An optimal algorithm for identifying a maximum-density segment

(1)

2004/12/13

2004/12/13 Maximum-Density Segment @ EE.NTUMaximum-Density Segment @ EE.NTU 11

An optimal algorithm for

identifying a maximum-density

segment

呂學一 ( 中央研究院資訊科學所 )

http://www.iis.sinica.edu.tw/~hil/

Microsoft Office

XP is needed to

see all the

animation

(2)

What do

algorithm people

do?

Inventing

efficient recipes

to solve

combinatorial problems

(3)

2004/12/13

A famous combinatorial

problem



The Factorization Problem

–

_Input:

_{a number}

_N

–

_Output:



“yes”

if

N

is a prime number;



A factorization of N

if

N

is not a prime number.

– For example,



N

= 323264989793317.

(4)

OPEN QUESTION

Is there an efficient recipe for the

(5)

2004/12/13

Why Factorization?

The security of many encryption schemes

is based upon the assumption that the

(6)

(7)

2004/12/13

RSA factorization

challenges

Challenge

Number

Prize ($US)

Challenge

Number

Prize ($US)

RSA-576

$10,000

RSA-896

$75,000

RSA-640

$20,000

RSA-1024

$100,000

RSA-704

$30,000

RSA-1536

$150,000

(8)

US$10,000 –– RSA-576



1881988129206079638386972394616504

3980716356337941382700763356422988

8597152346654853190606065047430453

1738801130339671619969232120573403

1879550656996213051687593076502570

59

(9)

2004/12/13

RSA-576 factored in

December 3, 2003



3980750864240649373971255005503864

9119906436234252670840638518957594

6388957261768583317



4727721461074353025362230719730482

2463291469530209711645985217113052

0711256363590397527



At the same time, Adi Shamir gave two

(10)

US$20,000 –– RSA-640



3107418240490043721350750035888567

9300373460228427275457201619488232

0644051808150455634682967172328678

2437916272838033415471073108501919

5485290073377248227835257423864540

14691736602477652346609

(11)

2004/12/13

US$200,000 –– RSA-2048



25195908475657893494027183240048398571429282126204

03202777713783604366202070759555626401852588078440

69182906412495150821892985591491761845028084891200

72844992687392807287776735971418347270261896375014

97182469116507761337985909570009733045974880842840

17974291006424586918171951187461215151726546322822

16869987549182422433637259085141865462043576798423

38718477444792073993423658482382428119816381501067

48104516603773060562016196762561338441436038339044

14952634432190114657544454178424020924616515723350

77870774981712577246796292638635637328991215483143

81678998850404453640235273819513786365643912120103

97122822120720357

(12)

Short of cash?

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2004/12/13

RSA 2003 (April ’03)

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2002 Turing Award

(June’03)

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2004/12/13

The awarded paper



Only 7 pages.

– “A Method for Obtaining Digital Signatures

and Public Key Cryptosystems”,

Communications of the ACM 21, 120-126,

(16)

“PRIMES is in P”

Agarwal, Kayal, and Saxena

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2004/12/13

PRIMES is in P



The PRIMES problem:

– Input: a number N.

– Output:



“yes”if N is a prime number.



“no” if N is not a prime number.



Only 9 pages!



Running time is O(n

12 ), where n is the

(18)

NEW YORK TIMES

, Aug.

_{, Aug.}

8, 2002



Previous algorithmic results that caught

the attention of the New York Times

– 1984, Karmarkar’s algorithm for solving

linear programs.

– 1979, Khachian’s algorithm for solving linear

programs.

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2004/12/13

The latest version (v.3) of

AKS’s paper



The running time is now improved from

(20)

What do algorithm people

do?



Looking for important/interesting

combinatorial problems



Coming up with efficient recipes to solve

(21)

2004/12/13

Bioinformatics

(22)

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2004/12/13

Finding a DNA segment with Max

GC-density in linear time

WABI  J. Comput. Sys. Sci.

ESA  SIAM J. Computing

(24)

DNA Sequences



[Chargaff and Vischer, 1949]

– DNA consisting of A, G, T, C



Adenine ( 腺嘌呤 )



Guanine ( 鳥糞嘌呤 )



Cytosine ( 胞嘧啶 )



Thymine ( 胸腺嘧啶 )

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2004/12/13



[Vischer, Zamenhof, Chargaff, 1949]

– Negative evidences for the widely believed

%A = %G = %T = %C.

(26)

Edwin Chargaff,

1905-

Observing

– %A ~ %T

– %G ~ %C



“A comparison of the

molar proportions

reveals certain

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2004/12/13

Double Helix



[Watson and Crick, Nature,

April 25, 1953]

– Biologist (age 23, fresh Ph.D.) +

Physicist (age 35, still a Ph.D.

student)

(28)

1962 Nobel Prize in

Physiology or Medicine

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2004/12/13

DNA’s picture



[Alexander Rich, 1973]

– Structure biologist at MIT.

(30)

Celebrating

50 years

_{50 years}

of Double

_{of Double}

Helix (April 25, 1953 – 2003)

(31)

2004/12/13

Francis Crick 1916-2004



Passed away on July 28, 2004

(32)

Maurice Wilkins 1916-2004

(33)

2004/12/13

GC-content



Non-uniformity of nucleotide composition

– 25% - 75% in genomes of all of organisms

– 40% - 50% in typical mammalian genomes

– 30% - 60% in human chromosomes

(34)

GC content



GC-content is positively correlated with

– gene length,

– gene density,

– patterns of coden usage,

(35)

2004/12/13

The Problem



Input:

– an n-bit string S,

– an integer L.



Output:

– a substring S[i, j] of S with maximum density

over all substrings of S with at least L bits.

(36)

Example



S = 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1



L = 1, 1

0 0 1 1 0 0 1 0 1 1 0 1 0 0 1



L = 2, 1 0 0 1 1

0 0 1 0 1 1 0 1 0 0 1



L = 3, 1 0 0 1 1 0 0 1 0 1 1

0 1 0 0 1

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2004/12/13

density of each segment

in O(1) time



prefix-sum(i) = S[1]+S[2]+…+S[i],

– all n prefix sums are computable in O(n) time.



sum(i, j) = prefix-sum(j) – prefix-sum(i-1)



density(i, j) = sum(i, j) / (j-i+1)

(38)

Good partners



Finding the

best ending position

g(i)

for

each i=1,2,…,n.

L

g(i)

maximing avg[i,

g(i)

]

(39)

2004/12/13

Previous Work



[Huang, CABIOS ’94]

–

_{O(nL) time}

_.



Key observation: no need to examine substrings

longer than

2L.

L

i+L

g(i)

L

(40)

Recent Progress



[Lin, Jiang, Chao, J. Computer Systems

and Science (JCSS), 2002]

–

_{O(n log L) time}

_.

– Techniques:



Right-skew decomposition

.

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2004/12/13

Our results

(42)

Reviewing Lin, Jiang, and

Chao’s Algorithm

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2004/12/13

Right-Skew Substring



S[i, j] is

right-skew

if for each k = i,…, j-1

– density[i, k] ≤ density[k+1, j].



S =

1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1



1 0 0 1

1 0 0 1 0 1 1 0 1 0 0 1



1 0

0 1 1

0 0 1 0 1 1 0 1 0 0 1



1 0 0 1 1 0

0 1 0 1 1

0 1 0 0 1

(44)

Right-Skew

Decomposition



Partition S into substrings S

₁

,S

₂

,…,S

_k

such

that

– each S

i

is a right-skew substring of S

– density(S

₁

)

>

density(S

₂

)

>

…

>

density(S

_k

)

(45)

2004/12/13

An example

1

0

1

0

1

0

1

0

1

1 >

2/3

>

3/5

>

1/3

(46)

Why RS-decomposition?

1. It suffices to search for g(i) among the

boundaries of RS-decomposition of

S[i,

n]

.

2. The boundaries’s “potential” of being a

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2004/12/13

Illustration

L

i+L

g(i)

i+L

L

(48)

Preprocessing steps

1. RS-decomposition of S[i, n] for each i.

2. Jumping table that enables binary search

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2004/12/13

First preprocessing:

All RS-decompositions



The RS-decomposition of each S[i, n]

– Linear time for each i = 1, …, n.



All n RS-decompositions

– [Lin et al.] O(n

2 ) time  O(n) time.

L

(50)

Key: nested structures

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2004/12/13

Second Preprocessing:

Jumping Table



Why?

– Need a table that enables jumping over 2

k

right-skew components in O(1) time for each

k.



[Lin et al.] O(n

2 ) time  O(n log L) time.

L

(52)

LJC’s Algorithm



Three main steps:

1. All RS-decompositions in O(n) time.

2. Jumping table in O(n log L) time.

3. For each i=1, 2,…, n

Binary searching g(i) in O(log L) time.

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2004/12/13

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A new important

observation



i < j < g(j) < g(i) implies



density(i, g(i)) is no more than

density(j, g(j))

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2004/12/13

(56)

Searching for all g(i) in

linear time

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2004/12/13

We still need RS-decomp.

L

(58)

A generalized version



Input:

– an n-bit string S

– an integer L

–

an integer U



Output: a substring S[i, j] of S with maximum

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2004/12/13

Difficulty

(60)

Idea: Partition the input

into blocks of length U-L



For each index i, g(i) can only be in two

consecutive blocks.

i

U-L

(61)

2004/12/13

Consider two cases

separately

L

i

L

i

L+d

i

(62)

Case 1

L

i



Taking care of all indices i with the same

left block together.



Just like no U is specified.

L

(63)

2004/12/13

Case 2



Taking care of all indices i with the same

left block together.

(64)

Need RS-decompositions for

all prefixes of each block

(65)

2004/12/13

ESA/SICOMP result



An Optimal Algorithm for the

Maximum-Density Segment Problem, with Kai-min

Chung, in Proceedings of the 11th

Annual European Symposium on

Algorithms, Budapest, Hungary,

(66)

Kai-min’s idea



For a segment

– i k j



For a feasible segment

lowest density

prefix of

lowest density

prefix of

lowest density

prefix of

max-density

segment got

(67)

2004/12/13

Our algorithm



For each possible j, we find a “good”

candidate S(i

_j

,j) and look for the max-density

segment over all S(i

_j

,j)

– i

j

-1 j

removable prefix

L

(68)

The features of our new

algorithm



No need of the clever but somewhat

complicated right-skew decomposition.



As a result, our algorithm can process the

(69)

2004/12/13

Some thoughts



Almost all algorithmic results start with

very simple observations.



Experience and skills (in analysis) are

certainly crucial, but being creative and

diligent is even more important in doing

good (algorithmic) research.

(70)

Would you like to join the algorithmic

adventure?

(71)

2004/12/13

Ads



_



vol 1

vol 2

one-way functions

pseudo-randomness

zero-knowledge proofs

encryption schemes

digital signatures

cryptographic protocols

(72)

Today’s slides



can be found at

An optimal algorithm for identifying a maximum-density segment