Periodic string comparison
|
|
|
- Ross Atkins
- 6 years ago
- Views:
Transcription
1 Periodi string omprison Alexnder Tiskin Deprtment of Computer Siene University of Wrwik Alexnder Tiskin (Wrwik) Periodi string omprison 1 / 51
2 1 Introdution 2 Semi-lol string omprison 3 The seweed lgorithm 4 Conlusions nd future work Alexnder Tiskin (Wrwik) Periodi string omprison 2 / 51
3 1 Introdution 2 Semi-lol string omprison 3 The seweed lgorithm 4 Conlusions nd future work Alexnder Tiskin (Wrwik) Periodi string omprison 3 / 51
4 Introdution String mthing: finding n ext pttern in string String omprison: finding similr ptterns in two strings (Also known s pproximte string mthing, no reltion to pproximtion lgorithms!) Applitions: omputtionl iology, imge reognition,... Alexnder Tiskin (Wrwik) Periodi string omprison 4 / 51
5 Introdution String mthing: finding n ext pttern in string String omprison: finding similr ptterns in two strings (Also known s pproximte string mthing, no reltion to pproximtion lgorithms!) Applitions: omputtionl iology, imge reognition,... Stndrd types of string omprison: glol: whole string vs whole string lol: sustrings vs sustrings Min fous of this work: semi-lol: whole string vs sustrings; prefixes vs suffixes Min tool: impliit unit-monge mtries Alexnder Tiskin (Wrwik) Periodi string omprison 4 / 51
6 Introdution Terminology nd nottion Integers:... 2, 1, 0, 1, 2,... Odd hlf-integers: , 3 2, 1 2, 1 2, 3 2, 5 2,... We onsider finite nd infinite integer mtries over integer nd odd hlf-integer indies. For simpliity, index rnge will usully e ignored. A permuttion mtrix is 0/1 mtrix with extly one nonzero per row nd per olumn Alexnder Tiskin (Wrwik) Periodi string omprison 5 / 51
7 Introdution Terminology nd nottion Given mtrix D, its distriution mtrix is D Σ (i, j) = i >i,j <j D(i, j ) In other words, D Σ (i, j) is the sum of ll D(i, j ), where (i, j ) is dominted y (i, j) Alexnder Tiskin (Wrwik) Periodi string omprison 6 / 51
8 Introdution Terminology nd nottion Given mtrix D, its distriution mtrix is D Σ (i, j) = i >i,j <j D(i, j ) In other words, D Σ (i, j) is the sum of ll D(i, j ), where (i, j ) is dominted y (i, j) Given mtrix E, its density mtrix is E (i, j) = E(i, j + ) E(i, j ) E(i +, j + ) + E(i +, j ) where i ± = i ± 1 2 ; DΣ, E over integers; D, E over odd hlf-integers Alexnder Tiskin (Wrwik) Periodi string omprison 6 / 51
9 Introdution Terminology nd nottion Given mtrix D, its distriution mtrix is D Σ (i, j) = i >i,j <j D(i, j ) In other words, D Σ (i, j) is the sum of ll D(i, j ), where (i, j ) is dominted y (i, j) Given mtrix E, its density mtrix is E (i, j) = E(i, j + ) E(i, j ) E(i +, j + ) + E(i +, j ) where i ± = i ± 1 2 ; DΣ, E over integers; D, E over odd hlf-integers (D Σ ) = D for ll D Mtrix E is simple, if (E ) Σ = E Alexnder Tiskin (Wrwik) Periodi string omprison 6 / 51
10 Introdution Terminology nd nottion Mtrix E is Monge, if E is nonnegtive Intuition: order-to-order distnes in (weighted) plnr grph Mtrix E is unit-monge, if E is permuttion mtrix Intuition: order-to-order distnes in grid-like grph Alexnder Tiskin (Wrwik) Periodi string omprison 7 / 51
11 Introdution Terminology nd nottion P P Σ (P Σ ) = P Alexnder Tiskin (Wrwik) Periodi string omprison 8 / 51
12 Introdution Impliit unit-monge mtries Impliit P Σ : rnge tree on nonzeros of P [Bentley: 1980] inry serh tree y i-oordinte under every node, inry serh tree y j-oordinte Alexnder Tiskin (Wrwik) Periodi string omprison 9 / 51
13 Introdution Impliit unit-monge mtries Impliit P Σ (ontd.) Every node of the rnge tree represents nonil rnge (retngulr region), nd stores its nonzero ount Overll, n log n nonil rnges re non-empty Rnge tree supports dominne ounting queries: how mny nonzeros re dominted y given point? Answered y deomposing query rnge into log 2 n disjoint nonil rnges. Totl size O(n log n), query time O(log 2 n) There re symptotilly more effiient (ut less prtil) dt strutures Alexnder Tiskin (Wrwik) Periodi string omprison 10 / 51
14 Introdution Mtrix -multiplition Mtrix -multiplition (.k.. distne, (min, +) or tropil multiplition) A B = C C(i, k) = min j ( A(i, j) + B(j, k) ) Alexnder Tiskin (Wrwik) Periodi string omprison 11 / 51
15 Introdution Mtrix -multiplition Mtrix -multiplition (.k.. distne, (min, +) or tropil multiplition) A B = C C(i, k) = min j ( A(i, j) + B(j, k) ) Mtrix lsses losed under -multiplition: generl numeril (integer, rel) mtries Monge mtries simple unit-monge mtries Simple unit-monge mtries of size n form n periodi monoid (i.e. monoid s fr s possile from group) under -multiplition We ll it the seweed monoid T n Alexnder Tiskin (Wrwik) Periodi string omprison 11 / 51
16 Introdution Mtrix -multiplition Simple unit-monge mtries: -multiplition = seweed omposition P A P B P C P Σ A PΣ B = PΣ C Alexnder Tiskin (Wrwik) Periodi string omprison 12 / 51
17 Introdution Mtrix -multiplition Simple unit-monge mtries: -multiplition = seweed omposition P A P B P C P Σ A PΣ B = PΣ C P A P B Alexnder Tiskin (Wrwik) Periodi string omprison 12 / 51
18 Introdution Mtrix -multiplition Simple unit-monge mtries: -multiplition = seweed omposition P A P B P C P Σ A PΣ B = PΣ C P A P B Alexnder Tiskin (Wrwik) Periodi string omprison 12 / 51
19 Introdution Mtrix -multiplition Simple unit-monge mtries: -multiplition = seweed omposition P A P B P C P Σ A PΣ B = PΣ C P A P C P B Alexnder Tiskin (Wrwik) Periodi string omprison 12 / 51
20 Introdution Mtrix -multiplition Seweeds: similr to rids, generted y strnd rossings Unlike in rids, ll seweed rossings re level (not underpss/overpss) idempotent, i.e. two seweeds n ross t most one Seweed omposition: ssoitive, no inverse ( rossing nnot e nelled) Identity: 1 x = x, no seweeds rossing Zero: 0 x = 0, ll seweeds rossing ) Σ 1 = 0 = ( ( ) Σ Alexnder Tiskin (Wrwik) Periodi string omprison 13 / 51
21 Introdution Mtrix -multiplition The seweed monoid T n : n! elements (permuttions of size n) n 1 genertors g 1, g 2,..., g n 1 (elementry rossings) g 2 i = g i for ll i (idempotene) g i g j = g j g i j i > 1 (fr ommuttivity) g i g j g i = g j g i g j j i = 1 (rid reltions) Computtion: onfluent rewriting system n e otined y softwre (Semigroupe, GAP) Generlistion: Coxeter monoids (sugroup monoids in groups) [Tsrnov: 90] Alexnder Tiskin (Wrwik) Periodi string omprison 14 / 51
22 Introdution Mtrix -multiplition The seweed monoid T 3 Genertors: 1, = g 1, = g 2 Other elements:,, = 0 Rewriting system: 0 0 Alexnder Tiskin (Wrwik) Periodi string omprison 15 / 51
23 Introdution Mtrix -multiplition The seweed monoid T 4 Genertors: 1, = g 1, = g 2, = g 3 Other elements:,,,,,,,,,,,,,,,,,,, = 0 Rewriting system: 0 Alexnder Tiskin (Wrwik) Periodi string omprison 16 / 51
24 Introdution Mtrix -multiplition The impliit mtrix -multiplition prolem Given permuttion mtries P A, P B, ompute P C, suh tht P Σ A PΣ B = PΣ C Alexnder Tiskin (Wrwik) Periodi string omprison 17 / 51
25 Introdution Mtrix -multiplition The impliit mtrix -multiplition prolem Given permuttion mtries P A, P B, ompute P C, suh tht P Σ A PΣ B = PΣ C Mtrix -multiplition: running time mtrix type time generl O(n 3 ) stndrd Monge O(n 2 ) y [Aggrwl+: 1987] impliit simple unit-monge (P Σ ) O(n 1.5 ) [T: 2006] O(n log n) [T: NEW] Alexnder Tiskin (Wrwik) Periodi string omprison 17 / 51
26 Introdution Mtrix -multiplition P B P A P C? Alexnder Tiskin (Wrwik) Periodi string omprison 18 / 51
27 Introdution Mtrix -multiplition P B,lo, P B,hi P A,lo, P A,hi Alexnder Tiskin (Wrwik) Periodi string omprison 19 / 51
28 Introdution Mtrix -multiplition P B,lo, P B,hi P A,lo, P A,hi Alexnder Tiskin (Wrwik) Periodi string omprison 19 / 51
29 Introdution Mtrix -multiplition P B,lo, P B,hi P A,lo, P A,hi Alexnder Tiskin (Wrwik) Periodi string omprison 19 / 51
30 Introdution Mtrix -multiplition P B,lo, P B,hi P A,lo, P A,hi P C,lo + P C,hi Alexnder Tiskin (Wrwik) Periodi string omprison 19 / 51
31 Introdution Mtrix -multiplition Impliit mtrix -multiplition: the lgorithm PC Σ(i, k) = min ( j P Σ A (i, j) + PB Σ (j, k)) Divide-nd-onquer on the rnge of j Divide P A horizontlly, P B vertilly; two suprolems of effetive size n/2: P Σ A,lo PΣ B,lo = PΣ C,lo P Σ A,hi PΣ B,hi = P Σ C,hi Conquer: most (ut not ll!) nonzeros of P C,lo, P C,hi pper in P C Missing nonzeros n e otined in time O(n) using the Monge property Overll time O(n log n) Alexnder Tiskin (Wrwik) Periodi string omprison 20 / 51
32 Introdution Mtrix -multiplition P B,lo, P B,hi P A,lo, P A,hi P C,lo + P C,hi Alexnder Tiskin (Wrwik) Periodi string omprison 21 / 51
33 Introdution Mtrix -multiplition P B,lo, P B,hi P A,lo, P A,hi P C Alexnder Tiskin (Wrwik) Periodi string omprison 21 / 51
34 1 Introdution 2 Semi-lol string omprison 3 The seweed lgorithm 4 Conlusions nd future work Alexnder Tiskin (Wrwik) Periodi string omprison 22 / 51
35 Semi-lol string omprison Semi-lol LCS nd edit distne Consider strings (= sequenes) over n lphet of size σ Distinguish ontiguous sustrings nd not neessrily ontiguous susequenes Speil ses of sustring: prefix, suffix Nottion: strings, of length m, n respetively Assume where neessry: m n; m, n resonly lose Alexnder Tiskin (Wrwik) Periodi string omprison 23 / 51
36 Semi-lol string omprison Semi-lol LCS nd edit distne Consider strings (= sequenes) over n lphet of size σ Distinguish ontiguous sustrings nd not neessrily ontiguous susequenes Speil ses of sustring: prefix, suffix Nottion: strings, of length m, n respetively Assume where neessry: m n; m, n resonly lose The longest ommon susequene (LCS) sore: length of longest string tht is susequene of oth nd equivlently, lignment sore, where sore(mth) = 1 nd sore(mismth) = 0 Alexnder Tiskin (Wrwik) Periodi string omprison 23 / 51
37 Semi-lol string omprison Semi-lol LCS nd edit distne The LCS prolem Give the LCS sore for vs Alexnder Tiskin (Wrwik) Periodi string omprison 24 / 51
38 Semi-lol string omprison Semi-lol LCS nd edit distne The LCS prolem Give the LCS sore for vs LCS: running time O(mn) [Wgner, Fisher: 1974] O ( ) mn log n O ( mn(log log n) 2 ) log n σ = O(1) [Msek, Pterson: 1980] [Crohemore+: 2003] [Pterson, Dnik: 1994] Alexnder Tiskin (Wrwik) Periodi string omprison 24 / 51
39 Semi-lol string omprison Semi-lol LCS nd edit distne LCS on the lignment grph (direted, yli) lue = 0 red = 1 LCS("", "") = "" LCS = highest-sore orner-to-orner pth Alexnder Tiskin (Wrwik) Periodi string omprison 25 / 51
40 Semi-lol string omprison Semi-lol LCS nd edit distne LCS: dynmi progrmming (DP) lgorithm [Wgner, Fisher: 1974] Sweep lignment grph, respeting node dependenies Running time O(mn) Alexnder Tiskin (Wrwik) Periodi string omprison 26 / 51
41 Semi-lol string omprison Semi-lol LCS nd edit distne LCS: dynmi progrmming (DP) lgorithm [Wgner, Fisher: 1974] Sweep lignment grph, respeting node dependenies Running time O(mn) LCS: miro-lok DP lgorithm [Msek, Pterson: 1980] Sweep lignment grph in squre loks, respeting lok dependenies Blok size: t = O(log n) Blok interfe: O(t) inputs/outputs, eh of size O(log σ) Use preomputed mpping of ll possile input/output omintions Running time O ( mn log n) when σ = O(1), even on log-ost RAM Alexnder Tiskin (Wrwik) Periodi string omprison 26 / 51
42 Semi-lol string omprison Semi-lol LCS nd edit distne The semi-lol LCS prolem Give the (impliit) mtrix of O(m 2 + n 2 ) LCS sores: string-sustring LCS: string vs every sustring of prefix-suffix LCS: every prefix of vs every suffix of symmetrilly, sustring-string nd suffix-prefix LCS Alexnder Tiskin (Wrwik) Periodi string omprison 27 / 51
43 Semi-lol string omprison Semi-lol LCS nd edit distne The semi-lol LCS prolem Give the (impliit) mtrix of O(m 2 + n 2 ) LCS sores: string-sustring LCS: string vs every sustring of prefix-suffix LCS: every prefix of vs every suffix of symmetrilly, sustring-string nd suffix-prefix LCS The three-wy semi-lol LCS prolem Give the (impliit) mtrix of O(n 2 ) LCS sores: string-sustring, prefix-suffix, suffix-prefix LCS no sustring-string LCS Suitle for m n Alexnder Tiskin (Wrwik) Periodi string omprison 27 / 51
44 Semi-lol string omprison Semi-lol LCS nd edit distne The semi-lol LCS prolem Give the (impliit) mtrix of O(m 2 + n 2 ) LCS sores: string-sustring LCS: string vs every sustring of prefix-suffix LCS: every prefix of vs every suffix of symmetrilly, sustring-string nd suffix-prefix LCS The three-wy semi-lol LCS prolem Give the (impliit) mtrix of O(n 2 ) LCS sores: string-sustring, prefix-suffix, suffix-prefix LCS no sustring-string LCS Suitle for m n Cf.: dynmi progrmming gives prefix-prefix LCS Alexnder Tiskin (Wrwik) Periodi string omprison 27 / 51
45 Semi-lol string omprison Semi-lol LCS nd edit distne Semi-lol LCS on the lignment grph lue = 0 red = 1 LCS("", "...") = "" Semi-lol LCS = ll highest-sore order-to-order pths (string-sustring = top-to-ottom, et.) Alexnder Tiskin (Wrwik) Periodi string omprison 28 / 51
46 Semi-lol string omprison Semi-lol LCS nd edit distne The LCS prolem is speil se of the lignment sore prolem with weighted mthes, mismthes nd gps LCS sore: w mth = 1, w mismth = w gp = 0 Levenshtein sore: w mth = 2, w mismth = 1, w gp = 0 An lignment sore is rtionl, if w mth, w mismth, w gp re rtionl; redues to LCS sore y onstnt-ftor low-up of lignment grph Alexnder Tiskin (Wrwik) Periodi string omprison 29 / 51
47 Semi-lol string omprison Semi-lol LCS nd edit distne The LCS prolem is speil se of the lignment sore prolem with weighted mthes, mismthes nd gps LCS sore: w mth = 1, w mismth = w gp = 0 Levenshtein sore: w mth = 2, w mismth = 1, w gp = 0 An lignment sore is rtionl, if w mth, w mismth, w gp re rtionl; redues to LCS sore y onstnt-ftor low-up of lignment grph The semi-lol lignment sore prolem: string-sustring, prefix-suffix, sustring-string, suffix-prefix lignment sores Edit distne: minimum ost to trnsform into y weighted hrter edits (insertion, deletion, sustitution) The semi-lol edit distne prolem: semi-lol lignment sore prolem with w mth = 0, w mismth = w su, w gp = w indel Alexnder Tiskin (Wrwik) Periodi string omprison 29 / 51
48 Semi-lol string omprison Highest-sore mtries The semi-lol LCS sore mtrix = "" = "" A(0, 13) = LCS(, ) = 8 = "..." A(4, 11) = LCS(, ) = 5 A(i, j) = j i if i > j Alexnder Tiskin (Wrwik) Periodi string omprison 30 / 51
49 Semi-lol string omprison Highest-sore mtries Semi-lol LCS: output representtion nd running time size query time O(n 2 ) O(1) trivil O(m 1/2 n) O(log n) string-sustring [Alves+: 2003] O(n) O(n) string-sustring [Alves+: 2005] O(n log n) O(log 2 n) [T: 2006] running time O(mn 2 ) nive O(mn) string-sustring [Shmidt: 1998] string-sustring [Alves+: 2005] O(mn) [T: 2006] O ( ) mn [T: 2006] log 0.5 n O ( mn(log log n) 2 ) log n [T: 2007] Alexnder Tiskin (Wrwik) Periodi string omprison 31 / 51
50 Semi-lol string omprison Highest-sore mtries A: the semi-lol LCS sore mtrix for vs A(i, j): the numer of mthed hrters for vs sustring of Q(i, j) = j i A(i, j): the numer of unmthed hrters Properties of mtrix Q: Q is simple unit-monge therefore, Q = P Σ for some permuttion mtrix P P = Q = A is n impliit representtion of A Rnge tree for P: memory O(n log n), query time O(log 2 n) Alexnder Tiskin (Wrwik) Periodi string omprison 32 / 51
51 Semi-lol string omprison Highest-sore mtries The semi-lol LCS sore mtrix = "" = "" = "..." A(4, 11) = LCS(, ) = 5 A(i, j) = j i if i > j Alexnder Tiskin (Wrwik) Periodi string omprison 33 / 51
52 Semi-lol string omprison Highest-sore mtries The semi-lol LCS sore mtrix = "" = "" = "..." A(4, 11) = LCS(, ) = 5 A(i, j) = j i if i > j lue: differene 0 red: differene 1 Alexnder Tiskin (Wrwik) Periodi string omprison 33 / 51
53 Semi-lol string omprison Highest-sore mtries The semi-lol LCS sore mtrix = "" = "" = "..." A(4, 11) = LCS(, ) = 5 A(i, j) = j i if i > j lue: differene 0 red: differene 1 green: P(i, j) = 1 A(i, j) = j i P Σ (i, j) Alexnder Tiskin (Wrwik) Periodi string omprison 33 / 51
54 Semi-lol string omprison Highest-sore mtries The semi-lol LCS sore mtrix = "" = "" = "..." A(4, 11) = LCS(, ) = 11 4 P Σ (i, j) = = 5 P gives n impliit representtion of A Alexnder Tiskin (Wrwik) Periodi string omprison 34 / 51
55 Semi-lol string omprison Highest-sore mtries The seweeds in the lignment grph = "" = "" = "..." A(4, 11) = LCS(, ) = 11 4 P Σ (i, j) = = 5 P gives n impliit representtion of A P(i, j) = 1 orresponds to seweed (top, i) (ottom, j) Alexnder Tiskin (Wrwik) Periodi string omprison 35 / 51
56 Semi-lol string omprison Highest-sore mtries The seweeds in the lignment grph = "" = "" = "..." A(4, 11) = LCS(, ) = 11 4 P Σ (i, j) = = 5 P gives n impliit representtion of A P(i, j) = 1 orresponds to seweed (top, i) (ottom, j) Also define top right, left right, left ottom seweeds Gives omplete order-to-order grph-theoreti mthing Alexnder Tiskin (Wrwik) Periodi string omprison 35 / 51
57 1 Introdution 2 Semi-lol string omprison 3 The seweed lgorithm 4 Conlusions nd future work Alexnder Tiskin (Wrwik) Periodi string omprison 36 / 51
58 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
59 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
60 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
61 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
62 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
63 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
64 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
65 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
66 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
67 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
68 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
69 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
70 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
71 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
72 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
73 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
74 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
75 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
76 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
77 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
78 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
79 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
80 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
81 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
82 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
83 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
84 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
85 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
86 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
87 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
88 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
89 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
90 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
91 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
92 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
93 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
94 The seweed lgorithm The seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 37 / 51
95 The seweed lgorithm The seweed lgorithm Semi-lol LCS: the seweed lgorithm [T: 2006] Iterte over lignment grph, tring seweeds Pik ells in ny order, respeting dependenies In every ell, the two entering seweeds ross, if mismth nd they hve not rossed efore end otherwise Running time O(mn) Alexnder Tiskin (Wrwik) Periodi string omprison 38 / 51
96 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
97 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
98 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
99 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
100 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
101 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
102 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
103 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
104 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
105 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
106 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
107 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
108 The seweed lgorithm The miro-lok seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 39 / 51
109 The seweed lgorithm The miro-lok seweed lgorithm Semi-lol LCS: the miro-lok seweed lgorithm [T: 2007] Iterte over lignment grph in loks, tring seweeds Use preomputed mpping of ll possile lok inputs to outputs Blok size: t = O ( log n ) log log n Blok interfe: O(t) vlues (input hrs nd seweeds), eh of size O(log n) ut n e ompressed to O(log log n) y reursive sheme Running time O ( m t n t t log log n) = O ( mn(log log n) 2 ) log n, even on log-ost RAM Alexnder Tiskin (Wrwik) Periodi string omprison 40 / 51
110 The seweed lgorithm Cyli LCS The yli LCS prolem Give the mximum LCS sore for vs ll yli rottions of Alexnder Tiskin (Wrwik) Periodi string omprison 41 / 51
111 The seweed lgorithm Cyli LCS The yli LCS prolem Give the mximum LCS sore for vs ll yli rottions of Cyli LCS: running time O ( ) mn 2 log n nive O(mn log m) [Mes: 1990] O(mn) [Bunke, Bühler: 1993; Lndu+: 1998; Shmidt: 1998] O ( mn(log log n) 2 ) log n [T: 2007] Cyli LCS: the lgorithm Run the miro-lok seweed lgorithm on vs, time O ( mn(log log n) 2 ) log n Mke n string-sustring LCS queries, time negligile Alexnder Tiskin (Wrwik) Periodi string omprison 41 / 51
112 The seweed lgorithm Longest repeting susequene The longest repeting susequene prolem Find the longest susequene of tht is squre ( repetition of two identil strings) Motivted y tndem repets in genome Alexnder Tiskin (Wrwik) Periodi string omprison 42 / 51
113 The seweed lgorithm Longest repeting susequene The longest repeting susequene prolem Find the longest susequene of tht is squre ( repetition of two identil strings) Motivted y tndem repets in genome Longest repeting susequene: running time O(n 3 ) nive O(n 2 ) [Kosowski: 2004] O ( n 2 (log log n) 2 ) log n [T: 2007] Longest repeting susequene: the lgorithm Run the miro-lok seweed lgorithm on vs, time O ( mn(log log n) 2 ) log n Mke n 1 suffix-prefix LCS queries, time negligile Alexnder Tiskin (Wrwik) Periodi string omprison 42 / 51
114 The seweed lgorithm Approximte mthing The pproximte pttern mthing prolem Give the sustring losest to y lignment sore, strting t eh position in Assume rtionl lignment sore Approximte pttern mthing: running time O(mn) [Sellers: 1980] O ( ) mn log n σ = O(1) vi [Msek, Pterson: 1980] O ( mn(log log n) 2 ) log n vi [Pterson, Dnik: 1994] Alexnder Tiskin (Wrwik) Periodi string omprison 43 / 51
115 The seweed lgorithm Approximte mthing Approximte pttern mthing: the lgorithm Run the miro-lok seweed lgorithm on vs under given lignment sore in time O ( mn(log log n) 2 ) log n The impliit semi-lol edit sore mtrix: n nti-monge mtrix pproximte pttern mthing row minim Row minim in O(n) element queries [Aggrwl+: 1987] Eh query in time O(log 2 n) using the rnge tree representtion, omined query time negligile Overll running time dominted y lok seweed lgorithm, sme s [Pterson, Dnik: 1994] Alexnder Tiskin (Wrwik) Periodi string omprison 44 / 51
116 The seweed lgorithm The periodi seweed lgorithm The periodi string-sustring LCS prolem Give (impliit) LCS sores for vs eh sustring of =... uuu... = u ± Let u e of length p; my ssume tht every hrter of ours in u The tndem LCS prolem Give LCS sore for vs = u k We hve n = kp; my ssume k m (otherwise LCS sore is m) Tndem LCS: running time O(mkp) nive O(m(k + p)) [Lndu, Ziv-Ukelson: 2001] O(mp) [NEW] Alexnder Tiskin (Wrwik) Periodi string omprison 45 / 51
117 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
118 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
119 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
120 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
121 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
122 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
123 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
124 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
125 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
126 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
127 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
128 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
129 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
130 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
131 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
132 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
133 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
134 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
135 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
136 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
137 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
138 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
139 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
140 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
141 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
142 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
143 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
144 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
145 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
146 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
147 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
148 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
149 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
150 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
151 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
152 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
153 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
154 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
155 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
156 The seweed lgorithm The periodi seweed lgorithm Alexnder Tiskin (Wrwik) Periodi string omprison 46 / 51
157 The seweed lgorithm The periodi seweed lgorithm Periodi string-sustring LCS: The periodi seweed lgorithm Iterte over lignment grph, tring seweeds row-y-row In every row, strt from mth ell nd move rightwrds, wrpping round t the grph edge In every ell, the two entering seweeds ross, if mismth nd they hve not rossed efore end otherwise t right edge of the grph, wrp round k to left edge Running time O(mn) Querying string-sustring LCS sore (inluding tndem LCS): ount eh nonzero dominted y query point with pproprite multipliity, either diretly or vi rnge tree Alexnder Tiskin (Wrwik) Periodi string omprison 47 / 51
158 The seweed lgorithm The periodi seweed lgorithm The tndem lignment prolem Give the sustring losest to y lignment sore mong ertin sustrings of = u ± : glol: sustrings of the form k ross ll k yli: sustrings of length kp ross ll k lol: sustrings of ny length Tndem lignment: running time O(m 2 p) ll nive O(mp) glol [Myers, Miller: 1989] O(mp log p) yli [Benson: 2005] O(mp) yli [NEW] O(mp) lol [Myers, Miller: 1989] Alexnder Tiskin (Wrwik) Periodi string omprison 48 / 51
159 The seweed lgorithm The periodi seweed lgorithm Cyli tndem lignment: the lgorithm Run periodi seweed lgorithm (under given lignment sore), time O(np) For eh k [1 : m]: solve tndem LCS (under given lignment sore) for ginst k otin p suessive string-sustring lignment sores y inrementl sore updting, eh in time O(1) Running time O(mp) Alexnder Tiskin (Wrwik) Periodi string omprison 49 / 51
160 1 Introdution 2 Semi-lol string omprison 3 The seweed lgorithm 4 Conlusions nd future work Alexnder Tiskin (Wrwik) Periodi string omprison 50 / 51
161 Conlusions nd future work Semi-lol LCS prolem: representtion y impliit unit-monge mtries generlistion to rtionl lignment sores open: rel lignment sores? The seweed nd miro-lok seweed lgorithms: simple lgorithm for semi-lol LCS semi-lol LCS in time o(mn) vi miro-loks improvements on relted prolems The periodi seweed lgorithm: strightforwrd extension of the seweed lgorithm periodi semi-lol LCS in time O(mp) nturl pplitions open: o(mp) vi miro-loks? Alexnder Tiskin (Wrwik) Periodi string omprison 51 / 51
Semi-local string comparison
Semi-lol string omprison Alexnder Tiskin http://www.ds.wrwik..uk/~tiskin Deprtment of Computer Siene University of Wrwik 1 The prolem 2 Effiient output representtion 3 The lgorithms 4 Conlusions nd future
Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18
Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re
Prefix-Free Regular-Expression Matching
Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings
Data Structures and Algorithm. Xiaoqing Zheng
Dt Strutures nd Algorithm Xioqing Zheng zhengxq@fudn.edu.n String mthing prolem Pttern P ours with shift s in text T (or, equivlently, tht pttern P ours eginning t position s + in text T) if T[s +... s
Computational Biology Lecture 18: Genome rearrangements, finding maximal matches Saad Mneimneh
Computtionl Biology Leture 8: Genome rerrngements, finding miml mthes Sd Mneimneh We hve seen how to rerrnge genome to otin nother one sed on reversls nd the knowledge of the preserved loks or genes. Now
Common intervals of genomes. Mathieu Raffinot CNRS LIAFA
Common intervls of genomes Mthieu Rffinot CNRS LIF Context: omprtive genomis. set of genomes prtilly/totlly nnotte Informtive group of genes or omins? Ex: COG tse Mny iffiulties! iology Wht re two similr
Algorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:
Eidgenössishe Tehnishe Hohshule Zürih Eole polytehnique fédérle de Zurih Politenio federle di Zurigo Federl Institute of Tehnology t Zurih Deprtement of Computer Siene. Novemer 0 Mrkus Püshel, Dvid Steurer
Computing the Optimal Global Alignment Value. B = n. Score of = 1 Score of = a a c g a c g a. A = n. Classical Dynamic Programming: O(n )
Alignment Grph Alignment Mtrix Computing the Optiml Globl Alignment Vlue An Introduction to Bioinformtics Algorithms A = n c t 2 3 c c 4 g 5 g 6 7 8 9 B = n 0 c g c g 2 3 4 5 6 7 8 t 9 0 2 3 4 5 6 7 8
Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.
I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the
Project 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
Computing the Cyclic Edit Distance for Pattern Classification by Ranking Edit Paths
Computing the Cyli Edit Distne for Pttern Clssifition y Rnking Edit Pths Vítor M. Jiménez, Andrés Mrzl, Viente Plzón, nd Guillermo Peris DLSI, Universitt Jume I, 27 Cstellón, Spin {vjimenez,mrzl,plzon,peris}@uji.es
Learning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
Powering a number. More Divide & Conquer
CS 4 -- Spring 29 Powering numer Prolem: Compute n, where n N. Nive lgorithm: Θ(n). ore Divide & Conquer Crol Wenk Slides ourtesy of Chrles Leiserson with smll hnges y Crol Wenk 2//9 CS 4 nlysis of lgorithms
Finite State Automata and Determinisation
Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions
Introduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
( ) { } [ ] { } [ ) { } ( ] { }
![( ) { } [ ] { } [ ) { } ( ] { }](/thumbs/96/129291861.jpg "( ) { } [ ] { } [ ) { } ( ] { }")
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
Fast index for approximate string matching
Fst index for pproximte string mthing Dekel Tsur Astrt We present n index tht stores text of length n suh tht given pttern of length m, ll the sustrings of the text tht re within Hmming distne (or edit
Hybrid Systems Modeling, Analysis and Control
Hyrid Systems Modeling, Anlysis nd Control Rdu Grosu Vienn University of Tehnology Leture 5 Finite Automt s Liner Systems Oservility, Rehility nd More Miniml DFA re Not Miniml NFA (Arnold, Diky nd Nivt
Linear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
The Word Problem in Quandles
The Word Prolem in Qundles Benjmin Fish Advisor: Ren Levitt April 5, 2013 1 1 Introdution A word over n lger A is finite sequene of elements of A, prentheses, nd opertions of A defined reursively: Given
NON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α
Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw
Nondeterministic Finite Automata
Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)
Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
11/3/13. Indexing techniques. Short-read mapping software. Indexing a text (a genome, etc) Some terminologies. Hashing
I9 Introdution to Bioinformtis, 0 Indeing tehniques Yuzhen Ye (yye@indin.edu) Shool of Informtis & Computing, IUB Contents We hve seen indeing tehnique used in BLAST Applitions tht rely on n effiient indeing
CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4
Am Blnk Leture 13 Winter 2016 CSE 332 CSE 332: Dt Astrtions Sorting Dt Astrtions QuikSort Cutoff 1 Where We Are 2 For smll n, the reursion is wste. The onstnts on quik/merge sort re higher thn the ones
Alpha Algorithm: Limitations
Proess Mining: Dt Siene in Ation Alph Algorithm: Limittions prof.dr.ir. Wil vn der Alst www.proessmining.org Let L e n event log over T. α(l) is defined s follows. 1. T L = { t T σ L t σ}, 2. T I = { t
Solutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
Lecture 2: Cayley Graphs
Mth 137B Professor: Pri Brtlett Leture 2: Cyley Grphs Week 3 UCSB 2014 (Relevnt soure mteril: Setion VIII.1 of Bollos s Moern Grph Theory; 3.7 of Gosil n Royle s Algeri Grph Theory; vrious ppers I ve re
Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
Chapter 8 Roots and Radicals
Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi
University of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
Learning Partially Observable Markov Models from First Passage Times
Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).
Chapter 4 State-Space Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
Computing with finite semigroups: part I
Computing with finite semigroups: prt I J. D. Mitchell School of Mthemtics nd Sttistics, University of St Andrews Novemer 20th, 2015 J. D. Mitchell (St Andrews) Novemer 20th, 2015 1 / 34 Wht is this tlk
Discrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
First Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
Lecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
Computing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt
Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:
Aperiodic tilings and substitutions
Aperioi tilings n sustitutions Niols Ollinger LIFO, Université Orléns Journées SDA2, Amiens June 12th, 2013 The Domino Prolem (DP) Assume we re given finite set of squre pltes of the sme size with eges
TIME AND STATE IN DISTRIBUTED SYSTEMS
Distriuted Systems Fö 5-1 Distriuted Systems Fö 5-2 TIME ND STTE IN DISTRIUTED SYSTEMS 1. Time in Distriuted Systems Time in Distriuted Systems euse eh mhine in distriuted system hs its own lok there is
Nondeterministic Automata vs Deterministic Automata
Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n
Abstraction of Nondeterministic Automata Rong Su
Astrtion of Nondeterministi Automt Rong Su My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 1 Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering,
A Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
Efficient High-Similarity String Comparison: The Waterfall Algorithm
Efficient High-Similarity String Comparison: The Waterfall Algorithm Alexander Tiskin Department of Computer Science University of Warwick http://go.warwick.ac.uk/alextiskin Alexander Tiskin (Warwick)
Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz
University of Southern Cliforni Computer Siene Deprtment Compiler Design Spring 7 Lexil Anlysis Smple Exerises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sienes Institute 47 Admirlty Wy, Suite
Can one hear the shape of a drum?
Cn one her the shpe of drum? After M. K, C. Gordon, D. We, nd S. Wolpert Corentin Lén Università Degli Studi di Torino Diprtimento di Mtemti Giuseppe Peno UNITO Mthemtis Ph.D Seminrs Mondy 23 My 2016 Motivtion:
Fast Frequent Free Tree Mining in Graph Databases
The Chinese University of Hong Kong Fst Frequent Free Tree Mining in Grph Dtses Peixing Zho Jeffrey Xu Yu The Chinese University of Hong Kong Decemer 18 th, 2006 ICDM Workshop MCD06 Synopsis Introduction
Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression
5-2 Dt Strutures n Algorithms Dt Compression n Huffmn s Algorithm th Fe 2003 Rjshekr Rey Outline Dt ompression Lossy n lossless Exmples Forml view Coes Definition Fixe length vs. vrile length Huffmn s
Fingerprint idea. Assume:
Fingerprint ide Assume: We cn compute fingerprint f(p) of P in O(m) time. If f(p) f(t[s.. s+m 1]), then P T[s.. s+m 1] We cn compre fingerprints in O(1) We cn compute f = f(t[s+1.. s+m]) from f(t[s.. s+m
Figure 1. The left-handed and right-handed trefoils
The Knot Group A knot is n emedding of the irle into R 3 (or S 3 ), k : S 1 R 3. We shll ssume our knots re tme, mening the emedding n e extended to solid torus, K : S 1 D 2 R 3. The imge is lled tuulr
y1 y2 DEMUX a b x1 x2 x3 x4 NETWORK s1 s2 z1 z2
BOOLEAN METHODS Giovnni De Miheli Stnford University Boolen methods Exploit Boolen properties. { Don't re onditions. Minimiztion of the lol funtions. Slower lgorithms, etter qulity results. Externl don't
QUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
Coalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
= state, a = reading and q j
4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those
Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
Lossless Compression Lossy Compression
Administrivi CSE 39 Introdution to Dt Compression Spring 23 Leture : Introdution to Dt Compression Entropy Prefix Codes Instrutor Prof. Alexnder Mohr mohr@s.sunys.edu offie hours: TBA We http://mnl.s.sunys.edu/lss/se39/24-fll/
GM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
A Differential Approach to Inference in Bayesian Networks
Dierentil pproh to Inerene in Byesin Networks esented y Ynn Shen shenyn@mi.pitt.edu Outline Introdution Oeriew o lgorithms or inerene in Byesin networks (BN) oposed new pproh How to represent BN s multi-rite
Necessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )
Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us
Similar Right Triangles
Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems
Introduction to Bioinformatics
Introdution to Bioinformtis Outline } Method without onsidering bkground distribution } Generl pproh onsidering bkground distribution } Wys to speed up the lgorithm Trnsription Ftor Binding Sites (TFBSs)
CSCI565 - Compiler Design
CSCI565 - Compiler Deign Spring 6 Due Dte: Fe. 5, 6 t : PM in Cl Prolem [ point]: Regulr Expreion nd Finite Automt Develop regulr expreion (RE) tht detet the longet tring over the lphet {-} with the following
Random subgroups of a free group
Rndom sugroups of free group Frédérique Bssino LIPN - Lortoire d Informtique de Pris Nord, Université Pris 13 - CNRS Joint work with Armndo Mrtino, Cyril Nicud, Enric Ventur et Pscl Weil LIX My, 2015 Introduction
General Suffix Automaton Construction Algorithm and Space Bounds
Generl Suffix Automton Constrution Algorithm nd Spe Bounds Mehryr Mohri,, Pedro Moreno, Eugene Weinstein, Cournt Institute of Mthemtil Sienes 251 Merer Street, New York, NY 10012. Google Reserh 76 Ninth
where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}
Lecture Summaries for Multivariable Integral Calculus M52B
These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition.
MATH Final Review
MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out
HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem
HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots
Håkan Lennerstad, Lars Lundberg
GENERALIZATIONS OF THE FLOOR AND CEILING FUNCTIONS USING THE STERN-BROCOT TREE Håkn Lennerstd, Lrs Lunderg Blekinge Institute of Tehnology Reserh report No. 2006:02 Generliztions of the floor nd eiling
Functions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:
PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
Comparing the Pre-image and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity
Three termination problems. Patrick Dehornoy. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen
Three termintion prolems Ptrick Dehornoy Lortoire de Mthémtiques Nicols Oresme, Université de Cen Three termintion prolems Ptrick Dehornoy Lortoire Preuves, Progrmmes, Systèmes Université Pris-Diderot
Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
Unit 4. Combinational Circuits
Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute
Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}
Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)
Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths
Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t
Algebra 2 Semester 1 Practice Final
Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers
CIT 596 Theory of Computation 1. Graphs and Digraphs
CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege
@#? Text Search ] { "!" Nondeterministic Finite Automata. Transformation NFA to DFA and Simulation of NFA. Text Search Using Automata
![@#? Text Search ] { ! Nondeterministic Finite Automata. Transformation NFA to DFA and Simulation of NFA. Text Search Using Automata](/thumbs/92/110360658.jpg "@#? Text Search ] { ! Nondeterministic Finite Automata. Transformation NFA to DFA and Simulation of NFA. Text Search Using Automata")
g Text Serh @#? ~ Mrko Berezovský Rdek Mřík PAL 0 Nondeterministi Finite Automt n Trnsformtion NFA to DFA nd Simultion of NFA f Text Serh Using Automt A B R Power of Nondeterministi Approh u j Regulr Expression
KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)
KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 6-7 CLASS - XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners: - A Reltion is sid to e Reflexive if.. every A
Hyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
Symmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent
Pre-Lie algebras, rooted trees and related algebraic structures
Pre-Lie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A pre-lie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers
Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition
Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt
A Non-parametric Approach in Testing Higher Order Interactions
A Non-prmetri Approh in Testing igher Order Intertions G. Bkeerthn Deprtment of Mthemtis, Fulty of Siene Estern University, Chenkldy, Sri Lnk nd S. Smit Deprtment of Crop Siene, Fulty of Agriulture University
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
Subsequence Automata with Default Transitions
Susequene Automt with Defult Trnsitions Philip Bille, Inge Li Gørtz, n Freerik Rye Skjoljensen Tehnil University of Denmrk {phi,inge,fskj}@tu.k Astrt. Let S e string of length n with hrters from n lphet
Logic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions Jie-Hong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 22 Reding: Logi Synthesis in Nutshell Setion 2 most