O(m) CPU 1 CPU Xeon LZEnd L1 1 L2 1.5 MiB 40 [10] CPU on-memory 2 2 LZEnd LZEnd 1 Xeon 5670 Xeon 5670 L1 data L2 LL Last Level 129 KiB 1
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1 DEIM Forum 2015 G3-3 LZE++: NTT {yamamuro.takeshi,honjo.toshimori}@lab.ntt.co.jp, onizuka@ist.osaka-u.ac.jp T[0...N-1] m LZE++ O(m) LZ77 LZEnd LZEnd O(m) CPU LZE++ O(m) mk/n+1 K K< = N while-loop LZEnd LZEnd on-memory 64 KiB KiB LZ77 1. LZ77 UNIX gzip 1 CPU Snappy 2 LZ77 DBMS [1] [2] [11] URI LZ N O(N) N 2 1 k Block-based Compression BC i i/k i%k O(k) Lucene 2 m O(m) LZEnd [3] LZEnd LZ ,. [5] [6] [7] [8].. BC k. k k k LZEnd
![O(m) CPU 1 CPU Xeon 5670 2 LZEnd L1 1 L2 1.5 MiB 40 [10] CPU on-memory 2 2 LZEnd 2 2. 2 LZEnd 1 Xeon 5670 Xeon 5670 L1 data L2 LL Last Level 129 KiB 1.](/docs-images/86/93646678/images/2-1.jpg "5 MiB 12 MiB 16 KiB PrefixSpan [12] O(m) mk/n+1 K K< = N while-loop LZE++ mk/n+1 LZEnd LZEnd on-memory 64 KiB 3.3 58.2 64 KiB 1439.0 LZE++ LZEnd LZE++ 2.")
2 O(m) CPU 1 CPU Xeon LZEnd L1 1 L2 1.5 MiB 40 [10] CPU on-memory 2 2 LZEnd LZEnd 1 Xeon 5670 Xeon 5670 L1 data L2 LL Last Level 129 KiB 1.5 MiB 12 MiB 16 KiB PrefixSpan [12] O(m) mk/n+1 K K< = N while-loop LZE++ mk/n+1 LZEnd LZEnd on-memory 64 KiB KiB LZE++ LZEnd LZE LZEnd 2 LZEnd 3. O(m) mk/n : LZEnd 2 LZEnd L1 m BC LZE++ LZE++ O(m) while-loop LZEnd N M M<<N) k LZEnd T[0...N-1]=t 0 t 1...t N 1 t i Σ 0< = i<n 3 Z[0...K-1]=z 0 z 1...z K 1 (K K< = N) T[0...i-1] Z[0...p-1] history Z[0...q] (q<p) T[i...i+l-1] T[i...N-1] T[i...i+l] z p =z q t i+l T[i...i+l-1] history z q source T[N-1] LZEnd z K 1 coarsely optimal [13] Theorem 1. ρ(t ) k H k (T ) [14] T 3 Σ
![T 0 k lim T f k ( T )=0 s.t. ρ(t ) < = H k (T )+f k ( T ) f k 2. 1 LZEnd LZEnd Z 3 c[0...k-1]: Z x[0...k-1]: z q source B[0.](/docs-images/86/93646678/images/3-1.jpg "..N-1]: c bit z p=z qt i+l c[p]=t i+l x[p]=q B[i+l] bit B rank select rank B(i) B[0...i] bit select B(i) i+1 bit rank select O(1) O(logN) bit B NH 0 [15] z p l B select B select B(p+1)-select B(p)-1 2.")
3 T 0 k lim T f k ( T )=0 s.t. ρ(t ) < = H k (T )+f k ( T ) f k 2. 1 LZEnd LZEnd Z 3 c[0...k-1]: Z x[0...k-1]: z q source B[0...N-1]: c bit z p=z qt i+l c[p]=t i+l x[p]=q B[i+l] bit B rank select rank B(i) B[0...i] bit select B(i) i+1 bit rank select O(1) O(logN) bit B NH 0 [15] z p l B select B select B(p+1)-select B(p) LZEnd Algorithm 1 extract(base, m) 1: /* IN base: start position to decompress */ 2: /* IN m: length */ 3: /* OUT extracted symbols in T[base...base+m-1] */ 4: if n > 0 then 5: end = base + m - 1; 6: r = rank B (end); 7: if B[end] == 1 then 8: extract(base, m - 1); 9: Append c[r] to an output; 10: else 11: pos = select B (r) + 1; 12: if base < pos then 13: extract(base, pos - base); 14: m = end - pos : base = pos; 16: end if 17: ref = select B (x[r+1]) - select B (r + 1) + base + 1; 18: extract(ref, m); 19: end if 20: end if LZEnd Algorithm 1 [3] extract(base, m) T[base...base+m-1] B[end] bit 7 c[r] 9 LZEnd T[i...i+l-1] history z q source source extract T[i...i+l-1] extract m O(m) extract 3. : LZE LZE++ O(m) T[0...N-1] k PrefixSpan [12] S[0...M-1] M<<N LZE++ LZEnd c extract 1 3 abcd LZEnd d c abc extract 3 extract LZE++ ab 2 LZE++ extract LZEnd x CPU LZE++ PrefixSpan CPU CPU 4 (extract seq) wlen extract LZE++ wlen wlen
4 4 wlen extract wlen agpxz LZE++ extract seq CPU 3. 4 CPU 3. 2 LZE S[0...M-1] M<<N CPU k PrefixSpan [12] T N r% B L 0 0, kn/l, 2kN/L,... L/k k PrefixSpan α β ξ Algorithm 2 PrefixSpan 5 ababaac I B L B[i...L-1] (i I) {0, 1,..., L-1} 7 15 Σ s Σ s 13 length α β 10 4 ξ 9 I s B[i...i+length] i I s S 11 S abcde abcd 9 B[i...i+length+1] /S α=2, β=3 2 ξ=2 I {0, 1,..., 6} a, aa, aaa, aab,... 1 ab {0, 2, 4, 5} 2 2 ab S ba S S={ab, ba} S abba 3. 3 LZE++ LZE++ LZEnd 2. 1 hisotry LZE++ LZEnd S[0...M-1]: x [0...K-1]: S Algorithm 2 PrefixSpan(I, length) 1: /* IN I: indices of T and initially I = {0...N-1} */ 2: /* IN length: length of common prefixes among suffixes */ 3: /* OUT S: set of frequent patterns (shared dictionary) */ 4: if length > β then 5: Return; 6: end if 7: while s Σ do 8: I s = {i i I T[i + length] == s}; 9: if I s > = ξ then 10: if length > α && T[i...i + length + 1] / S then 11: S = S {T[i...i + length] i I s}; 12: end if 13: PrefixSpan(I s, length + 1); 14: end if 15: end while 5 ababaac PrefixSpan α=2, β=3, ξ=2 S={ab, ba} R[0...K-1]: bit z p z p =S[j...j+l-1]t i+l c[p]=t i+l x [p]=j B[i+l] R[p] bit LZE++ z p source z q q<p Z[x[p]] S[x [p]] R Algorithm 3 LZEnd extract Algorithm Algorithm 3 LZEnd extract 1: if R[r] == 1 then 2: ref = x [rank R (r) + base - pos]; 3: Append S[ref...ref + m] to B; 4: else 5: ref = select B (x[r + 1]) - select B (r + 1) + base + 1; 6: extract(ref, m); 7: end if LZE++ LZEnd Theorem 2. Z[p] Z[p ] p<p Z[p ] 2 history Z[p]c c Σ S[i...i+k-1]c i+k-1<m Z[p ] Z[p] history 3. 4 extract seq
5 CPU LZE++ wlen wlen T[base...base+m-1] m>wlen T[base...wlen-1] extract T[base+wlen...base+m-1] extract seq wlen LZE++ x log(wlen)-bit extract seq CPU LZE++ B R bit LZEnd Algorithm 4 T[base+wlen...base+m-1] extract seq T[base...wlen-1] extract (r1, r2, pos) while-loop 1 1 B CPU bit x64 CPU 64bit B bit bsr 0 0 l!=0 R[r2]==0 14 x select B 13 R[r2]= O(m) mk/n+1 while-loop 3 K/N<1 CPU 1 rank/select 4. LZE++ LZEnd 4. 1 LZE++ LZEnd LZEnd 2. 2 c 1B x source 4B Algorithm 4 extract seq(base, m) 1: /* IN base: start position to decompress */ 2: /* IN m: length (m > wlen) */ 3: /* OUT extracted symbols in T[base + wlen...base + m - 1] */ 4: r1 = rank B (base + wlen); 5: r2 = rank R (r1); 6: pos = select B (r1) + 1; 7: while pos - base < m do 8: l = Count leading zeroes from pos in B 9: if l!= 0 then 10: if R[r2] == 0 then 11: r3 = r1 - x[r1] + 1; 12: ref = select B (r3); 13: Copy T[ref...ref + l - 1] to an output; 14: else 15: ref = x [r2++]; 16: Copy S[ref...ref + l - 1] to an output; 17: end if 18: end if 19: Print c[r1++] to an output; 20: pos += l + 1; 21: end while bit B [3] rank select RecRank [15] LZEnd c x B 41bit LZEnd Suffix Array LZE++ c B LZEnd x 3. 4 log(wlen)-bit wlen 65,536 2B S 1B S 4B 3. 2 PrefixSpan T 1% r=1 k=1024 B PrefixSpan α β ξ LZE++ c S x x B R 58bits 8B 64bits α=8 wlen LZ77 LZE LZEnd Suffix Array O(m) while-loop B R LZE++1 LZE++2 2 LZE++1 LZEnd RecRank B R 3. 4 while-loop 1 select B select LZE++2 R LZE++1 select B LZE++2
6 DNA (influenza Escherichia Coli) einstein.de.txt world leaders 4 2 enwiki8 5 gov2 6 3 Hadoop 7 Cassandra 8 PostgreSQL 9 (1) (2) enwiki Wikipedia 10 8 B gov gov Web 128 MiB (3) 128 MiB 4. 3 LZE++ LZEnd LZE++ PostgreSQL einstein.de.txt enwiki8 3 LZE++ mk/n+1 while-loop K 4. 5 LZEnd 6 gzip (Version 1.4) LZ77 bzip2 10 (Version 1.0.6) Burrows-Wheeler [14] gzip bzip2-9 lzip 11 (Version 1.12) LZ77 LZMA Snappy (Revision 73) LZ4 (Revision 90) LZ77 Re-Pair 12 [16] Xeon5670 Xeon5670 L1 L2 LL 129 KiB 1.5 MiB 12 MiB CPU perf v3.6.9 C++ gcc v O KiB B 2 16 B LZE++ LZEnd PostgreSQL einstein.de.txt mmahoney/compression/textdata.html 6 mmahoney/compression/textdata.html rwan/en/restore.html 7 T T[0...N-1] S enwik8 LZE++ LZEnd LZEnd LZE LZE select B LZE++ LZE T k B B PrefixSpan T[0...N-1] S T[i...i + l] PostgreSQL 44.57% 71.32% 63.8 KiB 4.83% einstein.de.txt 23.78% 22.85% KiB 13.07% enwik % 47.46% 27.8 KiB 2.84% einstein.de.txt enwik8 0.93% 18.74% CPU L1/LL 8 LZEnd baseline LZE++ LZEnd CPU LL LZEnd LL 16.01% 55.59% 39.37% LL LZE KiB B 2 23 B LZE++ LZEnd LZE++ LZEnd LZEnd LZE LZE
7 6 2 5 B (32 B) 2 23 B (8 MiB) PostgreSQ einstein.de.txt enwik KiB LZEnd LZE LZE++ 64 KiB 3. 3 O(m) mk/n+1 while-loop 4. 4 LZEnd MiB Cassandra PostgreSQL wlen LZEnd 1.52% 6.15% world leaders 36.41% LZE++ Suffix Array 17bit 6 gzip bzip2 lzip Re-Pair LZEnd LZE++ LZEnd LZE++ Re-Pair DNA bzip2 enwik8 gov2 Snappy LZ7 LZEnd LZE++ LZE++ LZ-End LZE++ gzip Escherichia Coli einstein.de.txt bzip2 2 bzip2 lzip LZE++ Snappy LZ4 Re-Pair 4. 5 LZEnd LZE++ 9 LZEnd LZE LZEnd LZE B LZE++ LZEnd PostgreSQL LZEnd 5. [1] [2] [11] DBMS
8 1 LZE++ Apache Cassandra PostgreSQL ( ) wlen LZEnd 1.52% 6.15% Source Size(B) LZE++ LZEnd gzip-9 bzip-9 lzip Snappy LZ4 Re-Pair Escherichia Coli 112,689, influenza 154,808, einstein.de.txt 92,758, world leaders 46,968, enwik8 100,000, gov2 135,268, Hadoop 135,268, Cassandra 135, PostgreSQL 135,268, MiB ms LZE++ LZ-End Source LZE++ LZEnd gzip-9 bzip-9 lzip Snappy LZ4 Re-Pair Escherichia Coli influenza einstein.de.txt world leaders enwik gov Hadoop Cassandra PostgreSQL [1] Run Length Encoding Bitmap Encoding [5] [6] [7] [8] [2] LZEnd [4] [17] N O(logN) 6. m LZE++ LZE++ O(m) while-loop [1] D. Abadi et al., The Design and Implementation of Modern Column-Oriented Database Systems, Foundations and Trends c in Databases, Vol. 5, No. 3, pp , [2] P. Ferragina and G. Manzini, On Compressing the Textual Web, Proceedings of WSDM, [3] S. Kreft and G. Navarro, LZ77-like Compression with Fast Random Access, Proceedings of DCC, [4] S. Kreft and G. Navarro, Self-indexing based on LZ77, Proceedings of CPM, [5] K.Sadakane, New Text Indexing Functionalities of the Compressed Suffix Arrays, J. Algorithms, Vol. 48, No. 2, [6] R. Grossi, A. Gupta, and J. S. Vitter, High-Order Entropy- Compressed Text Indexes, Proceedings of SODA, [7] R.Grossi and J.S.Vitter, Compressed Suffix Arrays and Suffix Trees with Applications to Text Indexing and String Matching, SIAM J. Comput., Vol. 35, No. 2, pp , [8] P. Ferragina and G. Manzini, Opportunistic Data Structures with Applications, Proceedings of 41st FOCS, [9] U. Manber and G. Myers, Suffix Arrays: a New Method for On-line String Searches, Proceedings of SODA, [10] C. Kim et al., Closing the Ninja Performance Gap through Traditional Programming and Compiler Technology, techresearch.intel.com, [11] C. Hoobin et al., Relative Lempel-Ziv Factorization for Efficient Storage and Retrieval of Web Collections, Proceedings of VLDB Endow., Vol. 5, No. 3, [12] J. Pei, et al., PrefixSpan: Mining Sequential Patterns Efficiently by Prefix-Projected Pattern Growth, Proceedings of 17th ICDE, [13] Kosaraju, S.R. and Manzini, G., Compression of Low Entropy Strings with Lempel-Ziv Algorithms, Proceedings of CCS, [14] G. Manzini, An Analysis of the Burrows-Wheeler Transform, J. ACM, Vol. 48, No. 3, pp , [15] D. Okanohara and K. Sadakane, Practical Entropy-Compressed rank/select Dictionary, Proceedings of ALENEX, pp , [16] N. Larsson and A. Moffat, Offline dictionary-based compression, Proceedings of DCC, 1999, pp , [17] B. Philip et al., Random access to grammar-compressed strings, Proceedings of SODA, 2011
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JAIST Reposi https://dspace.j Title 質問の曖昧性を考慮した質問応答システムに関する研 究 Author(s) 松本, 匡史 Citation Issue Date 2006-03 Type Thesis or Dissertation Text version author URL http://hdl.handle.net/10119/1986 Rights Description
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