Gad M' Landau

1
LCS Approximation via Embedding into Locally
Non-repetitive Strings

• Gad M. Landau
• Avivit Levy
• Ilan Newman

2
The Problem

• Longest Common Subsequence (LCS)
• important similarity measure
• Example
• X a a b a e c b b e a c d c e
• Y a b d d a e c b b e c d c e
• ED deletions, insertions, mismatches

3
Motivation

• Exact computations
• dynamic programming - O(n2)
• Hunt-Szymanski - O((r n)log n),
  rmatches
• Hirschberg - O(nLCS)
• Landau,Crochemore,Ziv-Ukelson - fast for well
  compressed strings
• Computing large LCS (of near-linear size) may
  still take quadratic time.

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Motivation

• Our direction approximation.
• Input?
• Small alphabet ?-approximation
• Small LCS exact algorithms

• Our input
• Relatively large alphabet ?(n?), 0lt??1
• Large LCS near linear
• Main tool low distortion embedding.

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What is embedding?
A
B
x
f(x)
f
pA(x,y)
pB(f(x),f(y))
y
f(y)
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LCS embedding
Definition Let A and B be classes of n-long
strings. A LCS preserving embedding from A into B
with distortion ? is an injective mapping fA?B,
such that for every x,y?A, ??LCS(x,y) ?
LCS(f(x),f(y)) ? LCS(x,y)
Our goal f RL(n,?)?LNR(n) where ???(n?),
?????.
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Locally Non-repetitive Strings
Definition A string S is (t,w)-non-repetitive if
every w successive t-substrings in S are
distinct. If t1 then S is locally
non-repetitive. Example n14 S1abbacbababbacb
S2abcdeabcdeabcd
(3,7)-non-rep. ?
(4,8)-non-rep.
(1,5)-non-rep. LNR
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Approximating LCS in RL strings

• Observation
• RL(n) imply (t,w)-non-rep. for good w.

• Property Let S be n-long string.
• If S has period length p, S is (p,p)-non-rep.
• If S is aperiodic, S is (n,w)-non-rep. n/2?w?n.

S1 is aperiodic (14,14)-non-rep. S2 has period
size 5 (5,5)-non-rep.
Example n14 S1abbacbababbacb S2abcdeabcdeabcd
c
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Approximating LCS in RL strings

• Basic idea
• Embed RL(n) into LNR with good w.
• Approximate LCS in LNR strings.

Why is it a good idea?
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Approximating LCS in LNR strings
Assumption (1,n/c)-non-rep. strings
x1
x2
x3
C3
x
LIS(xi,yj)
y
y1
y2
y3
Cost c2 (n/c) loglog(n/c)cnloglog n
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Approximating LCS in LNR strings
What to do with the c2 block results?

• choose the best
• assures ?(1/c)-apprx. ratio
• additional O(c2) time

• best combine non-crossing pairs
• assures ?(k/c)-apprx. ratio
• where LCS(x,y)?kn/c
• takes O(c2) time using DP

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Approximating LCS in LNR strings
This means
Large CS can be found much faster in LNR strings
with wn/c.

• The total time O(cnloglog nc2)
• if wO(n?) we get O(n2-?loglogn)
• The apprx. ratio
• ?(k/c) for LCS(x,y)?kn/c
• is constant for LCS of linear size.

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Embedding into LNR strings
How to embed RL(n) into LNR strings?
Naïve idea Use the fact that all p-substrings
are distinct in wp. Rename p-substrings.
abbacbab1 bbacbaba2 bacbabab3 acbababb4 cbabab
ba5 bababbac6 ababbacb7 babbacba8
Example n14 Sabbacbababbacb S12345678123456
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Embedding into LNR strings
How much of the LCS we lost?
LCS(x,y)?LCS(x,y) no expansion. LCS(x,y) ?
n - t?(n-LCS(x,y)) ?LCS(x,y) -
(t-1)?(n-LCS(x,y)) ?LCS(x,y) - (t-1)?ED(x,y)/2
Both t and ED may be too big. Unbearable
contraction.
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Embedding into LNR strings
How to overcome this obstacle?
The idea Use stronger property. All p-substrings
are distinct in wp in many locations. Can use
few coordinates to rename p-substrings and still
get different names.
Formally dgt2 arbitrary constant Fix a random
(t-1)-long binary vector every location is 1
with prob. 2dln t/? and 0 otherwise.
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Embedding into LNR strings
Example n14 Sabbacbababbacb S1234567812345
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V(1,0,1,1,0,0,1,0)
abbacbab1 bbacbaba2 bacbabab3 acbababb4 cbabab
ba5 bababbac6 ababbacb7 babbacba8
Our embedding f
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Embedding into LNR strings
What is the distortion now?
LCS(x,y)?LCS(f(x),f(y)) no expansion. LCS(f(x),f
(y)) ? n (12(t-1)dln t/?)?(n-LCS(x,y)) ?LC
S(x,y) -2(t-1)dln t/??(n-LCS(x,y)) ?LCS(x,y) -
(t-1)dln t/??ED(x,y)
If EDo(LCS(x,y)??/tlnt) the distortion is
1-o(1). Large LCS preserved unless ED is too
large.
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Conclusions

• Large LCS can be approximated also for RL(n).
• LNR accelerates computations.
• Embedding accelerates computations.
• Batu, Ergun, Sahinalp (SODA,06)
• Andoni Onak (STOC,09)

Thank You.