Small-Space Dictionary Matching (Dissertation Proposal)

Transcription

1 Small-Space Dictionary Matching (Dissertation Proposal) Graduate Center of CUNY 1/24/2012

2 Problem Definition Dictionary Matching Input: Dictionary D = P 1,P 2,...,P d containing d patterns. Text T of length n. Output: All positions in text at which a dictionary pattern occurs.

3 Applications Dictionary Matching Search for specific phrases in a book Scanning file for virus signatures Network intrusion detection systems Searching DNA sequence for a set of motifs

4 Small-Space Challenge: Limited storage capacity in devices. Massive proliferation of data. Goal: efficient algorithms with respect to both time and space.

5 Research Plan Thesis Goals: 1 Succinct 2D dictionary matching 2 Succinct dynamic 2D dictionary matching 3 for succinct 1D dictionary matching

6 Small-Space 1D One-Dimensional Two-Dimensional 1D single pattern matching in linear time and O(1) working space: Galil and Seiferas (1981) Crochemore and Perrin (1991) Rytter (2003) Gasieniec and Kolpakov (2004)

7 Empirical Entropy One-Dimensional Two-Dimensional How is space measured in succinct data structures and algorithms? empirical entropy Definition Empirical entropy of a string, H 0 or H k describes the minimum number of bits that are needed to encode the string within context. H 0 (S) = σ i=1 n i n log n i n

8 Empirical Entropy One-Dimensional Two-Dimensional H 0( s) = σ i= 1 ni ni log n n

9 Empirical Entropy One-Dimensional Two-Dimensional H k (S): lower bound to the compression we can achieve for each symbol using a code which depends on the k symbols preceding it. H k (S) = 1 w n S H 0 (w S ) w Σ k w S is the sequence of characters following occurrences of w in S.

10 Empirical Entropy One-Dimensional Two-Dimensional For any string S and k 0, H k (S) H k 1 (S) H 0 (S) log Σ

11 Small-Space 1D One-Dimensional Two-Dimensional 1D dictionary matching in small space : Space (bits) Search Time Reference O(l log l) O(n + occ) Aho-Corasick (1975) O(l) O((n+occ)log 2 l) Chan et al. (2007) lh k (D)+o(llogσ)+O(d logl) O(n(log ǫ l +logd)+occ) Hon et al. (2008) l(h 0 +O(1))+O(d log(l/d)) O(n+occ) Belazzougui (2010) lh k (D)+O(l) O(n+occ) Hon et al. (2010) d is the number of patterns in D. l is the total size of the dictionary.

12 Small-Space 1D Dictionary One-Dimensional Two-Dimensional Compressed AC automaton [Hon et al. (2010)]: No slowdown! Separates the three functions of the AC automaton. * goto * report * fail Encodes each function differently. Space complexity meets H k (D), kth order empirical entropy.

13 Dynamic Dictionary Matching One-Dimensional Two-Dimensional 1D dynamic dictionary matching: Preprocess Time Update Time Search Time Ref. O(llogl) O(plogl) O((n+occ)logl) AF91 O(llogσ) O(plogl) O((n+occ)logl) IS94 O(llogσ) O(p logl loglogl ) logl O((n+occ) loglogl ) AFIPS95 O(l) O(p) O(n + occ) SV96 d is the number of patterns in D. l is the total size of the dictionary. p is the size of a pattern to add / remove.

14 Dynamic Dictionary Matching One-Dimensional Two-Dimensional Previous dynamic algorithms require O(l log l) extra space. 1D dynamic dictionary matching in small space : Chan et al. (2007) Update Time: O(plog 2 l) Search Time: O((n+occ)log 2 l) Space: O(lσ) Hon et al. (2009) Update Time: O(plogσ +logl) Search Time: O(nlogl+occ) Space: meets kth order empirical entropy

15 Small-Space 2D One-Dimensional Two-Dimensional 2D linear-time single pattern matching Crochemore et al. (1995): Preprocessing: linear time within log space. Text Scanning: linear time, O(1) extra space. Use small-space 2D single pattern matching for set of patterns * requires several scans of text.

16 2D Dictionary Matching One-Dimensional Two-Dimensional Existing 2D dictionary matching algorithms: Bird (1977) / Baker (1978) Amir, Farach (1992) Giancarlo (1995) Idury, Schaffer (1995) Require working space proportional to input.

17 2D Dictionary Matching One-Dimensional Two-Dimensional Existing 2D dictionary matching algorithms: Bird (1977) / Baker (1978) * For rectangular patterns same size in one dimension. * Linearize dictionary and label text. * Use 1D dictionary matching. * Linear time and space. Amir, Farach (1992) Giancarlo (1995) Idury, Schaffer (1995)

18 2D Dictionary Matching One-Dimensional Two-Dimensional Existing 2D dictionary matching algorithms: Bird (1977) / Baker (1978) Amir, Farach (1992) * For square patterns of different sizes. * Linearization around diagonals. * Use suffix tree and AC automaton. * Linear time and space. Giancarlo (1995) Idury, Schaffer (1995)

19 2D Dictionary Matching One-Dimensional Two-Dimensional Existing 2D dictionary matching algorithms: Bird (1977) / Baker (1978) Amir, Farach (1992) Giancarlo (1995) * For square patterns of different sizes. * Uses 2D suffix tree. * Slower than linear time. * Linear space. Idury, Schaffer (1995)

20 2D Dictionary Matching One-Dimensional Two-Dimensional Existing 2D dictionary matching algorithms: Bird (1977) / Baker (1978) Amir, Farach (1992) Giancarlo (1995) Idury, Schaffer (1995) * For rectangular patterns of different sizes. * Multidemensional range searching and dictionary matching. * Slower than linear text processing. * Linear space.

21 Dynamic 2D Dictionary Matching One-Dimensional Two-Dimensional For square patterns: Update Time Search Time Ref. O(p 2 log D ) O((n 2 +occ)log D ) AFIPS (1995) O(p 2 log 2 D ) O((n 2 logd +occ)log D ) Giancarlo95 O(p 2 +plogd) O((n 2 +occ)logd) CL96 Pattern P i is a square of size p i p i. d d D = pi 2 and D = p i. i=1 i=1 P, of size p p, is added / removed.

22 Dynamic 2D Dictionary Matching One-Dimensional Two-Dimensional For rectangular patterns of different sizes: Idury, Schaffer (1995) Groups patterns by size. Multidemensional range searching and dictionary matching. O(log 4 D ) slowdown in all stages * Preprocess dictionary * Scan text * Update dictionary

23 Research Plan One-Dimensional Two-Dimensional Thesis Goals: 1 Succinct 2D dictionary matching Develop first efficient algorithm Linear Time Small-Space 2 Succinct dynamic 2D dictionary matching 3 for succinct 1D dictionary matching

24 Problem Definition One-Dimensional Two-Dimensional 2D Dictionary Matching Input: Dictionary of d patterns, each is m m in size. Text T of size n n. Output: All positions in text at which a dictionary pattern occurs.

25 2D Dictionary Matching One-Dimensional Two-Dimensional Bird / Baker Convert 2D data to 1D representation. Name patterns rows. Name text positions. Use 1D dictionary matching to find pattern occurrences.

26 2D Dictionary Matching One-Dimensional Two-Dimensional Bird / Baker Convert 2D data to 1D representation. Name patterns rows. Name text positions. Use 1D dictionary matching to find pattern occurrences. Our work: mimic Bird/Baker algorithm in small space.

27 1D Periodicity One-Dimensional Two-Dimensional Definition A string p is periodic in u if p = u k u where u is a proper prefix of u, u is primitive, and k 2. aabcaabcaabcaa

28 1D Periodicity One-Dimensional Two-Dimensional Definition A string p is periodic in u if p = u k u where u is a proper prefix of u, u is primitive, and k 2. We divide patterns into 2 groups based on 1D periodicity. In each case, different difficulties to overcome.

29 Types of Patterns One-Dimensional Two-Dimensional Case I: Patterns with ALL rows periodic, period m/4. Problem: can have more candidates than the space we allow. Case II: Patterns contain aperiodic row or row with period > m/4. Problem: several patterns can overlap in both directions.

30 Types of Patterns One-Dimensional Two-Dimensional Case I: Patterns with ALL rows periodic, period m/4. Problem: can have more candidates than the space we allow.

31 Lyndon Words One-Dimensional Two-Dimensional Definition Two words x, y are conjugate if x = uv, y = vu for some u, v. Definition A Lyndon word is a primitive string which is lexicographically smaller than any of its conjugates. Canonization computes the least conjugate of a word.

32 Naming One-Dimensional Two-Dimensional New technique for naming rows: same name if periods are conjugate

33 Naming One-Dimensional Two-Dimensional New technique for naming rows: same name if periods are conjugate How is this done in linear time, yet small space? witness tree Witness tree stores a distinction between two names. To name a new row, it is compared to only one other row.

34 Types of Patterns One-Dimensional Two-Dimensional Case II: Patterns contain aperiodic row or row with period > m/4. Problem: several patterns can overlap in both directions.

35 Types of Patterns One-Dimensional Two-Dimensional Case II: Patterns contain aperiodic row or row with period > m/4. Problem: several patterns can overlap in both directions. Many 1D names can overlap in a text block row. Identification of candidates is simpler. Identify candidates with aperiodic row of each pattern.

36 Types of Patterns One-Dimensional Two-Dimensional Case II: Patterns contain aperiodic row or row with period > m/4. Problem: several patterns can overlap in both directions. Difficulty: single pass verification. We introduce dynamic dueling. Eliminate vertically inconsistent candidates. 1 Use 1D representations in each column. 2 Use witness tree to compare row names in constant time. Then verify consistent candidates against text rows.

37 Research Plan One-Dimensional Two-Dimensional Thesis Goals: 1 Succinct 2D dictionary matching 2 Succinct dynamic 2D dictionary matching Develop dynamic algorithm Adapts to modifications of dictionary * Efficiently insert pattern * Efficiently delete pattern Linear Time Small-Space Dynamic succinct version of Bird/Baker algorithm. 3 for succinct 1D dictionary matching

38 Small-Space 1D Indexing Dictionary Matching 1D dictionary matching in small space : Space (bits) Search Time Reference O(l log l) O(n + occ) Aho-Corasick (1975) O(l) O((n+occ)log 2 l) Chan et al. (2007) lh k (D)+o(llogσ)+O(d logl) O(n(log ǫ l +logd)+occ) Hon et al. (2008) l(h 0 +O(1))+O(d log(l/d)) O(n+occ) Belazzougui (2010) lh k (D)+O(l) O(n+occ) Hon et al. (2010) d is the number of patterns in D. l is the total size of the dictionary.

39 1D Dictionary Matching Indexing Dictionary Matching For 1D data, Time-Space efficient dictionary matching has been achieved. * Only in the theoretical realm. * Implementations lag behind. * Rely on complex data structures that have not been implemented. Our goal: Efficient software for succinct dictionary matching that relies on popular succinct data structures.

40 Suffix Tree Indexing Dictionary Matching M i s s i s s i p p i $

41 Suffix Tree Indexing Dictionary Matching Applications of Suffix Tree: Single pattern matching Longest common substring of suffixes Longest repeated substring Longest palindrome

42 Suffix Array Indexing Dictionary Matching sort suffixes alphabetically

43 Suffix Tree Indexing Dictionary Matching M i s s i s s i p p i $ s 12 s 11 s 8 s 3 s 6 s 5 s 2 s 1 s10 s 9 s 7 s 4

44 Suffix Tree Indexing Dictionary Matching Dynamic Suffix Tree Choi, Lam (1997) Strings can be inserted or deleted efficiently. Update time: O(p) to insert/delete string of size p. No edge labeled by a deleted string. Additional Storage: two-way pointer for each edge.

45 Compressed Suffix Tree Indexing Dictionary Matching Compressed suffix tree (CST) Compressed self-index Replaces input data and answers queries No more space than underlying data Minor slowdown in compressed suffix array as well

46 Compressed Suffix Tree Indexing Dictionary Matching Compressed suffix tree (CST) is composed of several compressed data structures: * Compressed Suffix Array * Compressed LCP Array * Navigation between nodes

47 Compressed Suffix Tree Indexing Dictionary Matching Uncompressed Suffix Tree: O(n log n) bits of space. Compressed Suffix Tree: 1 Sadakane (2007): O(n log σ) bits, O(polylog(n)) slowdown. 2 Russo et al. (2011): kth order empirical entropy, O(log n) slowdown. 3 Fischer et al. (2009): kth order empirical entropy, sub-logarithmic slowdown. 4 Ohlebusch, Fischer, Gog (2011): some queries in O(1) time. Have all been implemented.

48 Compressed Suffix Tree Indexing Dictionary Matching Dynamic Compressed Suffix Tree 1 Chan et al. (2007): Adaptation of Sadakane (2007). O(n) bits O(log 2 n) slowdown. 2 Russo et al. (2011): nh k +o(nlogσ) bits. O(log 2 n) slowdown. Have not been implemented.

49 Suffix Tree Indexing Dictionary Matching Suffix Tree for set of patterns: 1 Concatenate patterns with unique delimiter between them. OR * Problem: indexes artificial suffixes. 2 Generalized suffix tree * Combines suffixes of each pattern in a single data structure. * Construction: Ukkonen s online suffix tree construction algorithm.

50 Suffix Links Indexing Dictionary Matching Definition A suffix link is a pointer from an internal node labeled xs to another internal node labeled S, where x is an arbitrary character and S is a possibly empty substring. Suffix links facilitate traversal of suffix tree during and after construction.

51 Suffix Links Indexing Dictionary Matching M i s s i s s i p p i $ s 12 s 11 s 8 s 3 s 6 s 5 s 2 s 1 s10 s 9 s 7 s 4

52 Indexing Dictionary Matching Algorithm for 1D dictionary matching on suffix tree: Generalized suffix tree: index of several strings Ukkonen can insert one string at a time Our algorithm: modeled after Ukkonen s suffix tree construction algorithm Online processing of text Linear time: as if inserting new pattern Skip-count trick uses suffix links

53 Indexing Dictionary Matching Algorithm for 1D dictionary matching on suffix tree: Generalized suffix tree: index of several strings Ukkonen can insert one string at a time Our algorithm: modeled after Ukkonen s suffix tree construction algorithm Online processing of text Linear time: as if inserting new pattern Skip-count trick uses suffix links Small-Space: compressed suffix tree

54 Skip-Count Trick Indexing Dictionary Matching M i s s i s s i p p i $ s 12 s 11 s 8 s 3 s 6 s 5 s 2 s 1 s10 s 9 s 7 s 4

55 Research Plan Indexing Dictionary Matching Thesis Goals: 1 Succinct 2D dictionary matching 2 Succinct dynamic 2D dictionary matching 3 for succinct 1D dictionary matching Uses compressed suffix tree Space meets entropy-compressed bounds Linear time, with slowdown to query compressed self-index

56 Evaluation Indexing Dictionary Matching Data Sets: Biological * Fly sequences: FlyBase * Flu sequences: Influenza Virus Sequence Database Security * Network intrusion detection signatures: ClamAV Virus detection * Virus signatures: Snort

57 Timeline Indexing Dictionary Matching Thesis Goals: 1 Succinct 2D dictionary matching 2 Succinct dynamic 2D dictionary matching 3 for succinct 1D dictionary matching

58 Timeline Indexing Dictionary Matching Thesis Goals: 1 Succinct 2D dictionary matching Case I: presented at CPM 2010 Case II: presented at WADS 2011 * With no slowdown! * Complete algorithm accepted to Algorithmica 2 Succinct dynamic 2D dictionary matching 3 for succinct 1D dictionary matching

59 Timeline Indexing Dictionary Matching Thesis Goals: 1 Succinct 2D dictionary matching 2 Succinct dynamic 2D dictionary matching Algorithm has been developed Complexity has been proven Manuscript is in final editing stages 3 for succinct 1D dictionary matching

60 Timeline Indexing Dictionary Matching Thesis Goals: 1 Succinct 2D dictionary matching 2 Succinct dynamic 2D dictionary matching 3 for succinct 1D dictionary matching Dictionary matching over generalized Ukkonen suffix tree. Ported code to run over compressed suffix tree, using SDSL. Assumption: no pattern is substring of another. Add lowest marked ancestor framework to overcome. Evaluate time/space trade-offs of CST implementations. Run over large, realistic data sets.

61 Indexing Dictionary Matching Thank you!