1.
Finite State Automata
in
Dawid WEISS

2.
.
Dawid Weiss
.
20+ years of coding
10 years assembly only
. Academia & Research
PhD in Information Retrieval, PUT
Open source
Carrot2 , HPPC, Lucene, …
Industry & Business
Carrot Search s.c.
. .

3.
Talk outline
State machines (automata)
FSAs, DFAs, FSTs and other XXXs.
Use cases in Lucene and Solr
Suggester. FuzzySearch. Index.
No API details
Still @experimental.

4.
(Non)? Deterministic Finite
State (Automata|Machines)

5.
HashSet
hash → slot → value
0x29384d34 → lucene
0xde3e3354 → lucid
0x00000666 → lucifer

6.
HashSet
hash → slot → value
0x29384d34 → lucene
0xde3e3354 → lucid
0x00000666 → lucifer
FSA (deterministic)
l u c e n e
i
d
r
f
e

7.
HashSet
hash → slot → value
0x29384d34 → lucene
0xde3e3354 → lucid
0x00000666 → lucifer
FSA (deterministic)
l u c e n e
i
d
exists(sequence) r
oor(pre x) f
ceil(pre x) e

8.
k i l l
b l deterministic, non-minimal
i l

9.
k i l l
b l deterministic, non-minimal
i l
k
i l l
deterministic, minimal
b

10.
k i l l
b l deterministic, non-minimal
i l
k
i l l
deterministic, minimal
b
k
i l
l non-deterministic,
i l
non-minimal
b

11.
(Sorted)Map
lucene → 1
lucid → 2
lucifer → 666

12.
(Sorted)Map
lucene → 1
lucid → 2
lucifer → 666
FST (transducer)
l u c e n e|1
i
d|2
r|666
f
e

13.
(Sorted)Map
lucene → 1
lucid → 2
lucifer → 666
FST (transducer)
l|1 u c e n e
i|1
d
r
f|664
e

14.
NFSAs and
Regular expressions
a
a
e1e2 e1 e1
Determinization e+
e
states explosion, not always possible
Backtracking
recursion explosion e*
e
e?
e

15.
a?nan

16.
a?nan
n=3 → a?a?a?aaa

17.
a?nan
n=3 → a?a?a?aaa
Source: Russ Cox, Regular Expression Matching Can Be Simple And Fast (re2).

18.
35000
30000
25000
Time [ms]
20000
15000
10000
5000
0
0 5 10 15 20 25 30
Time of matching an for pattern a?n an , depending on n. Java 1.6, modern hardware.

19.
Linear-time, minimal, deterministic
FSA construction
Linear algorithm from sorted input
by Daciuk, Mihov, et al.
Active path
states that still can change
States dictionary
nodes that will never change

20.
1) common AP pre x
2) freeze the rest of AP
3) add suffix → new AP
lucene

21.
1) common AP pre x
2) freeze the rest of AP
3) add suffix → new AP
l u c e n e
lucid

22.
1) common AP pre x
2) freeze the rest of AP
3) add suffix → new AP
l u c e n e
i
d

23.
1) common AP pre x
2) freeze the rest of AP
3) add suffix → new AP
l u c e n e
i
d
lucifer

24.
1) common AP pre x
2) freeze the rest of AP
3) add suffix → new AP
l u c e n e
i
d
f
e r

25.
1) common AP pre x
2) freeze the rest of AP
3) add suffix → new AP
l u c e n e
i
d
r
f
e

26.
FS(A|T)s in (Lucene|Solr)

27.
Automata in
Lucene|Solr
org.apache.lucene.util.automaton.*
partial port of brics, FuzzyQuery, AutomatonTermsEnum
org.apache.lucene.util.automaton.fst.FST
FSA and FSTs from sorted data, suggester, indexes

28.
org.apache.lucene.util.automaton.fst.*
FSA representation
Arc-based, not state-based
Moore vs. Mealy. Compact vs. intuitive
Input: abc, bd, bde.
a b c a b c
b d b
e d e
d

29.
org.apache.lucene.util.automaton.fst.*
FSA representation
a b c
Arc-based, not state-based s3 s2 s1
Moore vs. Mealy. Compact vs. intuitive b d e
s4 s5
Next-state chaining
requires unusual tricks during construction
s1 s2 s5 s4 s3
cFL bL eFL dL a bL
s1 s2 s5 s4 s3
cFL bL eFL dL bLN a

30.
org.apache.lucene.util.automaton.fst.*
FSA representation
a b c
Arc-based, not state-based s3 s2 s1
Moore vs. Mealy. Compact vs. intuitive b d e
s4 s5
Next-state chaining
requires unusual tricks during construction
s1 s2 s5 s4 s3
Everything in a byte[] cFL bL eFL dL a bL
traversals-ready, memory-efficient
s1 s2 s5 s4 s3
cFL bL eFL dL bLN a

31.
org.apache.lucene.util.automaton.fst.*
FSA representation
a b c
Arc-based, not state-based s3 s2 s1
Moore vs. Mealy. Compact vs. intuitive b d e
s4 s5
Next-state chaining
requires unusual tricks during construction
s1 s2 s5 s4 s3
Everything in a byte[] cFL bL eFL dL a bL
traversals-ready, memory-efficient
Dual transition storage format
lookup: bsearch or linear scan s1 s2 s5 s4 s3
cFL bL eFL dL bLN a

32.
Input size Compressed size (MB)
Input MB Terms Lucene morf. gzip
Wikipedia t.index 481 38 092 045 258 164 149
Polish in . 162 3 672 200 3.1 1.7 15.4
.

33.
Use Cases:
Solr's Autocomplete

34.
Solr's
Suggesters
Design choices
sort order (alpha, score), pre x vs. spelling, boost exact matches?
Weights
term→weight, lookup(term, onlyMorePopular)
org.apache.solr.spelling.suggest.Lookup
JaspellLookup, TSTLookup, FSTLookup

35.
flour|3
four|4
fourier|3
furious|2
.
.
Take 1 .
.

36.
flour|3
four|4
→fou*
fourier|3
furious|2
o u
l
i e r |
f o u r
| 3
u 4
r 2
i o u s |
Find pre x.
Depth-in traversal for completions.
PQ on score|alpha
.
.
Take 1 .
.

37.
2furious
3flour
3fourier
4four
.
.
Take 2 .
.

38.
2furious
3flour
→fou*
3fourier
4four
u
f r
i o u
2 s
l
3 f u r i e r
o
4 u
f o
From score roots, until N collected.
Find pre x.
Depth-in traversal for completions, stop if N collected.
Find/boost exact match. .
.
Take 2 .
.

39.
2furious
5urious|furious
5rious|furious
5ious|furious
5ous|furious
5us|furious
5s|furious
3flour
…
.
Take 3 (in xes) .
.
.

40.
2
i
o
o
u
i
r s
r s |
f
5 u
s
u
4
o u
6 r
u
r
r | . i o u
s
f
7 u r
o
l
f o i e
3 l u r
e f o
|
f
r |
r
i
l o u e
i
o | |
r
i
u r
u
.

41.
Constant time lookups!
Regardless of the terms dictionary size.
Regardless of pre x length.

42.
Constant time lookups!
Regardless of the terms dictionary size.
Regardless of pre x length.
Exact matches only.
Static snapshot (not incremental).
Discretized weights.

43.
Top50KWiki.utf8, 676 KB, 50 000 terms
Jaspell TST FST
.
RAM [B] .
7 869 415 .
7 914 524 300 .175
queries per second,. . . tpq
.
PREFIX [100-200] .
458 .
966 .
742
.
PREFIX [6-9] .
330 .
228 .
659
.
PREFIX [2-4] .
126 .
29
. .
501

44.
Summary

45.
Summary and Conclusions
Automata
compact, powerful, efficient data structure
Lucene/Solr bene ts
behind the scenes, but spreading: index, queries, suggesters
API in Lucene
…is shaped right now, still @experimental

46.
Acknowledgement
Michael McCandless
Robert Muir
committer: .+

47.
dawid.weiss@carrotsearch.com