1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
|
/* -*- mode: c++; c-basic-offset: 2; indent-tabs-mode: nil; -*-
* vim:expandtab:shiftwidth=2:tabstop=2:smarttab:
*
* Copyright (C) 2008-2009 Sun Microsystems, Inc.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; version 2 of the License.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
*/
#pragma once
namespace drizzled {
namespace optimizer {
/*
A construction block of the SEL_ARG-graph.
The following description only covers graphs of SEL_ARG objects with
sel_arg->type==KEY_RANGE:
One SEL_ARG object represents an "elementary interval" in form
min_value <=? table.keypartX <=? max_value
The interval is a non-empty interval of any kind: with[out] minimum/maximum
bound, [half]open/closed, single-point interval, etc.
1. SEL_ARG GRAPH STRUCTURE
SEL_ARG objects are linked together in a graph. The meaning of the graph
is better demostrated by an example:
tree->keys[i]
|
| $ $
| part=1 $ part=2 $ part=3
| $ $
| +-------+ $ +-------+ $ +--------+
| | kp1<1 |--$-->| kp2=5 |--$-->| kp3=10 |
| +-------+ $ +-------+ $ +--------+
| | $ $ |
| | $ $ +--------+
| | $ $ | kp3=12 |
| | $ $ +--------+
| +-------+ $ $
\->| kp1=2 |--$--------------$-+
+-------+ $ $ | +--------+
| $ $ ==>| kp3=11 |
+-------+ $ $ | +--------+
| kp1=3 |--$--------------$-+ |
+-------+ $ $ +--------+
| $ $ | kp3=14 |
... $ $ +--------+
The entire graph is partitioned into "interval lists".
An interval list is a sequence of ordered disjoint intervals over the same
key part. SEL_ARG are linked via "next" and "prev" pointers. Additionally,
all intervals in the list form an RB-tree, linked via left/right/parent
pointers. The RB-tree root SEL_ARG object will be further called "root of the
interval list".
In the example pic, there are 4 interval lists:
"kp<1 OR kp1=2 OR kp1=3", "kp2=5", "kp3=10 OR kp3=12", "kp3=11 OR kp3=13".
The vertical lines represent SEL_ARG::next/prev pointers.
In an interval list, each member X may have SEL_ARG::next_key_part pointer
pointing to the root of another interval list Y. The pointed interval list
must cover a key part with greater number (i.e. Y->part > X->part).
In the example pic, the next_key_part pointers are represented by
horisontal lines.
2. SEL_ARG GRAPH SEMANTICS
It represents a condition in a special form (we don't have a name for it ATM)
The SEL_ARG::next/prev is "OR", and next_key_part is "AND".
For example, the picture represents the condition in form:
(kp1 < 1 AND kp2=5 AND (kp3=10 OR kp3=12)) OR
(kp1=2 AND (kp3=11 OR kp3=14)) OR
(kp1=3 AND (kp3=11 OR kp3=14))
3. SEL_ARG GRAPH USE
Use get_mm_tree() to construct SEL_ARG graph from WHERE condition.
Then walk the SEL_ARG graph and get a list of dijsoint ordered key
intervals (i.e. intervals in form
(constA1, .., const1_K) < (keypart1,.., keypartK) < (constB1, .., constB_K)
Those intervals can be used to access the index. The uses are in:
- check_quick_select() - Walk the SEL_ARG graph and find an estimate of
how many table records are contained within all
intervals.
- get_quick_select() - Walk the SEL_ARG, materialize the key intervals,
and create QuickRangeSelect object that will
read records within these intervals.
4. SPACE COMPLEXITY NOTES
SEL_ARG graph is a representation of an ordered disjoint sequence of
intervals over the ordered set of index tuple values.
For multi-part keys, one can construct a WHERE expression such that its
list of intervals will be of combinatorial size. Here is an example:
(keypart1 IN (1,2, ..., n1)) AND
(keypart2 IN (1,2, ..., n2)) AND
(keypart3 IN (1,2, ..., n3))
For this WHERE clause the list of intervals will have n1*n2*n3 intervals
of form
(keypart1, keypart2, keypart3) = (k1, k2, k3), where 1 <= k{i} <= n{i}
SEL_ARG graph structure aims to reduce the amount of required space by
"sharing" the elementary intervals when possible (the pic at the
beginning of this comment has examples of such sharing). The sharing may
prevent combinatorial blowup:
There are WHERE clauses that have combinatorial-size interval lists but
will be represented by a compact SEL_ARG graph.
Example:
(keypartN IN (1,2, ..., n1)) AND
...
(keypart2 IN (1,2, ..., n2)) AND
(keypart1 IN (1,2, ..., n3))
but not in all cases:
- There are WHERE clauses that do have a compact SEL_ARG-graph
representation but get_mm_tree() and its callees will construct a
graph of combinatorial size.
Example:
(keypart1 IN (1,2, ..., n1)) AND
(keypart2 IN (1,2, ..., n2)) AND
...
(keypartN IN (1,2, ..., n3))
- There are WHERE clauses for which the minimal possible SEL_ARG graph
representation will have combinatorial size.
Example:
By induction: Let's take any interval on some keypart in the middle:
kp15=c0
Then let's AND it with this interval 'structure' from preceding and
following keyparts:
(kp14=c1 AND kp16=c3) OR keypart14=c2) (*)
We will obtain this SEL_ARG graph:
kp14 $ kp15 $ kp16
$ $
+---------+ $ +---------+ $ +---------+
| kp14=c1 |--$-->| kp15=c0 |--$-->| kp16=c3 |
+---------+ $ +---------+ $ +---------+
| $ $
+---------+ $ +---------+ $
| kp14=c2 |--$-->| kp15=c0 | $
+---------+ $ +---------+ $
$ $
Note that we had to duplicate "kp15=c0" and there was no way to avoid
that.
The induction step: AND the obtained expression with another "wrapping"
expression like (*).
When the process ends because of the limit on max. number of keyparts
we'll have:
WHERE clause length is O(3*#max_keyparts)
SEL_ARG graph size is O(2^(#max_keyparts/2))
(it is also possible to construct a case where instead of 2 in 2^n we
have a bigger constant, e.g. 4, and get a graph with 4^(31/2)= 2^31
nodes)
We avoid consuming too much memory by setting a limit on the number of
SEL_ARG object we can construct during one range analysis invocation.
*/
class SEL_ARG :public memory::SqlAlloc
{
public:
uint8_t min_flag,max_flag,maybe_flag;
uint8_t part; // Which key part
uint8_t maybe_null;
/*
Number of children of this element in the RB-tree, plus 1 for this
element itself.
*/
uint16_t elements;
/*
Valid only for elements which are RB-tree roots: Number of times this
RB-tree is referred to (it is referred by SEL_ARG::next_key_part or by
SEL_TREE::keys[i] or by a temporary SEL_ARG* variable)
*/
ulong use_count;
Field *field;
unsigned char *min_value,*max_value; // Pointer to range
/*
eq_tree() requires that left == right == 0 if the type is MAYBE_KEY.
*/
SEL_ARG *left,*right; /* R-B tree children */
SEL_ARG *next,*prev; /* Links for bi-directional interval list */
SEL_ARG *parent; /* R-B tree parent */
SEL_ARG *next_key_part;
enum leaf_color { BLACK,RED } color;
enum Type { IMPOSSIBLE, MAYBE, MAYBE_KEY, KEY_RANGE } type;
enum
{
MAX_SEL_ARGS = 16000
};
SEL_ARG() {}
SEL_ARG(SEL_ARG &);
SEL_ARG(Field *,const unsigned char *, const unsigned char *);
SEL_ARG(Field *field,
uint8_t part,
unsigned char *min_value,
unsigned char *max_value,
uint8_t min_flag,
uint8_t max_flag,
uint8_t maybe_flag);
SEL_ARG(enum Type type_arg)
:
min_flag(0),
elements(1),
use_count(1),
left(0),
right(0),
next_key_part(0),
color(BLACK),
type(type_arg)
{}
int size() const
{
return elements;
}
inline bool is_same(SEL_ARG *arg)
{
if (type != arg->type || part != arg->part)
return 0;
if (type != KEY_RANGE)
return 1;
return (cmp_min_to_min(arg) == 0 && cmp_max_to_max(arg) == 0);
}
inline void merge_flags(SEL_ARG *arg)
{
maybe_flag|= arg->maybe_flag;
}
inline void maybe_smaller()
{
maybe_flag= 1;
}
/* Return true iff it's a single-point null interval */
inline bool is_null_interval()
{
return (maybe_null && max_value[0] == 1);
}
inline int cmp_min_to_min(SEL_ARG *arg)
{
return sel_cmp(field,min_value, arg->min_value, min_flag, arg->min_flag);
}
inline int cmp_min_to_max(SEL_ARG *arg)
{
return sel_cmp(field,min_value, arg->max_value, min_flag, arg->max_flag);
}
inline int cmp_max_to_max(SEL_ARG *arg)
{
return sel_cmp(field,max_value, arg->max_value, max_flag, arg->max_flag);
}
inline int cmp_max_to_min(SEL_ARG *arg)
{
return sel_cmp(field,max_value, arg->min_value, max_flag, arg->min_flag);
}
SEL_ARG *clone_and(SEL_ARG *arg);
SEL_ARG *clone_first(SEL_ARG *arg);
SEL_ARG *clone_last(SEL_ARG *arg);
SEL_ARG *clone(RangeParameter *param, SEL_ARG *new_parent, SEL_ARG **next);
bool copy_min(SEL_ARG *arg);
bool copy_max(SEL_ARG *arg);
void copy_min_to_min(SEL_ARG *arg);
void copy_min_to_max(SEL_ARG *arg);
void copy_max_to_min(SEL_ARG *arg);
/* returns a number of keypart values (0 or 1) appended to the key buffer */
int store_min(uint32_t length, unsigned char **min_key, uint32_t min_key_flag);
/* returns a number of keypart values (0 or 1) appended to the key buffer */
int store_max(uint32_t length, unsigned char **max_key, uint32_t max_key_flag);
/* returns a number of keypart values appended to the key buffer */
int store_min_key(KEY_PART *key, unsigned char **range_key, uint32_t *range_key_flag);
/* returns a number of keypart values appended to the key buffer */
int store_max_key(KEY_PART *key, unsigned char **range_key, uint32_t *range_key_flag);
SEL_ARG *insert(SEL_ARG *key);
SEL_ARG *tree_delete(SEL_ARG *key);
SEL_ARG *find_range(SEL_ARG *key);
SEL_ARG *rb_insert(SEL_ARG *leaf);
friend SEL_ARG *rb_delete_fixup(SEL_ARG *root,SEL_ARG *key, SEL_ARG *par);
SEL_ARG *first();
SEL_ARG *last();
void make_root();
inline bool simple_key()
{
return (! next_key_part && elements == 1);
}
void increment_use_count(long count)
{
if (next_key_part)
{
next_key_part->use_count+= count;
count*= (next_key_part->use_count - count);
for (SEL_ARG *pos= next_key_part->first(); pos; pos= pos->next)
if (pos->next_key_part)
pos->increment_use_count(count);
}
}
void free_tree()
{
for (SEL_ARG *pos= first(); pos; pos= pos->next)
if (pos->next_key_part)
{
pos->next_key_part->use_count--;
pos->next_key_part->free_tree();
}
}
inline SEL_ARG **parent_ptr()
{
return parent->left == this ? &parent->left : &parent->right;
}
/*
Check if this SEL_ARG object represents a single-point interval
SYNOPSIS
is_singlepoint()
DESCRIPTION
Check if this SEL_ARG object (not tree) represents a single-point
interval, i.e. if it represents a "keypart = const" or
"keypart IS NULL".
RETURN
true This SEL_ARG object represents a singlepoint interval
false Otherwise
*/
bool is_singlepoint()
{
/*
Check for NEAR_MIN ("strictly less") and NO_MIN_RANGE (-inf < field)
flags, and the same for right edge.
*/
if (min_flag || max_flag)
return false;
unsigned char *min_val= min_value;
unsigned char *max_val= max_value;
if (maybe_null)
{
/* First byte is a NULL value indicator */
if (*min_val != *max_val)
return false;
if (*min_val)
return true; /* This "x IS NULL" */
min_val++;
max_val++;
}
return ! field->key_cmp(min_val, max_val);
}
SEL_ARG *clone_tree(RangeParameter *param);
private:
/*
Check if a compare is ok, when one takes ranges in account
Returns -2 or 2 if the ranges where 'joined' like < 2 and >= 2
*/
int sel_cmp(Field *in_field,
unsigned char *a,
unsigned char *b,
uint8_t a_flag,
uint8_t b_flag)
{
int cmp= 0;
/* First check if there was a compare to a min or max element */
if (a_flag & (NO_MIN_RANGE | NO_MAX_RANGE))
{
if ((a_flag & (NO_MIN_RANGE | NO_MAX_RANGE)) ==
(b_flag & (NO_MIN_RANGE | NO_MAX_RANGE)))
return 0;
return (a_flag & NO_MIN_RANGE) ? -1 : 1;
}
if (b_flag & (NO_MIN_RANGE | NO_MAX_RANGE))
return (b_flag & NO_MIN_RANGE) ? 1 : -1;
if (in_field->real_maybe_null()) // If null is part of key
{
if (*a != *b)
{
return *a ? -1 : 1;
}
if (*a)
goto end; // NULL where equal
a++; b++; // Skip NULL marker
}
cmp= in_field->key_cmp(a , b);
if (cmp) return cmp < 0 ? -1 : 1; // The values differed
// Check if the compared equal arguments was defined with open/closed range
end:
if (a_flag & (NEAR_MIN | NEAR_MAX))
{
if ((a_flag & (NEAR_MIN | NEAR_MAX)) == (b_flag & (NEAR_MIN | NEAR_MAX)))
return 0;
if (! (b_flag & (NEAR_MIN | NEAR_MAX)))
return (a_flag & NEAR_MIN) ? 2 : -2;
return (a_flag & NEAR_MIN) ? 1 : -1;
}
if (b_flag & (NEAR_MIN | NEAR_MAX))
return (b_flag & NEAR_MIN) ? -2 : 2;
return 0; // The elements where equal
}
};
SEL_ARG *rb_delete_fixup(SEL_ARG *root,
SEL_ARG *key,
SEL_ARG *par);
extern SEL_ARG null_element;
} /* namespace optimizer */
} /* namespace drizzled */
|