An Efficient Algorithm for Finding Similar Short Substrings

An Efficient Algorithm for Finding Similar Short Substrings from Large Scale String Data Takeaki Uno National Institute of Informatics, JAPAN & The Graduate University for Advanced Science May/23/2008 PAKDD 2008

Motivation: Analyzing Huge Data • Recent information technology gave us many huge database - Web, genome, POS, log, … • "Construction" and "keyword search" can be done efficiently • The next step is analysis; capture features of the data - statistics, such as size, #rows, density, attributes, distribution… • Can we get more? Database look at (simple) local structures but keep being simple and basic 実験 1 実験 2 実験 3 　● 　▲ 　▲ 　● 　● 　▲ 　▲ 　実験 4 　　▲ 　● 　● 　 Results of experiments ATGCGCCGTA TAGCGGGTGG TTCGCGTTAG GGATATAAAT GCGCCAAATA ATAATGTATTA TTGAAGGGCG ACAGTCTCTCA ATAAGCGGCT genome

Our Focus • Find all pairs of similar objects (or structures) (or binary relation, instead) • Maybe, this is very basic and fundamental 　　There would be many applications - detecting global similar structures, - constructing neighbor graphs, - detecting locally dense structures (groups of related objects) In this talk, we look at strings

Existing Studies • There are so many studies on similarity search (homology search) 　Given a database, construct a data structure which enables us to find the objects similar to the given a query object, quickly - strings with Hamming distance, edit distance - points in plane (k-d trees), Euclidian space - sets - constructing neighbor graphs (for smaller dimensions) - genome sequence comparison (heuristics) • Both exact and approximate approaches • All pairs comparison does not work for large scale data

Approach from Algorithm Theory • Parallel computation is a popular way to fast computation, but its high cost, including hardness of programming, is a disadvantage • Algorithm improvement decreases the increase against the database size by the derivation on the design of the way of computation size = 100 size = 1, 000 2 -3 times 10, 000 times Efficiency increases as the increase of database size We approach the problem from the algorithmic point

Our Problem • We address databases whose records are short strings Problem: For given a database composed of n strings of the fixed same length l, and a threshold d, find all the pairs of strings such that the Hamming distance of the two strings is at most d. • We propose an efficient algorithm SACHICA (Scalable Algorithm for Characteristic/Homogenous Interval Calculation), and a method to detect long similar substrings of input strings (especially efficient for genomic data) ATGCCGCG GCGTGTAC GCCTCTAT TGCGTTTC TGTAATGA 　　．．．・ ATGCCGCG , AAGCCGCC ・ GCCTCTAT , GCTTCTAA ・ TGTAATGA , GGTAATGG 　　　　．．．

Approaching Long-string Similarity • When two strings S 1 and S 2 are similar, they must have several pairs of similar short substrings • “Having several similar substrings” is a necessary condition to be similar strings Ex) for strings of length 3000 s. t. , Hamming distance 290 (=10%) they have at least 3 pairs of substrings of length 30 with Hamming distance at most 2 the position of these substrings must differ at most 30, if we allow deletion and insertion • It gives a condition that substrings of length β are similar only if “k pairs of their short substrings are similar, and their start positions differ at most α

Detecting Long Similar Substrings • Consider to find long similar substrings of given strings S 1 and S 2 • Comparison of all substrings of length β needs square time redundant overlapping pairs use our similarity condition (1) find all pairs of similar short substrings (2) scan diagonal belt of width 2α to find an interval of length β including k pairs (3) shift the diagonal belt by α, and repeat • We can always find substrings of length βsatisfying the condition approach from similar short substrings is possible

Related Works • Computing edit/Hamming distance is done in square/linear time the whole strings have to be similar can not detect local exchange • Heuristic homology search such as BLAST, Pattern Hunter usually finds exact match of short substrings (11 letters), and extend must find terrible number of pairs when input strings are huge lengthen 11 letters loses the accuracy heuristics, ignoring frequent substring, dealing only gene areas • Similarity search involves huge number of queries, taking much longer time than exact search

Trivial Bound of the Complexity • If all the strings are exactly the same, we have to output all the pairs, thus take Θ(n 2) time 　 simple all pairs comparison of O(l n 2) time is optimal, if l is a fixed constant 　 is there no improvement? • In practice, we would analyze only when output is small, otherwise the analysis is non-sense 　 consider complexity in the term of the output size M: #outputs We propose O(2 l(n+l. M)) time algorithm

Basic Idea: Solve Subproblem • Consider the partition of strings into k blocks, and a subproblem: for given k-d block positions, find all pairs of strings with distance at most d s. t. "the given blocks are the same" Ex) 2 nd, 4 th, 5 th blocks of S 1 and S 2 (length 30) are the same much fewer comparisons !! • We can solve by "radix sort" on combined blocks, in O(l n) time.

Examine All Cases • Solve the subproblem for all combinations of the positions if distance of two strings S 1 and S 2 is at most 2, letters on l-2 blocks are the same in at least one combination of blocks, the pair ”S 1 and S 2” is found (in the subproblem of combination P) • #combinations is k. Cd. When k=5 and d=2, it is 10 computation is "radix sorts +α", O(k. Cd ln ) time for sorting recursive radix sort to reducing to O(k. Cd n )

Example ・ Find all pairs of strings with Hamming distance at most 1 ABCDE ABDDE ADCDE CDEFG CDEFF CDEGG AAGAB A BCDE A BDDE A DCDE C DEFG C DEFF C DEGG A AGAB 　 A A A C C C A BC BD DC DE DE DE AG DE DE DE FG FF GG AB ABC ABD ADC CDE CDE AAG DE DE DE FG FF GG AB

Figure out Intuition • Finding pairs of similar records is something finding all certain cells in a matrix • All pairs comparison sweeps and looks at all cells • Our multi-classification algorithm recursively reduces the areas to be checked in many ways, thus the search route forms a tree, whose leaves corresponds to a group of strings to be compared

Avoid Duplications by Canonical Positions • For two strings S 1 and S 2, their canonical positions are the first l-d positions of the same letters • Only we output the pair S 1 and S 2 only in the subproblem of their canonical positions • Computation of canonical posisions takes O(l) time, "+α" needs O(M l k. Cd ) time Avoid duplications without keeping the solutions in memory O(l. Cd (n+d. M)) = O(2 l (n + l. M) ) time in total ( if we set k=l )

Difference from BLAST • The original “BLAST” algorithm finds pairs of the identical intervals of 11 letters roughly, classifies into 411 = 4 million groups may take long time for 100 million letters • Our method for length 30 with Hamming distance 3 (quality equal to finding same interval of 7 letters), with dividing into 6 blocks roughly, classifies into 415 = 1, 000 million groups, (20 times) may take long time for 2000 million letters, but we can increase the #blocks But, not good at searching a given short string (no difference of time between many strings and one string)

Experiments: l = 20 and d = 0, 1, 2, 3 Prefixes of Y chromosome of Human Note PC with Pentium M 1. 1 GHz, 256 MB RAM

Comparison of Chromosome • Grid lines detect "repetitions of similar structures" 20 min. by PC chimpanzee 22 nd chr. Human 21 st and chimpanzee 22 nd chromosomes • Take strings of 30 letters from both, with overlaps • Intensity is given by # pairs human 21 st chr. • White possibly similar • Black never similar

Homology Search on Mouse X Chr. Human X and mouse X chromosomes (150 M strings for each) human X chr. 1 hour by PC mouse X chr. • take strings of 30 letters beginning at every position ・ For human X, without overlaps ・ d=2, k=7 ・ dots if 3 points are in area of width 300 and length 3000

Comparison of Many Bacterias Comparison of the genomes of 30 bacteria • The genomes are concatenated and compared in the same way 1 hour by PC

Comparison of BAC clones • Sequencing a genome is done by detecting overlaps of fragments • When genome has complex repeating structures, detection is hard • We detected the overlaps in the mouse genome, and completed some undetermined complex repeating parts (joint research with Koide, Umemori of National Institute of Genetics, Japan) 1 sec. by PC for a pair

Extensions ? ? ? • Can we solve the problem for other objects? (sets, sequences, graphs, …) • For graphs, maybe yes, but not sure for the practical performance • For sets, Hamming distance is not preferable. for large sets, many difference should be allowed. • For continuous objects, such as points in Euclidian space, we can hardly bound the complexity in the same way. (In the discrete version, the neighbors are finite, actually classified into constant number of groups)

Conclusion • Output sensitive algorithm for finding pairs of similar strings ( in the term of Hamming distance) • Multi-classification for speeding up • Application to genome sequence comparison Future works • Models and algorithms for natural language text • Extension to other objects (sets, sequences, graphs) • Extension to continuous objects (points in Euclidian space) • Efficient spin-out heuristics for practice • Genome analyze tools and systems