(17941, 109)
+      114 | (28511, 208),(28479, 114)
+      118 | (19946, 217),(19941, 118)
+      139 | (16906, 191),(16816, 139)
+      163 | (759, 187),(662, 163)
+      181 | (22684, 266),(22656, 181)
+      213 | (24423, 255),(24360, 213)
+      222 | (45989, 249),(45910, 222)
 (15 rows)
 
-SELECT * FROM test_point ORDER BY c~>4 DESC LIMIT 15; -- descending by 1st coordinate
-              c               
-------------------------------
- (30746, 50040, 30727, 49992)
- (36311, 50073, 36258, 49987)
- (3531, 49962, 3463, 49934)
- (17954, 49975, 17865, 49915)
- (2168, 50012, 2108, 49914)
- (31287, 49923, 31236, 49913)
- (21551, 49983, 21492, 49885)
- (43925, 49912, 43888, 49878)
- (19128, 49932, 19112, 49849)
- (38266, 49852, 38233, 49844)
- (14913, 49873, 14849, 49836)
- (37595, 49849, 37581, 49834)
- (46151, 49848, 46058, 49830)
- (29261, 49910, 29247, 49818)
- (19233, 49824, 19185, 49794)
+SELECT c~>4, c FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by upper bound
+ ?column? |             c             
+----------+---------------------------
+        0 | (100000)
+        1 | (1, 1)
+       50 | (30333, 50),(30273, 6)
+       75 | (43301, 75),(43227, 43)
+      142 | (19650, 142),(19630, 51)
+      155 | (18037, 155),(17941, 109)
+      160 | (2424, 160),(2424, 81)
+      171 | (3449, 171),(3354, 108)
+      187 | (759, 187),(662, 163)
+      191 | (16906, 191),(16816, 139)
+      208 | (28511, 208),(28479, 114)
+      217 | (19946, 217),(19941, 118)
+      249 | (45989, 249),(45910, 222)
+      255 | (24423, 255),(24360, 213)
+      266 | (22684, 266),(22656, 181)
 (15 rows)
 
+RESET enable_indexscan;
diff --git a/contrib/cube/sql/cube.sql b/contrib/cube/sql/cube.sql
index eb245768959..efa1dbe9e89 100644
--- a/contrib/cube/sql/cube.sql
+++ b/contrib/cube/sql/cube.sql
@@ -389,20 +389,29 @@ SELECT c FROM test_cube WHERE c <@ '(3000,1000),(0,0)' ORDER BY c;
 SELECT c FROM test_cube WHERE c <@ '(3000,1000),(0,0)' ORDER BY c;
 RESET enable_bitmapscan;
 
--- kNN with index
+-- Test kNN
+INSERT INTO test_cube VALUES ('(1,1)'), ('(100000)'), ('(0, 100000)'); -- Some corner cases
+SET enable_seqscan = false;
+
+-- Test different metrics
 SELECT *, c <-> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <-> '(100, 100),(500, 500)'::cube LIMIT 5;
 SELECT *, c <=> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <=> '(100, 100),(500, 500)'::cube LIMIT 5;
 SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <#> '(100, 100),(500, 500)'::cube LIMIT 5;
 
--- kNN-based sorting
-SELECT * FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by 1st coordinate of lower left corner
-SELECT * FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by 2nd coordinate or upper right corner
-SELECT * FROM test_cube ORDER BY c~>1 DESC LIMIT 15; -- descending by 1st coordinate of lower left corner
-SELECT * FROM test_cube ORDER BY c~>4 DESC LIMIT 15; -- descending by 2nd coordinate or upper right corner
-
--- same thing for index with points
-CREATE TABLE test_point(c cube);
-INSERT INTO test_point(SELECT cube(array[c->1,c->2,c->3,c->4]) FROM test_cube);
-CREATE INDEX ON test_point USING gist(c);
-SELECT * FROM test_point ORDER BY c~>1, c~>2 LIMIT 15; -- ascending by 1st then by 2nd coordinate
-SELECT * FROM test_point ORDER BY c~>4 DESC LIMIT 15; -- descending by 1st coordinate
+-- Test sorting by coordinates
+SELECT c~>1, c FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by left bound
+SELECT c~>2, c FROM test_cube ORDER BY c~>2 LIMIT 15; -- ascending by right bound
+SELECT c~>3, c FROM test_cube ORDER BY c~>3 LIMIT 15; -- ascending by lower bound
+SELECT c~>4, c FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by upper bound
+
+-- Same queries with sequential scan (should give the same results as above)
+RESET enable_seqscan;
+SET enable_indexscan = OFF;
+SELECT *, c <-> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <-> '(100, 100),(500, 500)'::cube LIMIT 5;
+SELECT *, c <=> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <=> '(100, 100),(500, 500)'::cube LIMIT 5;
+SELECT *, c <#> '(100, 100),(500, 500)'::cube as dist FROM test_cube ORDER BY c <#> '(100, 100),(500, 500)'::cube LIMIT 5;
+SELECT c~>1, c FROM test_cube ORDER BY c~>1 LIMIT 15; -- ascending by left bound
+SELECT c~>2, c FROM test_cube ORDER BY c~>2 LIMIT 15; -- ascending by right bound
+SELECT c~>3, c FROM test_cube ORDER BY c~>3 LIMIT 15; -- ascending by lower bound
+SELECT c~>4, c FROM test_cube ORDER BY c~>4 LIMIT 15; -- ascending by upper bound
+RESET enable_indexscan;
diff --git a/doc/src/sgml/cube.sgml b/doc/src/sgml/cube.sgml
index b995dc7e2ad..9cda8cac972 100644
--- a/doc/src/sgml/cube.sgml
+++ b/doc/src/sgml/cube.sgml
@@ -186,10 +186,11 @@
       <entry><literal>a ~&gt; n</literal></entry>
       <entry><type>float8</type></entry>
       <entry>
-        Get <replaceable>n</replaceable>-th coordinate in <quote>normalized</quote> cube
-        representation, in which the coordinates have been rearranged into
-        the form <quote>lower left &mdash; upper right</quote>; that is, the
-        smaller endpoint along each dimension appears first.
+        Get <replaceable>n</replaceable>-th coordinate of cube in following way:
+        n = 2 * k - 1 means lower bound of <replaceable>k</replaceable>-th
+        dimension, n = 2 * k means upper bound of
+        <replaceable>k</replaceable>-th dimension.  This operator is designed
+        for KNN-GiST support.
       </entry>
      </row>