OBLIVIOUS STRING EMBEDDINGS AND EDIT DISTANCE APPROXIMATIONS

Transcription

1 OBLIVIOUS STRING EMBEDDINGS AND EDIT DISTANCE APPROXIMATIONS Tuğkan Batu a, Funda Ergun b, and Cenk Sahinalp b a LONDON SCHOOL OF ECONOMICS b SIMON FRASER UNIVERSITY LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 1

2 EDIT DISTANCE Let S and T be strings over alphabet Σ. Edit Distance D(S, T): number character insertions, deletions, and substitutions required to transform S into T. Many string similarity problems are based on edit distance. Used in text processing, analysis of genomic sequences,... Variants: non-uniform costs, block operations,... Exact Computation: [Masek Paterson 80] gave an O(n 2 / log n)-time algorithm for constant-size alphabets. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 2

3 EDIT DISTANCE APPROXIMATION γ-approximation: Given strings S,T, and γ > 1, output a value d such that D(S,T) d γ D(S,T). [Bar-Yossef et al. 04]: n 3/7 -approximation in near linear time (Õ(n) time). Our results: n 1 3 +o(1) -approximation in near linear time. n 1 ɛ 3 +o(1) -approximation in Õ(n1+ɛ ) time. Block edit distance variants: Almost logarithmic approximation factors [Muthukrishnan Sahinalp 00], [Cormode Muthukrishnan 02] LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 3

4 STRING (TO STRING) EMBEDDINGS Def. Let Σ and Γ be two alphabets. A string embedding with distortion d is a mapping φ : Σ Γ such that D(S,T) d 1 D(φ(S),φ(T)) d 2 D(S,T) for all S,T, and d 1 d 2 d. Interesting if φ reduces string length (dimensionality reduction): Def. A string embedding with reduction r > 1 maps a string of length n to a string of length at most n r. Warning: This is not a compression. We require Σ Γ. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 4

5 STRING EMBEDDINGS (CONT.) Lemma. A string embedding with reduction r has a distortion at least r. "Proof." Maximum edit distance reduces by a factor of r. Minimum edit distance stays at 1. Our result: A string embedding with reduction r and distortion r 1+o(1) can be computed in Õ(n) time. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 5

6 STRING EMBEDDINGS VIA PARTITIONING S : adebccebaaadebcbadec adebc cebaa adebc badec α β α δ T : dadebccebaaadebcbade dadeb cceba aadeb cbade ε γ κ λ We want: Consistency S : adebc cebaa adebc badec T : d adebc cebaa adebc bade Maybe too strong to ask! Need to use content! Locality LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 6

7 LOCALLY CONSISTENT PARSING (LCP)[SAHINALP VISHKIN 95,96] Partitions depend only on local string content. Consistency condition: If sufficiently long substring w occurs more than once in S, then most blocks in w will be set identically in each occurrence of w (in S or any other string). most blocks = all but some boundary blocks LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 7

8 PARSING CONSISTENTLY Convention: a > b > c > d > e >... > z Assume no character is repeated consecutively. S :... dcadecbgabcdefghkmopstuadecbhc... We will partition S into blocks of size 2 or Mark every local maximum in sliding window of 3 (= a character larger than immediate neighbors) (Primary markers) (No consecutive markers) S :... dc adec bg abcdef ghkmopstu adec bhc Partition the segments into blocks of size 2 (or 3 if necessary) (Secondary markers) S :... dc ad ec bg ab cd ef gh km op stu ad ec bhc... LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 8

9 CONSISTENT, BUT IS IT LOCAL? dcadecbgabcdef ghkmopstuadecbhc dc adec bg abcdef ghkmopstu adec bhc dc ad ec bg a b cd ef gh km op stu ad ec bhc Problem: Primary markers can be far apart (as far as twice the alphabet size). Hence, marker locations can depend on far away characters. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 9

10 ALPHABET REDUCTION Note that all we need for parsing are no repetitions and total order of characters. We will reduce alphabet size to avoid far-away primary markers. Σ = {a = 1111, b = 1110, c = 1101, d = 1011, e = 1010,...} Assign a tag (with shorter bit complexity) to each character. Tag of ith character: rightmost bit position where S[i] and S[i 1] differ concatenated with value of that bit in S[i] S : a e b tag : Remark. Still no repetitions in tags. Reduction in alphabet: k bits to log k + 1 bits LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 10

11 ITERATIVE ALPHABET REDUCTION Apply alphabet reduction log k times ( k: initial alphabet size) the tags are constant use the LCP procedure (on tags) to partition string into blocks of size 2 and 3 Each marker is set based on O(log k) locations in the string. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 11

12 OBTAINING LONGER BLOCKS Problem. Blocks are of size 2 and 3. What if we need longer blocks? 1. Label blocks (by a new alphabet) Use the same character for different occurences of the same block. 2. Partition this new string into blocks of size 2 and 3. Step 1 : Step 2 : ad ec bg ab cd ef gh ko mp stu ad ec α β χ δ ε φ ν ß κ λ α β αβ χδε φ ν ßκλ αβ adec bgabcd ef gh kompstu adec Block sizes between 2 2 = 4 and 3 2 = 9. After t repetitions: Block sizes between 2 t and 3 t. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 12

13 LCP(C) We generalize above technique to obtain blocks of size between c and 2c 1. Same underlying ideas Various periodicities are handled separately Alphabet reduction is achieved by comparing substrings of length 2c 3 instead Computed in O(c 2 n) time. Markers depend on roughly O(c 2c ) locations Lemma. One edit operation to a string can change at most O(c 2c ) markers in LCP(c). LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 13

14 ITERATIVE APPLICATION OF LCP(C) 1. Label blocks (by a new alphabet) Use the same character for different occurences of the same block. 2. Partition this new string into blocks of size between c and 2c 1. Step 1 : Step 2 : ad ec bg ab cd ef gh ko mp stu ad ec α β χ δ ε φ ν ß κ λ α β αβ χδε φ ν ßκλ αβ adec bgabcd ef gh kompstu adec Block sizes between c 2 and (2c 1) 2. After t repetitions: Block sizes between c t and (2c 1) t. LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 14

15 STRING EMBEDDINGS VIA LCP(C) String embedding φ(s) with reduction r = Ω(log n): 1. Choose c = log log n/ log log log n and t such that c t = r. 2. Apply LCP(c) t times to partition S. block sizes between c t = r and (2c 1) t = r 1+o(1). 3. Label the blocks with new alphabet Γ to obtain string φ(s). Lemma. D(S,T)/(2c 1) t D(φ(S),φ(T)) O(c 2c ) D(S,T). Corollary. Embedding φ has distortion r 1+o(1). LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 15

16 LCP(2) VS LCP(C) When we use LCP(c) to get reduction r, we set c t = r. The distortion is the largest block size: (2c 1) t (2c) t = 2 t r. Overhead of LCP(c) on the distortion: 2 t = r log c 2 LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 16

17 APPROXIMATING EDIT DISTANCE A naive idea: Given S, T, and γ > 1, 1. Let S = φ(s) and T = φ(t) for a suitable reduction r (see below). 2. Calculate D(S,T ) using dynamic programming (in O((n/r) 2 ) time). 3. Translate into an approximation to D(S,T) using distortion of φ. Setting (n/r) 2 = n, gives r = n. Does not yield better than n-approximation in linear time! LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 17

18 HOW TO CALCULATE D(S, T ) MORE EFFICIENTLY We can exploit properties of S and T to compute D(S, T ). If D(S, T) k, then the number of insertions and deletions (from S to T ) is less than k/r. Observation. During the computation of D(S, T ), we do not have to compare far away locations in S and T. Hence, we can restrict algorithm to look at a "narrow" band along diagonal of DP table. Hence, we can make S and T longer (read: smaller r) for better accuracy. Say, use r = n 1/3. Result: n (1 ɛ)/3+o(1) -approximation in Õ(n 1+ɛ ) time. (We can get even better if D(S, T) < n 2/3 ). LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 18

19 FUTURE DIRECTIONS Better edit distance approximation Low distortion L 1 embeddings for strings: φ : Σ L d 1 Ω(log n) distortion lower bound [Krauthgamer Ostrovsky 06] 2 O( log nlog log n) best known [Ostrovsky Rabani 05] Other string-similarity problems LSE CDAM Seminar Oblivious String Embeddings and Edit Distance Approximations October 19, 2006 p. 19