X86opti 05 s5yata

Remove Branches in
BitVector Select Operations
- marisa 0.2.2 -
Susumu Yata
@s5yata
Brazil, Inc.

1
30 March 2013 Brazil, Inc.

Who I Am
Job
Brazil, Inc. (groonga developer)
We need R&D software engineers.

Personal research & development
Tries
darts-clone, marisa-trie, etc.
Corpus
Nihongo Web Corpus 2010 (NWC 2010)
2

Relationships between BitVector and Marisa.

BitVector and Marisa

3

BitVector
What‟s BitVector?
A sequence of bits

Operations
BitVector::get(i)
BitVector::rank(i)
BitVector::select(i)

4

BitVector – Get Operations
Interface
BitVector::get(i)

Description
The i-th bit (“0” or “1”)

0 1 2 … i–1 i i+1 … n-2 n-1
0 0 1 … 0 1 1 … 0 0

Get!
5

BitVector – Rank Operations
Interface
BitVector::rank(i)

Description
The number of “1”s up to the i-th bit

0 1 2 … i–1 i i+1 … n-2 n-1
0 0 1 … 0 1 1 … 0 0

How many “1”s?
6

BitVector – Select Operations
Interface
BitVector::select(i)

Description
The position of the i-th “1”

0 1 2 … … … … … n-2 n-1
0 0 1 … … … … … 0 0

Where is the i-th “1”?
7

Marisa
 Who‟s Marisa?
An ordinary human magician

 What‟s Marisa?
A static and space-efficient dictionary

 Data structure
Recursive LOUDS-based Patricia tries

 Site
http://code.google.com/p/marisa-trie
8

Marisa – Patricia
Patricia is a labeled tree.
Keys = Tree + Labels

Node Label
ID Key 1 “Ar”
4
0 “Argentina” 1 2 “Brazil”
1 “Armenia” 5 3 „C‟
0 2
2 “Brazil” 4 “gentina”
6
3 “Canada” 3 5 “menia”
4 “Cyprus” 7 6 “anada”
7 “yprus”

9

Marisa – Recursiveness
Unfortunately, this margin is too small…
Keys = Tree + Labels
Labels = Tree + Labels
Labels = Tree + Labels <– Reasonable
…
10

Marisa – BitVector Usage
LOUDS
Level-Order Unary Degree Sequence

Terminal flags
A node is terminal (“1”) or not (“0”).

Link flags
A node has a link to its multi-byte label
(“1”) or has a built-in single-byte label (“0”).
11

Marisa – BitVector Usage
LOUDS
BitVector::get(), select()

Terminal flags
BitVector::get(), rank(), select()

Link flags
BitVector::get(), rank()

12

How to implement Rank/Select operations.

Implementations

13

Rank Dictionary
Index structures
r_idx[x].abs = rank(512・x)
x = 0, 1, 2, …
r_idx[x].rel[y] =
rank(512・x + 64・y) – rank(512・x)
Y = 1, 2, 3, … , 7

Calculation
abs + rel + popcnt()
14

Rank Operations
Time complexity = O(1)
512 512 512 512 512

r_idx.abs

64 64 64 64 64 64 64 64

r_idx.rel

64

popcnt()
15

Select Dictionary
Index structure
s_idx[x] = select(512・x)
i = 0, 1, 2, …

Calculation
Limit the range by using s_idx.
Limit the range by using r_idx[x].abs.
Limit the range by using r_idx[x].rel[y].
Find the i-th “1” in the range.
16

Select Operations
s_idx s_idx
512 512 512 512 512 512 512

r_idx.abs r_idx.abs

64 64 64 64 64 64 64 64

r_idx.rel r_idx.rel

64

Final round
17

Select Final Round
Binary search & table lookup
Three-level branches
if

if if

if if if if

8 8 8 8 8 8 8 8

Table lookup
18

How to remove the branches in the final round.

Improvements

19

Original
// x is the final 64-bit block (uint64_t).
x = x – ((x >> 1) & MASK_55);
x = (x & MASK_33) + ((x >> 2) & MASK_33);
x = (x + (x >> 4)) & MASK_0F;
x *= MASK_01; // Tricky popcount
if (i < ((x >> 24) & 0xFF)) { // The first-level branch
if (i < ((x >> 8) & 0xFF)) { // The second-level branch
if (i < (x & 0xFF)) { // The third-level branch
// The first byte contains the i-th “1”.
} else {
// The second byte contains the i-th “1”.
20

Tips – Tricky PopCount
0 1 1 1 0 0 1 0

x = x – ((x >> 1) & MASK_55);
1 2 0 1

x = (x & MASK_33) + ((x >> 2) & MASK_33);
3 1

x = (x + (x >> 4)) & MASK_0F;
4

21

Tips – Tricky PopCount
// MASK_01 = 0x0101010101010101ULL;
// x = x | (x << 8) | (x << 16) | (x << 24) | …;
x *= MASK_01;

4 1 3 5 2 6 3 4

28 23 15 7

24 20 13 4

22

+ SSE2 (After PopCount)
// y[0 … 7] = i + 1;
__m128i y = _mm_cvtsi64_si128((i + 1) * MASK_01);
__m128i z = _mm_cvtsi64_si128(x);

// Compare the 16 8-bit signed integers in y and z.
// y[k] = (y[k] > z[k]) ? 0xFF : 0x00;
y = _mm_cmpgt_epi8(y, z); // PCMPGTB

// The j-th byte contains the i-th “1”.
// TABLE is a 128-byte pre-computed table.
uint8_t j = TABLE[_mm_movemask_epi8(y)];
23

Tips – PCMPGTB
y = _mm_cvtsi64_si128((i + 1) * MASK_01);
20 20 20 20 20 20 20 20

z = _mm_cvtsi64_si128(x);
28 24 23 20 15 13 7 4

// y[k] = (y[k] > z[k]) ? 0xFF : 0x00;
y = _mm_cmpgt_epi8(y, z);

0x00 0x00 0x00 0x00 0xFF 0xFF 0xFF 0xFF

24

+ Tricks (After Comparison)
uint64_t j = _mm_cvtsi128_si64(y);

// Calculation without TABLE
j = ((j & MASK_01) * MASK_01) >> 56;

// Calculation with BSR
j = (63 – __builtin_clzll(j + 1)) / 8;

// Calculation with popcnt (SSE4.2 or SSE4a)
j = __builtin_popcountll(j) / 8;

25

– SSE2 (Simple and Fast)
// x is the final 64-bit block (uint64_t).
x = x – ((x >> 1) & MASK_55);
x = (x & MASK_33) + ((x >> 2) & MASK_33);
x = (x + (x >> 4)) & MASK_0F;
x *= MASK_01; // Tricky popcount

uint64_t y = (i + 1) * MASK_01;
uint64_t z = x | MASK_80;
// Compare the 8 7-bit unsigned integers in y and z.
z = (z – y) & MASK_80;
uint8_t j = __builtin_ctzll(z) / 8;
26

Tips – Comparison
uint64_t y = (i + 1) * MASK_01;
0x14 0x14 0x14 0x14 0x14 0x14 0x14 0x14

uint64_t z = x | MASK_80;
0x9C 0x98 0x97 0x94 0x8F 0x8D 0x87 0x84

// Compare the 8 7-bit unsigned integers in y and z.
z = (z – y) & MASK_80;

0x80 0x80 0x80 0x80 0x00 0x00 0x00 0x00

27

+ SSSE3 (For PopCount)
// Get lower nibbles and upper nibbles of x.
__m128i lower = _mm_cvtsi64_si128(x & MASK_0F);
__m128i upper = _mm_cvtsi64_si128(x & MASK_F0);
upper = _mm_srli_epi32(upper, 4);
// Use PSHUFB for counting “1”s in each nibble.
__m128i table =
_mm_set_epi8(4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0);
lower = _mm_shuffle_epi8(table, lower);
upper = _mm_shuffle_epi8(table, upper);
// Merge the counts to get the number of “1”s in each byte.
x = _mm_cvtsi128_si64(_mm_add_epi8(lower, upper));
x *= MASK_01;
28

Tips – PSHUFB
lower = _mm_cvtsi64_si128(x & MASK_0F);
12 8 7 4 15 13 7 4

table = _mm_set_epi8(4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, …);
4 3 3 2 3 2 2 1 3 2 2 1 2 1 1 0

// Perform a parallel 16-way lookup.
lower = _mm_shuffle_epi8(table, lower);

2 1 3 1 4 3 3 1

29

How effective the improvements are.

Evaluation

30

Environment
OS
Mac OSX 10.8.3 (64-bit)
CPU
Core i7 3720QM – Ivy Bridge
2.6GHz – up to 3.6GHz
Compiler
Apple LLVM version 4.2 (clang-425.0.24)
(based on LLVM 3.2svn)

31

Data
Source
Japanese Wikipedia page titles
gzip –cd jawiki-20130328-all-titles-in-
ns0.gz | LC_ALL=C sort –R > data

Details
Number of keys: 1,367,750
Average length: 21.14 bytes
Total length: 28,919,893 bytes
32

Binaries
marisa 0.2.1
./configure CXX=clang++ --enable-popcnt
make
tools/marisa-benchmark < data

marisa 0.2.2
./configure CXX=clang++ --enable-sse4
make
tools/marisa-benchmark < data
33

Results – marisa 0.2.1
Without improvements
#Tries Size Build Lookup Reverse Prefix Predict
[KB] [Kqps] [Kqps] [Kqps] [Kqps] [Kqps]
1 11,811 724 1,105 1,223 1,038 711
2 8,639 632 790 877 753 453
3 8,001 621 750 816 708 406
4 7,788 591 723 791 687 391
5 7,701 590 712 781 680 384

Baseline

34

Results – marisa 0.2.2
With improvements
[KB] [Kqps] [Kqps] [Kqps] [Kqps] [Kqps]
1 11,811 757 1,198 1,359 1,115 772
2 8,639 657 873 1,000 820 503
3 8,001 621 817 924 770 453
4 7,788 613 797 900 752 438
5 7,701 610 787 884 737 427

Same size
Faster operations
35

Results – Improvements
Improvement ratios
[%] [%] [%] [%] [%] [%]
1 0.00 +4.56 +8.42 +11.12 +7.42 +8.58
2 0.00 +3.96 +10.52 +14.03 +8.90 +11.04
3 0.00 0.00 +8.93 +13.24 +8.76 +11.58
4 0.00 +3.72 +10.24 +13.78 +9.46 +12.02
5 0.00 +3.39 +10.53 +13.19 +8.38 +11.20

Same size
Faster operations
36

Conclusion
“Any sufficiently advanced technology
is indistinguishable from magic.”

“Any sufficiently advanced technique
is indistinguishable from magic.”

“You are magician.”
37

X86opti 05 s5yata

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X86opti 05 s5yata