Fast and Simple Circular Pattern Matching

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New tabulation and dynamic
programming based techniques for sequence
similarity problems
Szymon Grabowski
Lodz University of Technology, Institute of
Applied Computer Science, Lódz,
Polandsgrabow_at_kis.p.lodz.pl
Sept. 2014
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Agenda

1. (Naïve) dynamic programming.
2. Four Russians.
3. Main LCS results.
4. Bille Farach-Coltontechnique.
5. Our improvement of the BFC alg.
6. Our LCS result with sparse DP.
7. Algorithmic apps (Lev distance, LCTS, MerLCS).
8. Concl open problems.

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Dynamic Programming (DP)

• Everybody knows
• Quadratic cost for 2 sequences (cant compute a
  cell "in a middle" before knowing the previous
  rows/cols),
• Speedup ideas tabulation (aka Four
  Russians),bit-parallelism, sparse dynamic
  programming,compressing the input sequences.

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DP made (slightly) faster
If we can process blocks of b ? b symbols in O(1)
time, we immediately obtain O(mn / b2) time. We
can do it (Masek Paterson, 1980) e.g. for
binary alphabet and b log n / 4 ? O(mn / log2
n) time.
The idea is to precompute all possible inputs
(short enough strings are guaranteed to repeat
and represent the DP values in differential
manner).
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LCS, selected results (time compl.)
Standard DP O(mn). Tabulation (Masek Paterson,
1980) O(mn / log2 n) for a constant
alphabet. Tabulation (Bille Farach-Colton,
2008) O(mn (log log n)2 / log2 n) for an
integer alphabet. Bit-parallelism (Allison Dix,
1986, ) O(mn / w), w ? log n is machine word
size (in bits). Sparse DP Hunt Szymanski,
1977 O(r log log n), r is the of
matches,Eppstein, Galil, Giancarlo Italiano,
1992 O(D log log(minD, mn / D)), D ? r is the
of dominant matches.
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LCS, selected results, contd
Sparse DP Sakai, 2012 O(m? minD?, p(m-q)
n),where p LCS(A, B), q LCS(A1m,
B). LZ78-compressed inputCrochemore, Landau
Ziv-Ukelson, 2003O(hmn / log n), for a constant
alphabet,where h ? 1 is the entropy of the
inputs (for a binary alph.). RLE-compressed
inputseveral results, incl. Liu, Wang Lee,
2008O(minnl, km), where l, m are
RLE-compressed seq lengths. SLP-compressed
inputGawrychowski, 2012 O(kn sqrt(log(n / k)),
where k is total length of SLP-compressed
sequences.
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The technique of Bille Farach-Colton
For an integer alphabet of size ?, the Masek
Patersonresult can easily be modified to have
O(mn log2 ? / log2 n) time. This is fine for
small ?, but not if ? nc, c gt 0.
Bille Farach-Colton use alphabet mapping in
superblocks.Use superblocks of size e.g. log3 n
? log3 n and divide each superblock into blocks
of size ?(log n / log log n) ? ?(log n / log log
n).
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BFC, contd
That is, for current text snippet from A of
length log3 nextract up to log3 n distinct
symbols and encode the current snippet of A and
current snippet of B accordingly (one extra
symbol for "smth else" in snippet B needed).
Easily, O(log log n) bits per encoded symbol
are enough, mapping times overall negligible (a
BST can be used with log(superblock)-factor per
symbol) and O(mn (log log n)2 / log2 n) total
time.
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BFC, alphabet mapping example
Blocks of size 3 ? 3, superblocks of size 9 ? 9.
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Our technique (Alg 1)
Use the BFC alphabet mapping in superblocks. But
use many LUTs (instead of 1), yet with modified
input.One LUT per horizontal stripe (of length
n).

• The LUT input
• snippet of A,
• left block border (1 bit per cell),
• upper block border (1 bit per cell).
• No snippet of B as part of the input, as it is
  fixed for a given LUT! (Re-use LUTs for
  repeating snippets of B.)
• Thanks to it, we work on rectangular (not square)
  "portrait"-oriented blocksof size ?(log n / log
  log n) ? ?(log n).

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One horizontal stripe (4 blocks of 5 ? 5)
seq A
seqB
Red arrows explicitly stored LCS values black
arrows diff-encoded LCS values.
05550 and 34023 text snippets encoded with ref
to a superblock (not shown).
The diagonally shaded cells are the block output
cells.
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LCS, first result (Alg 1)
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Output-dependent algorithm
We work in blocks of (b1) ? (b1), but divide
theminto sparse ones, which have ? K
matches,and dense ones with gt K matches.
Key observationknowing the top row and leftmost
column for the blockplus the location of all
matches in itis enough to compute this
block.That is, the text snippets are not needed!
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Where sparse DP meets tabulation

• A sparse block input
• top row b bits (diff encoding),
• leftmost column b bits (diff encoding),
• match locations each in log(b2) bits,totalling
  O(K log b) bits.
• (Output even less.)

Hence, if K log b b O(log n) (with a small
enough constant), we can use a LUT for all sparse
blocks and compute each of them in constant time.
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Dense blocks
Dense blocks are partitioned into smaller
blockswhich then will be processed by our
technique from Alg 1. The smaller block sizes
are?(log n / log log n) ? ?(b).
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Choosing the parameters
b O(log n) (otherwise the LUT build costs will
be dominating), but also b ?(log n / sqrt(log
log n)) (otherwise this alg will never beat Alg
1). This implies K ?(log n / log log n), with
an appropriate constant.
For a small enough r ( total of matches in the
matrix) we may have O(mn / log2 n) from the above
formula, alas in the pp we have to find and
encode all matches in all sparse blocks, in O(n
r) time.
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LCS, second result (Alg 2)
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Alg 2 niche

• Considering the results of
• Eppstein et al., 1992,
• Sakai, 2012,
• Alg 1,
• we obtain the following niche in which Alg 2 is
  the winner

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Simple generalization of Th. 1 and 2
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Longest common transposition-invariant
subsequence (LCTS)
LCTS LCS in the best key transposition (in
music, transposition is shifting a sequence of
notes (pitches) up or down by a constant
interval).
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LCTS, known results and a new one
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Merged LCS (MerLCS)
A bioinformatics problem on 3 sequencesgiven
sequences A, B and P, return a longest seq. T
that is a subsequence of Pand can be split into
two subsequences T and Tsuch that T is a
subsequence of Aand T is a subsequence of
B.A n, B m, P u.
Known resultsPeng, Yang, Huang, Tseng Hor,
2010 O(lmn) time,where l ? n is the result
length. Deorowicz Danek, 2013 O(??u / w? mn
log w) time.
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Our result for MerLCS
DP matrix propertyDeorowicz and Danek noticed
thatM(i, j, k) is equal to or larger by 1
thanany of the three neighhbors M(i 1, j,
k), M(i, j 1, k), M(i, j, k 1).
We generalize our result on 2 sequences to 3
sequences (input 3 text snippetsplus 3 2-dim
walls instead of 1-dim borders!)to obtain O(mnu
/ log3/2 n) for MerLCS,if u ?(nc) for some c gt
0.
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Conclusions

• Tabulation ( Four Russians) is a classic
  DP-boosting technique. Interestingly, we managed
  to (slightly) improve its application to the LCS
  / edit distance problem.

• Applying tabulation may be even better for a
  sparse matrix.

• These techniques work also for a few other
  problems than LCS and edit distance.

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Open problems

• Can we improve the tabulation based result on
  compressible sequences?

• Can we adopt our technique(s) to problemsin
  which the conditions from Lemma 3 (or Lemma 7,
  involving 3 sequences) are relaxed, that is,
  consecutive DP cells may (sometimes) differ more
  than by a constant?Exemplary problem
  SEQ-EC-LCS (Chen Chao, 2011 Deorowicz
  Grabowski, 2014).

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