DFA Minimization in Map-Reduce
1 like600 views
The document presents a study on DFA (Deterministic Finite Automata) minimization using Map-Reduce frameworks, specifically comparing Moore's and Hopcroft's algorithms. It discusses the structure of the minimization problem, the algorithms' complexities, and their implementation in a Map-Reduce context, including communication costs and experimental results. The conclusion highlights that Hopcroft's algorithm outperforms Moore's in terms of communication cost and execution time under certain conditions, with suggestions for future work to further improve performance.
![DFA Minimization in Map-Reduce Moore-MR
Moore-MR: Moore’s algorithm in Map-Reduce
MAPPER REDUCER
PreProcessing
Pre-Processing
MAPPER REDUCER
First job
MAPPER REDUCER
PPHF-MR
Iterate till Q is not empty Construct
minimal DFA
δ
K = h(p)
∆+
0 , ∆−
0 ∆+
i−1 ∆−
i−1 p, a, q, πi
p, +
p, a, q, πi
p, −
[ p, true ]
K = h(q)
K = h(p) K = h(πi
p)
p, a, q, πi
p, +
p, a, q, πi
p, −
h(p) h(p)
h(p)
h(q)
h(πi
p)
∆+
0 = {(p, a, q, π0
p, +) : (p, a, q) ∈ δ}
∆−
0 = {(p, a, q, π0
q , −) : (p, a, q) ∈ δ}
πi
p = πi−1
p · πi−1
δ(p,a1)
· . . . · πi−1
δ(p,ak)
PPHF-MR in reducer j maps πp to {j · n, . . . , j · n + n − 1}
sub(πi
p): set of sub-strings obtained by dividing πi
p into (k + 1)
substrings. Thus h(p) emits p, true if |sub(πi
p)| > |sub(πi−1
p )|
Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 8 / 21](https://image.slidesharecdn.com/fqxn3wggsou4hh5v02an-signature-6ef91e549fa4467d3f1054b08ce651e7b110bbb17dce98afd89fe2177b4671af-poli-160619044613/85/DFA-Minimization-in-Map-Reduce-8-320.jpg)
DFA Minimization in Map-Reduce
- 1. DFA Minimization in Map-Reduce G¨osta Grahne Shahab Harrafi Iraj Hedayati Ali Moallemi Concordia University {grahne,s harraf,h iraj,moa ali}@cs.concordia.ca BeyondMR, SIGMOD/PODS, July 2016 Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 1 / 21
- 2. Introduction Outline 1 Introduction Preliminaries Minimization Algorithms 2 DFA Minimization in Map-Reduce Moore-MR Hopcroft-MR Communication Cost 3 Experimental Results 4 Conclusion Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 2 / 21
- 3. Introduction Preliminaries Preliminaries Finite Automata A = (Q, Σ, δ, qs, F) DFA minimization Motivation: Importance and wide use in applications Iterative and multi-round structure of problem Simialr problems: Coarsest partition Bi-Simulation Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 3 / 21
- 4. Introduction Minimization Algorithms DFA Minimization Algorithms - Moore Moore’s algorithm iteratively computes equivalence class of each state as p ≡i q ⇔ p ≡i−1 q AND ∀a ∈ Σ δ(p, a) ≡i−1 δ(q, a) p1 p2 p3 q1 q2 q3 a, b a, b a b p1 p2 p3 q1 q2 q3 a, b a, b a b Signatures: a b p1 q1 q1 p2 q2 q2 p3 q2 q3 Complexity: O(kn2) Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 4 / 21
- 5. Introduction Minimization Algorithms DFA Minimization Algorithms - Hopcroft Hopcroft’s algorithm works with splitters. At each iteration, for any splitter S, a , P ÷ S, a = {P1, P2} where P1 = {p ∈ P : δ(p, a) ∈ S} and P2 = {p ∈ P : δ(p, a) /∈ S}. It uses a data structure Γ(S, a), a subset of states from block S having incoming transition labeled with a. p1 p2 p3 q1 q2 q3 a, b a, b a b p1 p2 p3 q1 q2 q3 a, b a, b a b Here Γ(S, a) = {q1, q2} Complexity: O(kn log n) Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 5 / 21
- 6. Introduction Minimization Algorithms Moore vs. Hopcroft Moore’s algorithm: 0start 0 1 0 2 0 3 1 a, b a, b a, b a, b 0start 00 1 00 2 01 3 11 a, b a, b a, b a, b 0start 0000 1 0001 2 0111 3 1111 a, b a, b a, b a, b 0start 0000 1 0001 2 0111 3 1111 a, b a, b a, b a, b number of operations= 4 × (2 × 4) = 32 Hopcroft’s algorithm: 0start 0 1 0 2 0 3 1 a, b a, b a, b a, b 0start 0 1 0 2 2 3 1 a, b a, b a, b a, b 0start 0 1 3 2 2 3 1 a, b a, b a, b a, b number of operations= 3 × 2 = 6 Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 6 / 21
- 7. DFA Minimization in Map-Reduce Outline 1 Introduction Preliminaries Minimization Algorithms 2 DFA Minimization in Map-Reduce Moore-MR Hopcroft-MR Communication Cost 3 Experimental Results 4 Conclusion Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 7 / 21
- 8. DFA Minimization in Map-Reduce Moore-MR Moore-MR: Moore’s algorithm in Map-Reduce MAPPER REDUCER PreProcessing Pre-Processing MAPPER REDUCER First job MAPPER REDUCER PPHF-MR Iterate till Q is not empty Construct minimal DFA δ K = h(p) ∆+ 0 , ∆− 0 ∆+ i−1 ∆− i−1 p, a, q, πi p, + p, a, q, πi p, − [ p, true ] K = h(q) K = h(p) K = h(πi p) p, a, q, πi p, + p, a, q, πi p, − h(p) h(p) h(p) h(q) h(πi p) ∆+ 0 = {(p, a, q, π0 p, +) : (p, a, q) ∈ δ} ∆− 0 = {(p, a, q, π0 q , −) : (p, a, q) ∈ δ} πi p = πi−1 p · πi−1 δ(p,a1) · . . . · πi−1 δ(p,ak) PPHF-MR in reducer j maps πp to {j · n, . . . , j · n + n − 1} sub(πi p): set of sub-strings obtained by dividing πi p into (k + 1) substrings. Thus h(p) emits p, true if |sub(πi p)| > |sub(πi−1 p )| Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 8 / 21
- 9. DFA Minimization in Map-Reduce Hopcroft-MR Hopcroft-MR: Hopcroft’s algorithm in Map-Reduce MAPPER REDUCER PreProcessing Pre-Processing MAPPER REDUCER First job MAPPER REDUCER Second job MAPPER REDUCER PPHF-MR Iterate till Q is not empty Construct minimal DFA δ ∆0, Γ0 ∆i−1 Γi−1 update tuple ∆i Γi ∆i Γi h(q) h(q) h(p) h(πi p) ∆0 = {(p, a, q, π0 p, +) : (p, a, q) ∈ δ} Γ0 = Bj ∈π0 {(p, a, q, π0 q , −) : (p, a, q) ∈ δ, q ∈ Bj } update tuples (pj , πi−1 pj , βa pj , πi−1 q ) Aggregated tuple in second job: βp = k j=1 β aj p Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 9 / 21
- 10. DFA Minimization in Map-Reduce Hopcroft-MR Sample run of Hopcroft-MR Splitters in queue: Q = { S1, a , S1, b , S3, a , S3, b } p1 p2 p3 q1 q2 q3 P S1 S2 S3 a b a, b a b p1, a, q1, P S1, q1, a First Job ⇒ p1, P, 01, S1 p3, b, q3, P p1, b, q3, P S3, q3, b ⇒ p1, P, 10, S3 p3, P, 10, S3 Second Job h(p1) :h(p1) : p1, a, q1, P p1, b, q3, P p1, P, 10, S3 p1, P, 01, S1 ⇒ p1, a, q1, P11S3S1 p1, b, q3, P11S3S1 h(p3) :h(p3) : p3, a, q2, P p3, b, q3, P p3, P, 10, S3 ⇒ p3, a, q2, P10S3 p3, b, q3, P10S3 After applying PPHF-MR, p1 ∈ P1 = P 11S3S1 and p3 ∈ P2 = P 10S3 while p2 remains in P . Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 10 / 21
- 11. DFA Minimization in Map-Reduce Communication Cost Communcation Cost Communication cost can be calculated as: Number of rounds × (Replication rate × Input size + Output size) Number of rounds: O(n) Replication rate: O(1) Input size = Output size Moore-MR: Record size of output of first job is Θ(k log n). Thus communication cost of each round is Θ(k2n log n). Therefor total comunication cost is O(k2n2 log n). Hopcroft-MR: There are O(n log n) updates in parallel execution at each round. Thus it requires O(kn2 log n) bits of communication. Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 11 / 21
- 12. Experiments Outline 1 Introduction Preliminaries Minimization Algorithms 2 DFA Minimization in Map-Reduce Moore-MR Hopcroft-MR Communication Cost 3 Experimental Results 4 Conclusion Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 12 / 21
- 13. Experiments Data sets - Slow A member of DFA family know slow for Hopcroft and Moore 0start 1 2 . . . n a a a a a After minimization 0start 1 2 . . . n a a a a a Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 13 / 21
- 14. Experiments Data sets - Circular Equal number of incoming and outgoing transitions 0start 1 2 3 4 56 7 a b c d a b c d a b c d a b cd ab c d a b c d a b c d a b c d After minimization 0, 4start 1, 5 2, 6 3, 7 a b, d c a b, d c a b, d c a b, d c Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 14 / 21
- 15. Experiments Data sets - Star Most of the transitions’ target is one state 0start 1 2 3 4 5 6 7 a b, c, d a b, c, d a b, c, d a b, c, d a b, c, d a b, c, d a b, c, d a, b, c, d After minimization 0, 1, 2, 3, 4, 5, 6start 7 a, b, c, d a, b, c, d Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 15 / 21
- 16. Experiments Experimental Results - The affect of number of rounds Slow DFA k = 2, n = {2 − 512} 0 200 400 600 0 25 50 75 100 Number of States CommunicationCost(MB) Moore-MR Hopcroft-MR 0 200 400 600 0 500 1,000 1,500 Number of States ExecutionTime(sec) Moore-MR Hopcroft-MR Slow DFA 0start 1 2 . . . n a a a a a Minimized: 0start 1 2 . . . n a a a a a Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 16 / 21
- 17. Experiments Experimental Results - Evenly distributed transitions Circular DFA k = 4, n = {24 , 25 , . . . , 217 } 23 28 213 218 0.1 1 10 100 Number of States CommunicationCost(MB) Moore-MR Hopcroft-MR 23 28 213 218 10 100 Number of States ExecutionTime(sec) Moore-MR Hopcroft-MR Circular DFA 0start 1 2 3 4 56 7 a b c d a b c d a b c d a b cd ab c d a b c d a b c d a b c d Minimized: 0, 4start 1, 5 2, 6 3, 7 a b, d c a b, d c a b, d c a b, d c Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 17 / 21
- 18. Experiments Experimental Results - The affect of alphabet size Circular DFA k = {22 , . . . , 27 }, n = 210 0 50 100 0 1 2 3 4 5 Alphabet Size CommunicationCost(GB) Moore-MR Hopcroft-MR 0 50 100 0 500 1,000 Alphabet Size ExecutionTime(sec) Moore-MR Hopcroft-MR Circular DFA 0start 1 2 3 4 56 7 a b c d a b c d a b c d a b cd ab c d a b c d a b c d a b c d Minimized: 0, 4start 1, 5 2, 6 3, 7 a b, d c a b, d c a b, d c a b, d c Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 18 / 21
- 19. Experiments Experimental Results - The affect of skewness Star DFA k = 4, n = 217 0-15 16-31 32-47 48-63 64-79 80-95 96-111 112-126 127 0 1 2 3 4 5 Reducer Number NumberofTransitions(·105 ) Circular Random Star 0-15 16-31 32-47 48-63 64-79 80-95 96-111 112-126 127 0 1 2 3 4 5 Reducer Number NumberofTransitions(·105 ) Circular Random Star Star DFA 0start 1 2 3 4 5 6 7 a b, c, d a b, c, d a b, c, d a b, c, d a b, c, d a b, c, d a b, c, d a, b, c, d Minimized: 0, 1, 2, 3, 4, 5, 6start 7 a, b, c, d a, b, c, d Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 19 / 21
- 20. Conclusion Outline 1 Introduction Preliminaries Minimization Algorithms 2 DFA Minimization in Map-Reduce Moore-MR Hopcroft-MR Communication Cost 3 Experimental Results 4 Conclusion Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 20 / 21
- 21. Conclusion Conclusions and Future Works Hopcroft-MR outperforms Moore-MR in communication cost when the cardinality of the alphabet is at least 16, in wall-clock time when the cardinality is at least 32 Both algorithms are equally sensitive to skewness in the input data. Future work, There is potential to reduce skew-sensitiveness in Moore-MR. Investigate the average communication cost Reducer capacity vs. Number of rounds Grahne, Harrafi, Hedayati, Moallemi DFA Minimization in Map-Reduce BeyondMR’16 21 / 21