An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k...

45
An exploration of the size of the neighborhood of a word Yoann Dufresne, Hélène Touzet Seqbio 2017

Transcript of An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k...

Page 1: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

An exploration of the size of the neighborhood of a

wordYoann Dufresne, Hélène Touzet

Seqbio 2017

Page 2: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Levenshtein distance• Alphabet (Σ): {A, B}

• Words: AAABA, ABA, AAA

• Edit operations

• Levenshtein distance: smallest number of operations needed to transform one word to another

ABL| |ALL

A-L| |ALL

ABL| |A-L

Substitution Insertion Deletion

BALLAD- || ||SAL-ADS

distance = 3

Page 3: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Neighborhood of a word• a fixed number k

• a word P over Σ

• what is the number of words over Σ whose distance to P is bounded by k ?

Lev(AAA,1) → 17 elementsAAA, AA, AAB, AAL, ABA, ALA, BAA, LAA, AAAA, BAAA, LAAA, ABAA, ALAA,AABA, AALA, AAAB, AAAL

Lev(LAB,1) → 19 elementsLAB, LA, AB, LB, AAB, BAB, LBB, LLB, LAA, LAL, ALAB, BLAB, LLAB, LAAB,LBAB, LALB, LABA, LABB, LABL

Page 4: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Neighborhood of a word• a fixed number k

• a word P over Σ

• what is the number of words over Σ whose distance to P is bounded by k ?

Lev(AAA,1) → 17 elementsAAA, AA, AAB, AAL, ABA, ALA, BAA, LAA, AAAA, BAAA, LAAA, ABAA, ALAA,AABA, AALA, AAAB, AAAL

Lev(LAB,1) → 19 elementsLAB, LA, AB, LB, AAB, BAB, LBB, LLB, LAA, LAL, ALAB, BLAB, LLAB, LAAB,LBAB, LALB, LABA, LABB, LABL

Page 5: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Levenshtein automaton

Page 6: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Universal automaton

Page 7: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Universal automaton

Page 8: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Deterministic universal automaton

Page 9: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Wordborhood

Page 10: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Wordborhood

Page 11: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Wordborhood

Page 12: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Exhaustive executionsAAA218AAC240AAG240AAT240ACA256ACC240ACG258ACT258AGA256AGC258AGG240AGT258ATA256ATC258ATG258ATT240

CAA240CAC256CAG258CAT258CCA240CCC218CCG240CCT240CGA258CGC256CGG240CGT258CTA258CTC256CTG258CTT240

GAA240GAC258GAG256GAT258GCA258GCC240GCG256GCT258GGA240GGC240GGG218GGT240GTA258GTC258GTG256GTT240

TAA240TAC258TAG258TAT256TCA258TCC240TCG258TCT256TGA258TGC258TGG240TGT256TTA240TTC240TTG240TTT218

Σ=4, n=3, k=2

Page 13: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C T σ = |Σ|

Page 14: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

Page 15: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

subs

(σ-1)n

Page 16: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

Page 17: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss

Page 18: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

Page 19: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

ith blockxi ins

p

Page 20: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

ith blockxi ins

p

inside ith block≠xi ins

(n-p)(σ-1)

Page 21: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

ith blockxi ins

p

inside ith block≠xi ins

(n-p)(σ-1)

1st or last letter≠xi ins

2(σ-1)

Page 22: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

ith blockxi ins

p

inside ith block≠xi ins

(n-p)(σ-1)

1st or last letter≠xi ins

2(σ-1)

between ith and jth blocks≠xi ≠xj ins

(p-1)(σ-2)

Page 23: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

ith blockxi ins

p

inside ith block≠xi ins

(n-p)(σ-1)

1st or last letter≠xi ins

2(σ-1)

between ith and jth blocks≠xi ≠xj ins

(p-1)(σ-2)

(σ−1)(2n+1)+p+2

Page 24: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1A C C T T A A G C C C C Tp blocks σ = |Σ|

dels

p

subs

(σ-1)n

inss xi: letter of theith block

ith blockxi ins

p

inside ith block≠xi ins

(n-p)(σ-1)

1st or last letter≠xi ins

2(σ-1)

between ith and jth blocks≠xi ≠xj ins

(p-1)(σ-2)

(σ−1)(2n+1)+p+2 (σ−1)(2n+1)+2 + p

Page 25: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k=1 conclusion

• Proportional to σn

• All values between (σ−1)(2n+1)+2 and (σ−1)(2n+1)+2+p for fixed n and σ.

• Min values for p=1AAAA, AA, TTTTTT

• Max values for p=nACTAGC, ACACACA, ACTGACTGAC

(σ−1)(2n+1)+2+p

Page 26: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k \ σ 2 3 4 5 6

1 16 29 42 55 68

2 95 313 659 1 133 1 735

3 351 2 085 6 379 14 403 27 327

4 1 003 10 653 47 325 140 329 329 919

5 2 503 47 161 303 875 1 183 039 3 446 311

Focus on minimum

Page 27: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k \ σ 2 3 4 5 6

1 16 29 42 55 68

2 95 313 659 1 133 1 735

3 351 2 085 6 379 14 403 27 327

4 1 003 10 653 47 325 140 329 329 919

5 2 503 47 161 303 875 1 183 039 3 446 311

Focus on minimum

AAAAAABBBBBBCCCCCC

…

Patterns:

Page 28: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k \ σ 2 3 4 5 6

1 16 29 42 55 68

2 95 313 659 1 133 1 735

3 351 2 085 6 379 14 403 27 327

4 1 003 10 653 47 325 140 329 329 919

5 2 503 47 161 303 875 1 183 039 3 446 311

Focus on minimum

AAAAAABBBBBBCCCCCC

…

Patterns: Minimum: An

Page 29: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

k \ σ 2 3 4 5 6

1 16 29 42 55 68

2 95 313 659 1 133 1 735

3 351 2 085 6 379 14 403 27 327

4 1 003 10 653 47 325 140 329 329 919

5 2 503 47 161 303 875 1 183 039 3 446 311

Focus on minimum

AAAAAABBBBBBCCCCCC

…

Patterns: Minimum: An We havea proof !

Page 30: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

Page 31: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

A A A A A A A A A A A

Page 32: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

A A A A A A A A A A A

subs

Page 33: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

A A A A A A A A A A A

dels subs

Page 34: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

A A A A A A A A A A A

dels insssubs

Page 35: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

A A A A A A A A A A A

dels insssubs

Page 36: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Calculate the minimum

A A A A A A A A A A A

dels insssubs

•n-k≤x• p≤k• x+p≤n-k

Neighborhood for an. All neighbors are formed with x a letter and p other letters.

Page 37: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

And the maximum ?k \ σ 2 3 4 5 6

1 21 34 47 60 73

2 146 463 924 1 543 2 320

3 498 3 443 10 486 23 465 44 234

4 1 257 17 378 83 039 252 425 601 336

5 2 883 72 691 532 653 2 194 036 6 601 814

Page 38: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

And the maximum ?k \ σ 2 3 4 5 6

1 21 34 47 60 73

2 146 463 924 1 543 2 320

3 498 3 443 10 486 23 465 44 234

4 1 257 17 378 83 039 252 425 601 336

5 2 883 72 691 532 653 2 194 036 6 601 814

ABABAB ABCABC ABCDAB ABCDEA ABCDEF

Page 39: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

And the maximum ?k \ σ 2 3 4 5 6

1 21 34 47 60 73

2 146 463 924 1 543 2 320

3 498 3 443 10 486 23 465 44 234

4 1 257 17 378 83 039 252 425 601 336

5 2 883 72 691 532 653 2 194 036 6 601 814

ABABAB ABCABC ABCDAB ABCDEA ABCDEF

Maximum: each windowof size k with different letters

Page 40: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

And the maximum ?k \ σ 2 3 4 5 6

1 21 34 47 60 73

2 146 463 924 1 543 2 320

3 498 3 443 10 486 23 465 44 234

4 1 257 17 378 83 039 252 425 601 336

5 2 883 72 691 532 653 2 194 036 6 601 814

ABABAB ABCABC ABCDAB ABCDEA ABCDEF

Maximum: each windowof size k with different letters No formal proof yet :(

Page 41: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

And the maximum ?k \ σ 2 3 4 5 6

1 21 34 47 60 73

2 146 463 924 1 543 2 320

3 498 3 443 10 486 23 465 44 234

4 1 257 17 378 83 039 252 425 601 336

5 2 883 72 691 532 653 2 194 036 6 601 814

ABABAB ABCABC ABCDAB ABCDEA ABCDEF

Maximum: each windowof size k with different letters No formal proof yet :(k=2: ABCB, ABABA

k=3: ABCA, CABDAC

Page 42: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

BCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Block positions

Page 43: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Positions

Page 44: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Perspectives

• Use the automata to prove the upper bound

• Find a formula to compute the upper bound neighborhood without the DULA for k>7

• Discover influence of each parameter (n,p,σ,…) in general case

• Create a method to generate formulas for any kind of word

Page 45: An exploration of the size of the neighborhood of a wordNeighborhood of a word • a fixed number k • a word P over Σ • what is the number of words over Σ whose distance to

Questions ?


Form 2 Proofreadingtf/S2 2017-2018/S2 proofreading... · Web viewa word that is incorrect and write the correct word above it. Mark the position of a missing word with a ‘Λ’

Form 2 Proofreadingtf/S2 2017-2018/S2 proofreading... · Web viewa word that is incorrect and write the correct word above it. Mark the position of a missing word with a ‘Λ’

No Slide Titledigilab.teipir.gr/site_data/pega/memory.pdf · Word 0 Word 1 Word 2 Word N-2 Word N-1 Storage cell M bits M bits S0 S1 S2 SN-2 A0 A1 AK-1 K = log2N SN-1 Word 0 Word

No Slide Titledigilab.teipir.gr/site_data/pega/memory.pdf · Word 0 Word 1 Word 2 Word N-2 Word N-1 Storage cell M bits M bits S0 S1 S2 SN-2 A0 A1 AK-1 K = log2N SN-1 Word 0 Word

Stable over-expression of PPARβ/δ and PPARγ to …facstaff.bloomu.edu/mborland/Borland_2011.pdfStable over-expression of PPARβ/δ and PPARγ to examine receptor signaling in human

Stable over-expression of PPARβ/δ and PPARγ to …facstaff.bloomu.edu/mborland/Borland_2011.pdfStable over-expression of PPARβ/δ and PPARγ to examine receptor signaling in human

Thassos word

Thassos word

Nutrient Controls Over Cyanobacterial Synthesis of the ...

Nutrient Controls Over Cyanobacterial Synthesis of the ...

Word formation

Word formation

Word Provenance - University of British Columbia

Word Provenance - University of British Columbia

IS THERE A PROSODIC WORD IN VIETNAMESE? - JSEALSjseals.org/seals24/brunelle2014prosodic.pdf · • Traditional Vietnamese grammar has no notion of word ... –Yet morphemic structure

IS THERE A PROSODIC WORD IN VIETNAMESE? - JSEALSjseals.org/seals24/brunelle2014prosodic.pdf · • Traditional Vietnamese grammar has no notion of word ... –Yet morphemic structure

Equidistributions of Mahonian statistics over pattern ...

Equidistributions of Mahonian statistics over pattern ...

σημειώσεις Word ppt

σημειώσεις Word ppt

Word/phrase (with Macmillan English Dictionary star rating) Part of ...

Word/phrase (with Macmillan English Dictionary star rating) Part of ...

Word final print

Word final print

Lines Over Classes

Lines Over Classes

Confusion over Measures of Evidence (p's) versus … › ... › Hubbard-Bayarri-2003.pdfConfusion over Measures of Evidence (p's) versus Errors (α's) in Classical Statistical Testing

Confusion over Measures of Evidence (p's) versus … › ... › Hubbard-Bayarri-2003.pdfConfusion over Measures of Evidence (p's) versus Errors (α's) in Classical Statistical Testing

Multi-word expressions in Modern Greek: present and future · Multi-word expressions in Modern Greek Definitions A multi-word expression is a sequence of words which constitutes a

Multi-word expressions in Modern Greek: present and future · Multi-word expressions in Modern Greek Definitions A multi-word expression is a sequence of words which constitutes a

Testing of Circuit Breaker and over Current Relay ... · PDF filecurrent relay and inverse definite minimum time (IDMT) over current relay and also testing of over current relay with

Testing of Circuit Breaker and over Current Relay ... · PDF filecurrent relay and inverse definite minimum time (IDMT) over current relay and also testing of over current relay with

Tomas Riad Stockholms universitet - su.se · morphology • Word class ... word (ω. min) • Rythmic grouping ... Align a morphological word with a prosodic word (edge alignment)

Tomas Riad Stockholms universitet - su.se · morphology • Word class ... word (ω. min) • Rythmic grouping ... Align a morphological word with a prosodic word (edge alignment)

Over-expression of a γ-tocopherol methyltransferase gene in vitamin … · 2020. 5. 19. · RESEARCH ARTICLE Open Access Over-expression of a γ-tocopherol methyltransferase gene

Over-expression of a γ-tocopherol methyltransferase gene in vitamin … · 2020. 5. 19. · RESEARCH ARTICLE Open Access Over-expression of a γ-tocopherol methyltransferase gene

African Origin of the word “Nike - Asar Imhotep Online ... Origin of the word “Nike ... 1 See for instance The American Heritage Dictionary of the English Language, ... Gaulish

African Origin of the word “Nike - Asar Imhotep Online ... Origin of the word “Nike ... 1 See for instance The American Heritage Dictionary of the English Language, ... Gaulish

δ1 ρομποτ word

δ1 ρομποτ word