CS 360 Exam 2 Fall 2014 Name

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "CS 360 Exam 2 Fall 2014 Name"

Transcription

1 CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singly-linke list n stk. Write three ifferent O(n)-time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output the t ontine within the noes of the list in reverse orer, tht is, from k to front. Exmple: if the list ontins elements [4, 9, 1, 6] then the output shoul e lss Noe { int t; Noe next; } lss List { Noe he; voi reverse_print( ); } lss Stk { Stk( ); oolen isempty( ); voi push (int x); int pop( ); }. Use stk s n uxiliry t struture. Do not use ny other t struture, o not use reursion, n o not moify the linke list. [5 points]. Use reursion. Do not use n expliit stk or ny other t struture, n o not moify the linke list. Hint: use helper funtion tht is reursive. [5 points]

2 . Do not use stk or ny other t struture, n o not use reursion. You my use only O(1) extr spe eyon the linke list. Your metho my temporrily moify the linke list, ut when the metho ens, the linke list must e ore extly the sme s it ws efore the metho ws lle. Hint: reverse the list in ple. [10 points] 2. Drw one inry tree tht yiels ll three of the trversls s shown elow. [8 points] Pre-orer: Post-orer: In-orer: L G Z J T X C M P R A N Y E B H V T J M C P X Z G Y B E N V H A R L G T J Z C M X P L Y N B E A H V R

3 3. A stnr Queue implementtion is provie elow. Complete the lss MultipleQueues, whih represents ritrrily mny queues, y using one instne Q of lss Queue s shown. In lss MultipleQueues, istint queues re hieve y speifying istint qnum vlues. The expete ehvior of methos enqueue n equeue is illustrte in n exmple elow. The enqueue metho must tke O(1) time, ut equeue my tke ω(1) time. [15 points] lss Queue { Queue( ); int size ( ); voi enqueue (int z); int equeue( ); } lss MultipleQueues { Queue Q; MultipleQueues( ) { Q = new Queue( ); } voi enqueue (int qnum, int z); // write this metho int equeue (int qnum); // write this metho } MultipleQueues M = new MultipleQueues( ); M.enqueue (7, 30); M.enqueue (2, 40); M.enqueue (7, 50); M.enqueue (2, 60); print (M.equeue (2)); print (M.equeue (2)); // outputs: print (M.equeue (7)); print (M.equeue (7)); // outputs: 30 50

4 4. Answer the following questions out running times of list opertions. [25 points]. Stte the worst-se θ running time for n effiient implementtion of eh list opertion shown elow, when using eh speifie t struture. (x is vlue, n p is position.) Fin (x) After (p) Before (p) Insert (x) InsertAfter (p,x) InsertBefore (p,x) InsertFirst (x) InsertLst (x) Remove (x) Remove (p) RemoveFirst ( ) RemoveLst ( ) Unsorte singly-linke list, with link to he ut no link to til noe Unsorte ouly-linke list, with links to oth he n til noes Unsorte non-irulr rry, where rry inex k yiels the rnk k element. For whih opertion(s) oes the θ running time hnge if we use singly-linke list with links to oth he n til noes, rther thn singly-linke list with link to only the he noe? Also stte eh suh opertions running time for singly-linke list with links to oth he n til noes.. For whih opertion(s) oes the θ running time hnge if we use irulr rry rther thn non-irulr rry? Also stte eh suh opertions running time for irulr rry.. Whih opertion(s) shoul not e provie if the elements in the t struture must e mintine in sorte orer? e. For whih opertion(s) oes the θ running time hnge if we use sorte rry rther thn n unsorte rry? Also stte eh suh opertions running time for sorte rry.

5 5. Consier hsh tle with N slots, where N 5. [6 points]. Use liner proing n insert 2 rnom keys. Determine the proility tht these 2 keys will oupy 2 jent slots (possily with wrproun).. Use liner proing n insert 3 rnom keys. Determine the proility tht these 3 keys will oupy 3 jent slots (possily with wrproun).. Use seprte hining n insert 3 rnom keys. Determine the proility tht ll 3 keys re ple in the sme hin.. Use seprte hining n insert 3 rnom keys. Determine the proility tht ll 3 keys re ple in ifferent hins. e. Use seprte hining n insert 3 rnom keys. Determine the proility tht extly 2 of these 3 keys re ple in the sme hin.

6 6. Strt with the -lk tree shown elow whih hs keys 3, 4, 5. First insert keys 6, 7, 8, 9, 10, 11, 12 (in tht orer) n rw the tree fter eh insert. Next remove keys 3, 4, 5, 6, 7 (in tht orer) n rw the tree fter eh remove. Clerly inite the olor of eh noe (=lk, r=). Mke sure tht eh tree you rw is properly forme -lk tree. If you rw ny intermeite steps, irle the trees tht shoul e gre. [12 points] 4 3 r 5 r

7 7. Strt with the AVL tree shown elow whih hs keys 5, 10. First insert keys 6, 7, 8, 9, 0, 4, 3, 2, 1 (in tht orer) n rw the tree fter eh insert. Next remove keys 3, 10, 9 (in tht orer) n rw the tree fter eh remove. It is permitte ut not requi to show the height of eh noe. Mke sure tht eh tree you rw is properly forme AVL tree. If you rw ny intermeite steps, irle the trees tht shoul e gre. [12 points] 5 10

8 8. Strt with the sply tree shown elow whih hs key 5. First insert keys 6, 3, 4, 1, 2, 9, 7, 8 (in tht orer) n rw the tree fter eh insert. Next remove keys 1, 9, 6, 7 (in tht orer) n rw the tree fter eh remove. Mke sure tht eh tree you rw is properly forme sply tree. If you rw ny intermeite steps, irle the trees tht shoul e gre. [12 points] 5

9 Insertion into -lk tree Getting strte: Rehing the root: 0. x 3. root root lef lef lef Restruturing (single or oule rottion): Reoloring:

10 Removl from -lk tree Getting strte: 0. x lef 0. x lef efiient lef lef lef lef 0. x w 0. x y w lef lef lef lef y lef lef lef lef lef lef Restruturing (single or oule rottion): 1. ny sme ny 1. sme efiient efiient 1. ny sme 1. ny sme e efiient e e efiient e `

11 Removl from -lk tree (ontinue) Reoloring: efiient efiient 2. efiient 2. efiient efiient efiient Ajustment (single rottion): efiient efiient efiient efiient [Alwys followe y 1, 1, 1, 1, 2, or 2] Rehing the root: 4. root efiient root

12 AVL tree: insertion n removl 1. Y, h+3 1. U, h+3 Right rotte t Y U, h+2 Z, h T, h Y, h+2 Left rotte t U T, h+1 W W Z, h+1 2. Y, h+3 Left rotte t U, 2. U, h+3 Right rotte t Y, U, h+2 Z, h h Right rotte t Y T, h Y, h+2 th Left rotte t U T, h W, h+1 W, h+1 Z, h Sply tree: insertion n removl 1. Zig root X P Right rotte t P 1. Zg root P X Left rotte t P 2. G P Right rotte t G, 2. Zigzig Zgzg G P Left rotte t G, then Left rotte t P X Right rotte then t P X 3. P G X Left rotte t P, Right th rotte t G 3. Zigzg Zgzig G X P Right rotte t P, th Left rotte t G

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

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

Solutions to Problem Set #1

Solutions to Problem Set #1 CSE 233 Spring, 2016 Solutions to Prolem Set #1 1. The movie tse onsists of the following two reltions movie: title, iretor, tor sheule: theter, title The first reltion provies titles, iretors, n tors

More information

Pre-Lie algebras, rooted trees and related algebraic structures

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

More information

Automata and Regular Languages

Automata and Regular Languages Chpter 9 Automt n Regulr Lnguges 9. Introution This hpter looks t mthemtil moels of omputtion n lnguges tht esrie them. The moel-lnguge reltionship hs multiple levels. We shll explore the simplest level,

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

Nondeterministic Finite Automata

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)

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.) CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

Exam 2 Solutions ECE 221 Electric Circuits

Exam 2 Solutions ECE 221 Electric Circuits Nme: PSU Student ID Numer: Exm 2 Solutions ECE 221 Electric Circuits Novemer 12, 2008 Dr. Jmes McNmes Keep your exm flt during the entire exm If you hve to leve the exm temporrily, close the exm nd leve

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

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints) C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

More information

Solutions to Assignment 1

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

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chapter Five - Eigenvalues, Eigenfunctions, and All That Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

More information

On-Line Construction. of Suffix Trees. Overview. Suffix Trees. Notations. goo. Suffix tries

On-Line Construction. of Suffix Trees. Overview. Suffix Trees. Notations. goo. Suffix tries On-Line Cnstrutin Overview Suffix tries f Suffix Trees E. Ukknen On-line nstrutin f suffix tries in qudrti time Suffix trees On-line nstrutin f suffix trees in liner time Applitins 1 2 Suffix Trees A suffix

More information

Lecture 11 Binary Decision Diagrams (BDDs)

Lecture 11 Binary Decision Diagrams (BDDs) C 474A/57A Computer-Aie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, K-mp, iruit, et

More information

APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS

APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex

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

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

Nondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA

Nondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn

More information

expression simply by forming an OR of the ANDs of all input variables for which the output is

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

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

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorin Certificte of Euction 07 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written emintion Thursy 8 June 07 Reing time:.00 pm to.5 pm (5 minutes) Writing

More information

COMP 250. Lecture 20. tree traversal. Oct. 25/26, 2017

COMP 250. Lecture 20. tree traversal. Oct. 25/26, 2017 COMP 250 Lecture 20 tree trversl Oct. 25/26, 2017 1 2 Tree Trversl How to visit (enumerte, iterte through, trverse ) ll the nodes of tree? 3 depthfirst (root){ // preorder if (root is not empty){ visit

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

More information

Matrix & Vector Basic Linear Algebra & Calculus

Matrix & Vector Basic Linear Algebra & Calculus Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS 3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

More information

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

More information

MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM. May 30, 2007 MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

More information

2. Properties of Functions

2. Properties of Functions 2. PROPERTIES OF FUNCTIONS 111 2. Properties of Funtions 2.1. Injetions, Surjetions, an Bijetions. Definition 2.1.1. Given f : A B 1. f is one-to-one (short han is 1 1) or injetive if preimages are unique.

More information

Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1

Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1 Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016 CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple

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

Lecture 9: LTL and Büchi Automata

Lecture 9: LTL and Büchi Automata Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

Data Compression Techniques (Spring 2012) Model Solutions for Exercise 4

Data Compression Techniques (Spring 2012) Model Solutions for Exercise 4 58487 Dt Compressio Tehiques (Sprig 0) Moel Solutios for Exerise 4 If you hve y fee or orretios, plese ott jro.lo t s.helsii.fi.. Prolem: Let T = Σ = {,,, }. Eoe T usig ptive Huffm oig. Solutio: R 4 U

More information

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous

More information

Mining Frequent Web Access Patterns with Partial Enumeration

Mining Frequent Web Access Patterns with Partial Enumeration Mining Frequent We Aess Ptterns with Prtil Enumertion Peiyi Tng Deprtment of Computer Siene University of Arknss t Little Rok 2801 S. University Ave. Little Rok, AR 72204 Mrkus P. Turki Deprtment of Computer

More information

C/CS/Phys C191 Bell Inequalities, No Cloning, Teleportation 9/13/07 Fall 2007 Lecture 6

C/CS/Phys C191 Bell Inequalities, No Cloning, Teleportation 9/13/07 Fall 2007 Lecture 6 C/CS/Phys C9 Bell Inequlities, o Cloning, Teleporttion 9/3/7 Fll 7 Lecture 6 Redings Benenti, Csti, nd Strini: o Cloning Ch.4. Teleporttion Ch. 4.5 Bell inequlities See lecture notes from H. Muchi, Cltech,

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

Functions. mjarrar Watch this lecture and download the slides

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

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

What else can you do?

What else can you do? Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

More information

INTRODUCTION TO AUTOMATA THEORY

INTRODUCTION TO AUTOMATA THEORY Chpter 3 INTRODUCTION TO AUTOMATA THEORY In this hpter we stuy the most si strt moel of omputtion. This moel els with mhines tht hve finite memory pity. Setion 3. els with mhines tht operte eterministilly

More information

3 Regular expressions

3 Regular expressions 3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

Semantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4

Semantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4 Semnti nlysis SI 16 Priniples of Progrmming Lnguges Fulty of omputer Siene Dlhousie University Winter 2012 Reding: hpter 4 Motivtion Soure progrm (hrter strem) Snner (lexil nlysis) Front end Prse tree

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

Solutions to assignment 3

Solutions to assignment 3 D Sruure n Algorihm FR 6. Informik Sner, Telikeplli WS 03/04 hp://www.mpi-.mpg.e/~ner/oure/lg03/inex.hml Soluion o ignmen 3 Exerie Arirge i he ue of irepnie in urreny exhnge re o rnform one uni of urreny

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

Trigonometry. Trigonometry. Curriculum Ready ACMMG: 223, 224, 245.

Trigonometry. Trigonometry. Curriculum Ready ACMMG: 223, 224, 245. Trgonometry Trgonometry Currulum Rey ACMMG: 223, 22, 2 www.mthlets.om Trgonometry TRIGONOMETRY Bslly, mny stutons n the rel worl n e relte to rght ngle trngle. Trgonometry souns ffult, ut t s relly just

More information

Math 211A Homework. Edward Burkard. = tan (2x + z)

Math 211A Homework. Edward Burkard. = tan (2x + z) Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

More information

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages 5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

More information

Introduction to Graphical Models

Introduction to Graphical Models Introution to Grhil Moels Kenji Fukumizu The Institute of Sttistil Mthemtis Comuttionl Methoology in Sttistil Inferene II Introution n Review 2 Grhil Moels Rough Sketh Grhil moels Grh: G V E V: the set

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

4. Approximation of continuous time systems with discrete time systems

4. Approximation of continuous time systems with discrete time systems 4. Approximtion of continuous time systems with iscrete time systems A. heory (it helps if you re the course too) he continuous-time systems re replce by iscrete-time systems even for the processing of

More information

Foundation of Diagnosis and Predictability in Probabilistic Systems

Foundation of Diagnosis and Predictability in Probabilistic Systems Foundtion of Dignosis nd Preditility in Proilisti Systems Nthlie Bertrnd 1, Serge Hddd 2, Engel Lefuheux 1,2 1 Inri Rennes, Frne 2 LSV, ENS Chn & CNRS & Inri Sly, Frne De. 16th FSTTCS 14 Dignosis of disrete

More information

Tries and suffixes trees

Tries and suffixes trees Trie: A dt-structure for set of words Tries nd suffixes trees Alon Efrt Comuter Science Dertment University of Arizon All words over the lhet Σ={,,..z}. In the slides, let sy tht the lhet is only {,,c,d}

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata

5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata CSC4510 AUTOMATA 5.1 Definitions nd Exmples 5.2 Deterministic Pushdown Automt Definitions nd Exmples A lnguge cn be generted by CFG if nd only if it cn be ccepted by pushdown utomton. A pushdown utomton

More information

4.1. Probability Density Functions

4.1. Probability Density Functions STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

More information

B Veitch. Calculus I Study Guide

B Veitch. Calculus I Study Guide Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

More information

Where did dynamic programming come from?

Where did dynamic programming come from? Where did dynmic progrmming come from? String lgorithms Dvid Kuchk cs302 Spring 2012 Richrd ellmn On the irth of Dynmic Progrmming Sturt Dreyfus http://www.eng.tu.c.il/~mi/cd/ or50/1526-5463-2002-50-01-0048.pdf

More information

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

CSCI FOUNDATIONS OF COMPUTER SCIENCE

CSCI FOUNDATIONS OF COMPUTER SCIENCE 1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not

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

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:

More information

Automatic Synthesis of New Behaviors from a Library of Available Behaviors

Automatic Synthesis of New Behaviors from a Library of Available Behaviors Automti Synthesis of New Behviors from Lirry of Aville Behviors Giuseppe De Giomo Università di Rom L Spienz, Rom, Itly degiomo@dis.unirom1.it Sestin Srdin RMIT University, Melourne, Austrli ssrdin@s.rmit.edu.u

More information

The Riemann-Stieltjes Integral

The Riemann-Stieltjes Integral Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus

Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus Forml Query Lnguges: Reltionl Alger nd Reltionl Clculus Chpter 4 Lerning Gols Given dtse ( set of tles ) you will e le to express dtse query in Reltionl Alger (RA), involving the sic opertors (selection,

More information

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

More information

(Lec 9) Multi-Level Min III: Role of Don t Cares

(Lec 9) Multi-Level Min III: Role of Don t Cares Pge 1 (Le 9) Multi-Level Min III: Role o Don t Cres Wht you know 2-level minimiztion l ESPRESSO Multi-level minimiztion: Boolen network moel, Algeri moel or toring Retngle overing or extrtion Wht you on

More information

State Minimization for DFAs

State Minimization for DFAs Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

More information

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the

More information

Automata and Languages

Automata and Languages Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

1 Online Learning and Regret Minimization

1 Online Learning and Regret Minimization 2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants. Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

More information

interatomic distance

interatomic distance Dissocition energy of Iodine molecule using constnt devition spectrometer Tbish Qureshi September 2003 Aim: To verify the Hrtmnn Dispersion Formul nd to determine the dissocition energy of I 2 molecule

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( )

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( ) 0-9-0 he First Lw of hermoynmis Effiieny When severl lterntive proesses involving het n work re ville to hnge system from n initil stte hrterize y given vlues of the mrosopi prmeters (pressure p i, temperture

More information

Arc Consistency during Search

Arc Consistency during Search Ar Consisten uring Serh Chvlit Likitvivtnvong Shool of Computing Ntionl Universit of Singpore Yunlin Zhng Sott Shnnon Dept of Computer Siene Tes Teh Universit, USA Jmes Bowen Eugene C Freuer Cork Constrint

More information

CSC2542 Representations for Classical Planning

CSC2542 Representations for Classical Planning 1 Aknowlegements CSC2542 Representtions for Clssil Plnning Sheil MIlrith Deprtment of Computer Siene University of Toronto Winter 2009 Some the slies use in this ourse re moifitions of Dn Nu s leture slies

More information