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Variations of Forward-SBNDM
Hannu Peltola
Jorma Tarhio
Aalto University
Finland
Aims
Tuning algorithms for exact string matching.
Studying the effect of simultaneous 2-byte read.
Aug. 29, 2011
SBNDM
Simple Backward Nondeterministic DAWG Matching
SBNDM [18] is a simplification of BNDM [17].
Both are bit-parallel algorithms.
Text T = t1...tn, pattern P = p1...pm.
At each alignment window of P in T, scan T
from right to left until the suffix of the window
is not a factor of P or an occurrence of P is found.
Aug. 29, 2011
Shift of SBNDM
No factor: m
P found: 1
Else: next alignment starts at the last factor
Aug. 29, 2011
SBNDM, example
P = banana, T = antanabadbanana...
alignment:
antanabadbanana
a
na
ana
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SBNDM, example
P = banana, T = antanabadbanana...
alignment:
not a factor:
next alignment:
antanabadbanana
a
na
ana
tana
antanabadbanana
Aug. 29, 2011
SBNDM, example
P = banana, T = antanabadbanana...
alignment:
not a factor:
next alignment:
not a factor:
next alignment:
antanabadbanana
a
na
ana
tana
antanabadbanana
d
antanabadbanana
Aug. 29, 2011
SBNDMq
SBNDMq [6] is a tuned version of SBNDM.
Processing of an alignment starts with checking
a q-gram.
Let q = 4. Consider an alignment at antana.
Instead of testing four suffixes a, na, ana, tana,
only tana is tested.
Testing is done in a fast loop.
Aug. 29, 2011
Forward-SBNDM
Forward-SBNDM (FSB for short) by Faro & Lecroq [7] is
a lookahead version of SBNDM2.
Both FSB and SBNDM2 read a 2-gram x1x2 before
a factor test.
x1x2 is matched with the end of P in SBNDM2.
Only x1 is matched with the end of P in FSB, and x2 is a
lookahead character following the current alignment.
FSB is faster than SBNDM2 for large alphabets.
Aug. 29, 2011
Generalization of FSB: FSB(q,f)
FSB(q,f) (= Forward-SBNDM(q,f)) is SBNDMq with
f lookahead characters, f = 0, 1, ..., q-1.
FSB(2,1) = FSB and FSB(q,0) = SBNDMq.
Motivation: SBNDMq works well on modern processors
also for q>2.
Aug. 29, 2011
FSB(q,f)
Let UV be a q-gram, where |V| = f.
After reading UV there are 3 alternatives:
i. If U is a suffix of P, reading continues leftwards.
ii. Else if UV is a factor of P, reading continues leftwards.
iii. Else the state vector is zero and P is shifted m-q+f+1 positions
(f positions more than in SBNDMq).
Aug. 29, 2011
Occurrence vectors in FSB(q,2)
Example: P = banana
SBNDMq:
banana
B[n] = 00001010
extra bits
FSB(q,2): B[n] =
B[a] =
B[x] =
Aug. 29, 2011
00101011
01010111
00000011
State vectors in FSB(q,2) for q=4
4-gram nanx:
x
n
a
n
00000011
00101011
01010111
00101011
00001000
nanx is not
a factor
4-gram
nanx
xana
anan
State vector
00001000
00000000
01000000
Aug. 29, 2011
Conclusion
na is a suffix of P
not a factor
factor of P
Benefits / drawbacks of
lookahead characters and extra bits
Benefits
• Longer shifts  more speed
• Combined suffix/factor test
Drawback
• More q-grams accepted  less speed
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Greedy skip loop for SBNDM2
(GSB2 = Greedy-SBNDM2)
Factor tests of two 2-grams are done in one round.
Let B2[x,y] denote the combined occurrence vector of
characters x and y.
B2[x,y] = B[x] & (B[y]<<1)
next:
D  B2[ti,ti+1]
if D = 0 then
if B2[ti+m-1,ti+m] = 0 then
i  i+2*m-2
goto next
Aug. 29, 2011
2-byte read
Read two characters (= 2 bytes = 16 bits) in one
instruction (in a skip loop).
Suits well q-gram algorithms with even q.
For experiments we made two versions of the
algorithms:
• Standard (1-byte read)
• b-version using 2-byte read
Aug. 29, 2011
2-byte read
(cont.)
Advantage: a part of computation can moved to
preprocessing phase
• Example: B2[x,y] = B[x] & (B[y]<<1)
Speed-up factor even more than 2
Drawback: extra 0.1 ms for preprocessing.
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4-byte read?
Many border crosses happen => slow down
232 tables too big for practice
Aug. 29, 2011
Experimental results/KJV Bible
In the recent comparison S. Faro, T. Lecroq: The Exact String
Matching Problem: a Comprehensive Experimental Evaluation
(2010), the algorithms EBOM and Hash3 were the fastest
in the bible text for m = 4,...,20.
4
8
16
Hash3
14.6
5.42
2.79
EBOM
6.53
3.87
2.91
Aug. 29, 2011
KJV: EBOM & Hash3 (on ThinkPad X61s)
4
3,5
3
GB/s
2,5
2
EBOM
Hash3
1,5
1
0,5
0
4
8
12
m
Aug. 29, 2011
16
20
KJV: EBOMb & Hash3b (with 2-byte read) added
4
3,5
3
GB/s
2,5
EBOM
2
EBOMb
Hash3
1,5
Hash3b
1
0,5
0
4
8
12
m
Aug. 29, 2011
16
20
KJV: SBNDM2b = FSB(2,0)b added
4
3,5
3
EBOM
GB/s
2,5
EBOMb
2
Hash3
1,5
Hash3b
FSB(2,0)b
1
0,5
0
4
8
12
m
Aug. 29, 2011
16
20
KJV: GSB2b added
4
3,5
3
EBOM
GB/s
2,5
EBOMb
2
Hash3
Hash3b
1,5
FSB(2,0)b
1
GSB2b
0,5
0
4
8
12
m
Aug. 29, 2011
16
20
KJV: FSB(4,i)b added, i = 0,1,2
4
3,5
EBOM
3
EBOMb
Hash3
GB/s
2,5
Hash3b
2
FSB(2,0)b
1,5
GSB2b
FSB(4,0)b
1
FSB(4,1)b
0,5
FSB(4,2)b
0
4
8
12
m
Aug. 29, 2011
16
20
KJV: Speed-up factors of 2-byte read
GSB2
FSB(2,0)
FSB(2,1)
FSB(4,0)
FSB(4,1)
FSB(4,2)
Hash3
EBOM
1.32
1.34
1.24
1.72
2.15
2.03
1.05
1.17
Aug. 29, 2011
Other experiments
DNA and binary data was also tested.
• Gain of lookahead characters or the greedy loop was smaller
than with the bible data.
Gain of 2-byte read was smaller with 64-bit code
than with 32-bit code.
Aug. 29, 2011
Conclusions
Two new algorithms were presented:
• FSB(q,f)
• GSB2
The new algorithms are faster than earlier algorithms
on English data:
• GSB2 for m = 4, …, 8
• FSB(q,f) for m = 8, …, 20
2-byte read makes most string algorithms faster.
Aug. 29, 2011
Web site for practical speed comparison
cse.aalto.fi/stringmatching
Aug. 29, 2011
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