Mid-labeled Partial Digest

Student: 蕭禕廷Advisor: 傅恆霖 教授

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Contents

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1. Introduction 2. Partial Digest 3. Mid-labeled Partial Digest

1. Introduction

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DNA

T C A G G T C A C AA G T C C A G T G T

. . .

. . .. . .. . .

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Restriction Enzyme (限制內切酶 )

G A A T T CC T T A A G

EcoRI

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Restriction Sites (切位 )

EcoRI

G A A T T CC T T A A G

G A A T T CC T T A A G

G A A T T CC T T A A G

Restriction Sites

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2. Partial Digest

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Partial Digest Problem

Partial digest problem is to find , by knowing .

𝑥1 𝑥2 𝑥𝑛. . .. . .8

Partial Digest Problem Example

0 2 104 72 345

67

810

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Partial Digest Problem Algorithm

Skiena et al. gave an algorithm

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max ∆ 𝑋  

11

22334567810

8

0 10

12

0 10

223345678

8?

13

0 10

223345678

88

82

14

0 10

2334567

87

7 3

1 ?

15

0 10

2334567

83

3 7

5

16

⨉ o

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. . . . .

... .... . . 𝑛

124

2𝑛− 1

2𝑛 (+

2𝑛+1−1

...

Partial Digest ProblemAlgorithm

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∆ 𝑋

?

𝑛

Each node needs time.

The total cost is .

3. Mid-labeled Partial Digest

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Mid-Labeled Partial Digest

𝑥1 𝑥2 𝑥𝑛. . .. . .

partial digest

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Mid-Labeled Partial Digest

,

contains exactly labels

𝑥1 𝑥2 𝑥𝑛. . .. . .

𝑙1 𝑙2 𝑙𝑘. . .

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Mid-Labeled Partial Digest

𝑥1 𝑥2 𝑥6

𝑙1 𝑙2 𝑙3

𝑥3 𝑥4𝑥5𝑙1

∆ 𝑋 𝑙 1, 𝑙1 ∆ 𝑋 𝜙𝑙1 𝑙2 𝑙3

∆ 𝑋 𝑙 1, 𝑙322

Mid-Labeled Partial Digest Example

𝑙1

0 2 104 7

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22343567810

𝑙1

𝑑1 𝑑2

¿ {𝑑1,𝑑2 }={2 ,3 }

*

$

$

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22343567 10

𝑙1

*

0 104 7

$

4 33

67 $

$ $

$

8

8

$

Mid-Labeled Partial Digest Algorithm

𝑙1

𝑑1 𝑑2

2𝑛(𝑑1+𝑑2𝑑1 )$

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Mid-Labeled Partial Digest Algorithm

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𝑑2 𝑑𝑖

𝑙1 𝑙2 𝑙𝑖 −1 𝑙𝑖

𝑑1. . .

. . .

. . .

. . .𝑙𝑘

𝑑𝑘+1

Mid-Labeled Partial Digest Algorithm

. . .𝑙𝑖 −1 𝑙𝑖 𝑙 𝑗 𝑙 𝑗+1. . . . . .

𝑑𝑖 𝑑 𝑗+1

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. . .

Mid-Labeled Partial Digest Algorithm

. . .𝑙𝑖 −1 𝑙 𝑗+1𝑙𝑖 𝑙 𝑗 . . .

𝑑𝑖 𝑑 𝑗+1

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𝑙𝑖 , 𝑙𝑖+1 , …, 𝑙 𝑗

𝑙1

𝑑1 𝑑2

∆ 𝑋 𝑙 1, 𝑙1

∆ 𝑋 𝑙 𝑖 ,𝑙 𝑗

Mid-Labeled Partial Digest Algorithm

𝑙1 𝑙𝑖𝑙𝑖 −1. . .. . .. . .

𝑚𝑎𝑥 ∆𝑋 𝑙 1, 𝑙𝑖 −1

$𝑚𝑎𝑥 ∆𝑋 𝑙 1, 𝑙𝑖 −1

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done?

Mid-Labeled Partial Digest Algorithm. .

. . . .. . .

. . .

𝑛

(𝑑𝑚 1+𝑑𝑚2

𝑑𝑚1)

. . .

. . .

. . .

. . .

. . .

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𝑙𝑚 1,…, 𝑙𝑚 2−1

𝑑𝑚1𝑑𝑚2

𝑚𝑎𝑥 ∆𝑋 𝑙 1, 𝑙𝑖 −1

𝑙1 𝑙𝑖𝑙𝑖 −1

Mid-Labeled Partial Digest Algorithm. .

. . . .. . .

. . .

𝑛

(𝑑𝑚 1+𝑑𝑚2

𝑑𝑚1)

. . .

. . .

. . .

. . .

. . .

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≤𝑛(𝑑𝑚1+𝑑𝑚2

𝑑𝑚1)

Mid-Labeled Partial Digest Algorithm

Each node needs . The total time is .

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1 2 𝑘

𝑛

. . .𝑑𝑚1

+𝑑𝑚2≤ 2𝑛𝑘+1

Stirling`s approximation : The total time is

Conclusion

For partial digest problem, Skiena et al. gave an algorithm.

For mid-labeled partial digest problem, there is an algorithm for adding labels inside DNA .

for .

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References

T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, Second Edition, 2001.

B. Lewin, Genes VII, 2000.

S. S. Skiena, W. D. Smith and P. Lemke, Reconstructing Sets From Interpoint Distances, SOCG, 1990.

D. B. West, Introduction to graph theory, 1996.

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