Semi-local string comparison

Size: px
Start display at page:

Download "Semi-local string comparison"

Transcription

1 Semi-lol string omprison Alexnder Tiskin Deprtment of Computer Siene University of Wrwik

2 1 The prolem 2 Effiient output representtion 3 The lgorithms 4 Conlusions nd future work

3 1 The prolem 2 Effiient output representtion 3 The lgorithms 4 Conlusions nd future work

4 The prolem String mthing: find n ext pttern in string String omprison: find similr ptterns in two strings glol: ompre whole string ginst whole string lol: ompre sustrings ginst sustrings semi-lol: ompre whole string ginst sustrings (will extend this definition lter) Often lled pproximte string mthing (no reltion to pproximtion lgorithms!) Applitions: omputtionl iology, imge reognition,...

5 The prolem Consider strings (= sequenes) over finite or infinite lphet Distinguish ontiguous sustrings nd not neessrily ontiguous susequenes Speil ses of sustring: prefix, suffix Usul nottion: strings, of length m, n respetively

6 The prolem Longest ommon susequene (LCS) prolem: determine the LCS length for string ginst string A vrint of the edit distne prolem O(mn) y dynmi progrmming [Needlemn, Wunsh, 1970] O ( ) mn log(m+n) [Msek, Pterson, 1980] extended y [Crohemore+, 2003]

7 The prolem LCS("", "") = "" m n lue = 0 red = 1 Alignment grph Longest ommon susequene longest soure-to-sink pth

8 The prolem All string-sustring LCS prolem: determine the mximum LCS length for string ginst ll sustrings of Nive pproh: O ( mn 3 log(m+n)) to ompute Θ(n 2 ) outputs O(mn) [Shmidt, 1998] extended y [Alves+, 2005] New result: O ( mn ), with outputs represented impliitly log 0.5 (m+n)

9 The prolem LCS prolem on yli strings: determine the mximum LCS length for string ginst ll yli rottions of Nive pproh: O ( mn 2 log(m+n)) to ompute Θ(n) prtil results O(mn log m) [Mes, 1990] O(mn) [Bunke, Bühler, 1993] extended y [Lndu+, 1998], [Shmidt, 1998] New result: O ( ) mn log 0.5 (m+n)

10 The prolem Semi-lol susequene reognition prolem on ompressed text (ginst plin pttern): given stright-line progrm of size M generting, determine sustrings of ontining s susequene Covers vrious ompression prdigms, e.g. Lempel-Ziv Note m n e O( M ). Assume ddress rithmeti on still O(1). Nive pproh: unompress. Cn e O( M ). O(Mn 2 log n) [Cegielski+, 2006] New result: O(Mn 1.5 )

11 The prolem Longest inresing susequene (LIS) prolem: determine the LCS length for permuttion of length n ginst id = (1, 2,..., n) Nive pproh: O ( n 2 log n) y generi LCS O(n log n) impliit in [Erdös, Szekeres, 1935] lso [Roinson, 1938] extended y [Knuth, 1970] lso [Dijkstr, 1980] In stronger model: O(n log log n) [Hunt, Szymnski, 1977] lso [Chng, Wng, 1992] extended y [Bespmytnikh, Segl, 2000]

12 The prolem Window LIS prolem: given permuttion, determine the LIS length for every sustring of of fixed length r Nive pproh: O(n 2 log n) to ompute Θ(n) outputs O(n 2 ), omputes every LIS ompletely [Alert+, 2004] extended y [Chen+, 2005] New result: O(n 1.5 ), even for vrile r

13 The prolem Mx-lique prolem in irle grph: given irle nd n hords, determine the mximum-size suset of pirwise interseting hords O(n 3 ) [Gvril, 1973] O(n 2 ) [Rotem, Urruti, 1981] extended y [Hsu, 1985] lso [Msud+, 1990], [Apostolio+, 1992] New result: O(n 1.5 ), even to find (impliitly) ll mximl susets

14 The prolem All of the ove n e ptured y single prolem sttement!

15 The prolem All semi-lol longest ommon susequenes (LCS) prolem: determine the LCS length for string ginst every sustring of every prefix of ginst every suffix of s ove ut with nd swpped Totl Θ(n 2 ) outputs, llowed to e represented impliitly

16 The prolem LCS("", "...") = "" m n lue = 0 red = 1 All semi-lol LCS ll longest order-to-order pths (string-sustring top-to-ottom, et.)

17 The prolem All semi-lol LCS prolem: impliit output representtion Trivil: size O(n 2 ) query time O(1) size O(m 1/2 n) query time O(log n) [Alves+, 2003] size O(n) query time O(n) [Alves+, 2005] New result: size O(n log n) query time O(log 2 n) In stronger model: size O(n) query time O( log n log log n )

18 The prolem All semi-lol LCS prolem: omputtion time O(mn) for ll string-sustring LCS [Alves+, 2005] New result: O ( ) mn on strings over onstnt-size lphet log 0.5 (m+n) Implies LCS on yli strings, ll string-sustring LCS New result: O(n 1.5 ) on permuttions Implies Window LIS, Mx-lique in irle grph

19 1 The prolem 2 Effiient output representtion 3 The lgorithms 4 Conlusions nd future work

20 Effiient output representtion Nottion Integers 0, 1, 2,... Odd hlf-integers 1 2, 3 2, 5 2,... x y y x = 1 x y y x = 1 2 Definition Point (i 0, j 0 ) domintes point (i, j), if i 0 < i nd j < j 0

21 Effiient output representtion Definition Point (i, j) is A-ritil, if A(i, j ) A(i, j + ) = A(i +, j ) = A(i +, j + ) where i i i + j j j + Nottion d A (i 0, j 0 ) is the numer of A-ritil points dominted y (i 0, j 0 )

22 Effiient output representtion Lemm A(i 0, j 0 ) = j 0 i 0 d A (i 0, j 0 ) j 0 i 0 : input sustring length d A (i 0, j 0 ): unmthed hrters Proof: simple indution More generlly: A is Monge mtrix its density mtrix hppens to e the permuttion mtrix of ritil points

23 Effiient output representtion Full top-to-ottom sore mtrix: A(i, j) i, j n

24 Effiient output representtion Full top-to-ottom sore mtrix: A(i, j) i, j n A(i +, j) A(i, j) A(i, j ) A(i, j + ) A totlly monotone: A(i +, j + ) A(i, j + ) A(i +, j ) A(i, j ) A T totlly monotone: A(i, j ) A(i, j + ) A(i +, j ) A(i +, j + ) where i i +, j j + lue = 0 red = 1

25 Effiient output representtion Full top-to-ottom sore mtrix: A(i, j) i, j n A(i +, j) A(i, j) A(i, j ) A(i, j + ) A totlly monotone: A(i +, j + ) A(i, j + ) A(i +, j ) A(i, j ) A T totlly monotone: A(i, j ) A(i, j + ) A(i +, j ) A(i +, j + ) where i i +, j j + lue = 0 red = 1 green = ritil

26 Effiient output representtion Full top-to-ottom sore mtrix: A(i, j) 0 i, j n 5 j 0 i 0 d A (i 0, j 0 ) = = 5 lue = 0 red = 1 green = ritil

27 Effiient output representtion Full top-to-ottom sore mtrix: A(i, j) 0 i, j n 5 j 0 i 0 d A (i 0, j 0 ) = = 5 lue = 0 red = 1 green = ritil

28 Effiient output representtion Critil point (i, j) in sore mtrix gives ritil pir (top, i) (ottom, j) in lignment grph Also define top right, left right, left ottom ritil pirs Gives omplete order-to-order grph-theoreti mthing

29 Effiient output representtion Gudi s seweeds

30 Effiient output representtion Estlishing d A (i 0, j 0 ): dominne ounting Rnge tree: [Bentley, 1980] inry serh tree y i-oordinte for ll nodes rooted t its every node, inry serh tree y j-oordinte for relevnt nodes Every node represents nonil rnge (retngulr region), nd stores its point ount Theorem (Bentley, 1980) A rnge tree on n points hs size O(n log n) dominne ounting query time O(log 2 n)

31 Effiient output representtion Rnge tree: Every rnge n e deomposed into log 2 n nonil rnges Overll, n log n nonil rnges re non-empty

32 Effiient output representtion Advned dominne ounting [JJ+, 2004] Requires RAM model Theorem (JJ+,2004) There exists dt struture on n points with size O(n) dominne ounting query time O( log n log log n ) Corollry (new) All semi-lol LCS lengths n e represented in size O(n log n) query time O(log 2 n) size O(n) query time O( log n log log n )

33 1 The prolem 2 Effiient output representtion 3 The lgorithms 4 Conlusions nd future work

34 The lgorithms Min suroutine: sore mtrix multiplition AB = C Generl mtries: time O(n 3 ) Monge (= plnr distne) mtries: time O(n 2 ) Sore mtries: time O(n 1.5 ) proof lter

35 The lgorithms Prtition lignment grph into horizontl strips of width n Eh strip is initilly of height 1 represented y O(n) ritil points Min pttern: merge strips A, B C y sore mtrix multiplition A-ritil points, B-ritil points C-ritil points Eventully otin ritil points for the whole lignment grph Post-proess to effiient output representtion (e.g. rnge tree)

36 The lgorithms Lemm (Alves+, 2005, sed on Shmidt) A strip n e merged with strip of height 1 in time O(n). Proof: preliminry merge ll points (i, j), (j, k) (i, k) Resulting triples (i, j, k) must not ross twie (f. longest pths) Sn the nrrower strip left-to-right, removing doule rossings Theorem (Alves+, sed on Shmidt) Algorithm for ll semi-lol LCS: time O(mn) memory O(n) outputs O(n) ritil points Proof: suessive pplition of the lemm

37 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

38 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

39 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

40 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

41 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

42 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

43 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

44 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

45 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

46 The lgorithms Algorithm y Shmidt/Alves+: inrementl merging of strips lue = 0 red = 1 green = ritil

47 The lgorithms Lemm (new) Two strips of ny height n e merged in time O(n 1.5 ). Proof: divide-nd-onquer on sore mtrix C Count C-ritil points in squre loks, eginning with full mtrix For lok of size r r, only r A- nd B-ritil points re relevnt If the lok hs > 0 C-ritil points, reurse into hlf-sized loks Worst se: n loks of size n 0.5 n 0.5 for eh need to onsider t most n 0.5 A- nd B-ritil points Overll time n n 0.5 = n 1.5

48 The lgorithms New lgorithm: divide-nd-onquer on strips lue = 0 red = 1 green = ritil

49 The lgorithms New lgorithm: divide-nd-onquer on strips lue = 0 red = 1 green = ritil

50 The lgorithms New lgorithm: divide-nd-onquer on strips lue = 0 red = 1 green = ritil

51 The lgorithms New lgorithm: divide-nd-onquer on strips lue = 0 red = 1 green = ritil

52 The lgorithms New lgorithm: divide-nd-onquer on strips lue = 0 red = 1 green = ritil

53 The lgorithms New lgorithm: divide-nd-onquer on strips lue = 0 red = 1 green = ritil

54 The lgorithms Theorem (new) Algorithm for ll semi-lol LCS on permuttions of length n: time O(n 1.5 ) memory O(n) outputs O(n) ritil points Proof: in strip of height k, t most k ritil points non-trivil Two suh strips n e merged in time O(k 1.5 ) In every reursion level: numer of suprolems up y ftor of 2 time per suprolem down y ftor of Hene, top level domintes with time O(n 1.5 )

55 The lgorithms Applition: Mx-lique in irle grph with n hords Stndrd redution to strings: irle line, hords segments Segment endpoints represented y permuttion of size 2n Chords interset segments interset without ontinment Helly property: suset of segments interset pirwise they ll interset t ommon point

56 The lgorithms Theorem (new) Algorithm for Mx-lique in irle grph of size n: time O(n 1.5 ) memory O(n) Proof: hek ll 2n + 1 possile ommon intersetion points For eh ndidte intersetion point, need to find the lrgest suset of overing segments without pirwise ontinment Equivlent to prefix-suffix LCS on permuttions, id Run semi-lol LCS on, id nd uild rnge tree: time O(n 1.5 ) Query prefix-suffix LCS for eh ndidte intersetion point: time (2n + 1) O(log 2 n) = O(n log 2 n) Overll time O(n 1.5 )

57 The lgorithms Theorem (new) Algorithm for ll semi-lol LCS (over onstnt-size lphet Σ): time O ( ) mn memory O(n) log 0.5 (m+n) outputs O(m + n) full-order ritil points Must ssume m nd n resonly lose, e.g. (log m) 2.5 n m Proof: divide-nd-onquer in lternte diretions Strips repled y nerly-squre loks Clssil tehnique y Arlzrov+: when loks suffiiently smll, esier to preompute ll possile loks in dvne Threshold lok size 0.5 log Σ m reursion time = preomputtion time = O( mn log 0.5 (m+n) )

58 1 The prolem 2 Effiient output representtion 3 The lgorithms 4 Conlusions nd future work

59 Conlusions nd future work Current stte of the rt for ll semi-lol LCS Output representtion: size O(n log n) query time O(log 2 n) size O(n) query time O( log n log log n ) in stronger model Computtion time: O ( ) mn on strings over onstnt-size lphet log 0.5 (m+n) O(n 1.5 ) on permuttions Memory: O(m + n) Implies improvements for severl existing prolems

60 Conlusions nd future work Esy improvements (undergrdute exerises): extension to rtionl-weighted edit distne ommunition-effiient prllelistion More sophistited improvements (grdute projets): generlistion to sprse string omprison removing ssumption of onstnt lphet size Potentil further improvements (open prolems): effiient reovery of full optiml susequenes extension to rel-weighted edit distne new interesting speil ses nd pplitions etter dominne ounting lol string omprison: lterntive to Smith Wtermn?

61 Referenes I C. E. R. Alves, E. N. Cáeres, nd S. W. Song. An ll-sustrings ommon susequene lgorithm. Eletroni Notes in Disrete Mthemtis, 19: , J. L. Bentley. Multidimensionl divide-nd-onquer. Communitions of the ACM, 23(4): , J. JJ, C. Mortensen, nd Q. Shi. Spe-effiient nd fst lgorithms for multidimensionl dominne reporting nd ounting. In Proeedings of the 15th ISAAC, volume 3341 of Leture Notes in Computer Siene, pges , 2004.

62 Referenes II A. Tiskin. All semi-lol longest ommon susequenes in suqudrti time. In Proeedings of CSR, volume 3967 of Leture Notes in Computer Siene, pges , A. Tiskin. Longest ommon susequenes in permuttions nd mximum liques in irle grphs. In Proeedings of CPM, To pper.

Periodic string comparison

Periodic string comparison Periodi string omprison Alexnder Tiskin Deprtment of Computer Siene University of Wrwik http://www.ds.wrwik..uk/~tiskin Alexnder Tiskin (Wrwik) Periodi string omprison 1 / 51 1 Introdution 2 Semi-lol string

More information

Prefix-Free Regular-Expression Matching

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

More information

Computational Biology Lecture 18: Genome rearrangements, finding maximal matches Saad Mneimneh

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

More information

Common intervals of genomes. Mathieu Raffinot CNRS LIAFA

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

More information

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 )

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

More information

Data Structures and Algorithm. Xiaoqing Zheng

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

More information

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18

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

More information

Powering a number. More Divide & Conquer

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

More information

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. 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

More information

Finite State Automata and Determinisation

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

More information

Lecture 6: Coding theory

Lecture 6: Coding theory Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those

More information

CIT 596 Theory of Computation 1. Graphs and Digraphs

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

More information

Computing the Cyclic Edit Distance for Pattern Classification by Ranking Edit Paths

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

More information

Algorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:

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

More information

Technische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution

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:

More information

CSE 332. Sorting. Data Abstractions. CSE 332: Data Abstractions. QuickSort Cutoff 1. Where We Are 2. Bounding The MAXIMUM Problem 4

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

More information

Linear choosability of graphs

Linear choosability of graphs Liner hoosility of grphs Louis Esperet, Mikel Montssier, André Rspud To ite this version: Louis Esperet, Mikel Montssier, André Rspud. Liner hoosility of grphs. Stefn Felsner. 2005 Europen Conferene on

More information

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

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

More information

Chapter 4 State-Space Planning

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

More information

Fast index for approximate string matching

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

More information

Logic Synthesis and Verification

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

More information

Learning Partially Observable Markov Models from First Passage Times

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).

More information

( ) { } [ ] { } [ ) { } ( ] { }

( ) { } [ ] { } [ ) { } ( ] { } 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

More information

A Differential Approach to Inference in Bayesian Networks

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

More information

Figure 1. The left-handed and right-handed trefoils

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

More information

Counting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs

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

More information

NON-DETERMINISTIC FSA

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

More information

Discrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α

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

More information

Fast Frequent Free Tree Mining in Graph Databases

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

More information

Lecture Notes No. 10

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

More information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into

More information

CONTROLLABILITY and observability are the central

CONTROLLABILITY and observability are the central 1 Complexity of Infiml Oservle Superlnguges Tomáš Msopust Astrt The infiml prefix-losed, ontrollle nd oservle superlnguge plys n essentil role in the reltionship etween ontrollility, oservility nd o-oservility

More information

Nondeterministic Automata vs Deterministic Automata

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

More information

CS 491G Combinatorial Optimization Lecture Notes

CS 491G Combinatorial Optimization Lecture Notes CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,

More information

First Midterm Examination

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

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

Subsequence Automata with Default Transitions

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

More information

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

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:

More information

QUADRATIC EQUATION. Contents

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,

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

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 )

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

More information

Lecture Summaries for Multivariable Integral Calculus M52B

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.

More information

Learning Objectives of Module 2 (Algebra and Calculus) Notes:

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

More information

Håkan Lennerstad, Lars Lundberg

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

More information

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

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

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition

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

More information

Suffix Trays and Suffix Trists: Structures for Faster Text Indexing

Suffix Trays and Suffix Trists: Structures for Faster Text Indexing Suffix Trys nd Suffix Trists: Strutures for Fster Text Indexing Rihrd Cole Tsvi Kopelowitz Moshe Lewenstein rxiv:1311.1762v1 [s.ds] 7 Nov 2013 Astrt Suffix trees nd suffix rrys re two of the most widely

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

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)}

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)

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

Chapter 2 Finite Automata

Chapter 2 Finite Automata Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht

More information

Abstraction of Nondeterministic Automata Rong Su

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,

More information

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014 CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA

More information

Introduction to Olympiad Inequalities

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......................

More information

Alpha Algorithm: Limitations

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

More information

A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA

A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA A CLASS OF GENERAL SUPERTREE METHODS FOR NESTED TAXA PHILIP DANIEL AND CHARLES SEMPLE Astrt. Amlgmting smller evolutionry trees into single prent tree is n importnt tsk in evolutionry iology. Trditionlly,

More information

Introduction to Bioinformatics

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)

More information

8 THREE PHASE A.C. CIRCUITS

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),

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

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

More information

Computing with finite semigroups: part I

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

More information

Hybrid Systems Modeling, Analysis and Control

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

More information

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6

CS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6 CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized

More information

Graph width-parameters and algorithms

Graph width-parameters and algorithms Grph width-prmeters nd lgorithms Jisu Jeong (KAIST) joint work with Sigve Hortemo Sæther nd Jn Arne Telle (University of Bergen) 2015 KMS Annul Meeting 2015.10.24. YONSEI UNIVERSITY Grph width-prmeters

More information

#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS

#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS #A42 INTEGERS 11 (2011 ON THE CONDITIONED BINOMIAL COEFFICIENTS Liqun To Shool of Mthemtil Sienes, Luoyng Norml University, Luoyng, Chin lqto@lynuedun Reeived: 12/24/10, Revised: 5/11/11, Aepted: 5/16/11,

More information

Lossless Compression Lossy Compression

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/

More information

Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression

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

More information

CS 573 Automata Theory and Formal Languages

CS 573 Automata Theory and Formal Languages Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple

More information

Logic Synthesis and Verification

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 2010 Reding: Logi Synthesis in Nutshell Setion 2 most

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

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

More information

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es

More information

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment

More information

General Suffix Automaton Construction Algorithm and Space Bounds

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

More information

Alpha Algorithm: A Process Discovery Algorithm

Alpha Algorithm: A Process Discovery Algorithm Proess Mining: Dt Siene in Ation Alph Algorithm: A Proess Disovery Algorithm prof.dr.ir. Wil vn der Alst www.proessmining.org Proess disovery = Ply-In Ply-In event log proess model Ply-Out Reply proess

More information

Bisimulation, Games & Hennessy Milner logic

Bisimulation, Games & Hennessy Milner logic Bisimultion, Gmes & Hennessy Milner logi Leture 1 of Modelli Mtemtii dei Proessi Conorrenti Pweł Soboiński Univeristy of Southmpton, UK Bisimultion, Gmes & Hennessy Milner logi p.1/32 Clssil lnguge theory

More information

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

Unit 4. Combinational Circuits

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

More information

MATH Final Review

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

More information

Monochromatic Plane Matchings in Bicolored Point Set

Monochromatic Plane Matchings in Bicolored Point Set CCCG 2017, Ottw, Ontrio, July 26 28, 2017 Monohromti Plne Mthings in Biolore Point Set A. Krim Au-Affsh Sujoy Bhore Pz Crmi Astrt Motivte y networks interply, we stuy the prolem of omputing monohromti

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

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,,,,,,

More information

Reflection Property of a Hyperbola

Reflection Property of a Hyperbola Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the

More information

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz

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

More information

Propositional models. Historical models of computation. Application: binary addition. Boolean functions. Implementation using switches.

Propositional models. Historical models of computation. Application: binary addition. Boolean functions. Implementation using switches. Propositionl models Historil models of omputtion Steven Lindell Hverford College USA 1/22/2010 ISLA 2010 1 Strt with fixed numer of oolen vriles lled the voulry: e.g.,,. Eh oolen vrile represents proposition,

More information

TIME AND STATE IN DISTRIBUTED SYSTEMS

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

More information

Foundations of Computer Science Comp109

Foundations of Computer Science Comp109 Reding Foundtions o Computer Siene Comp09 University o Liverpool Boris Konev konev@liverpool..uk http://www.s.liv..uk/~konev/comp09 Prt. Funtion Comp09 Foundtions o Computer Siene Disrete Mthemtis nd Its

More information

NFAs continued, Closure Properties of Regular Languages

NFAs continued, Closure Properties of Regular Languages lgorithms & Models of omputtion S/EE 374, Spring 209 NFs continued, losure Properties of Regulr Lnguges Lecture 5 Tuesdy, Jnury 29, 209 Regulr Lnguges, DFs, NFs Lnguges ccepted y DFs, NFs, nd regulr expressions

More information

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:

22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of: 22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)

More information

Formal Languages and Automata

Formal Languages and Automata Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University

More information

Fingerprint idea. Assume:

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

More information

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

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

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

Linear Algebra Introduction

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 +

More information

Model Reduction of Finite State Machines by Contraction

Model Reduction of Finite State Machines by Contraction Model Reduction of Finite Stte Mchines y Contrction Alessndro Giu Dip. di Ingegneri Elettric ed Elettronic, Università di Cgliri, Pizz d Armi, 09123 Cgliri, Itly Phone: +39-070-675-5892 Fx: +39-070-675-5900

More information

Computing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt

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:

More information