The Minimum Label Spanning Tree Problem: Illustrating the Utility of Genetic Algorithms
|
|
- Ursula Maxwell
- 5 years ago
- Views:
Transcription
1 The Minimum Lel Spnning Tree Prolem: Illustrting the Utility of Genetic Algorithms Yupei Xiong, Univ. of Mrylnd Bruce Golden, Univ. of Mrylnd Edwrd Wsil, Americn Univ. Presented t BAE Systems Distinguished Speker Series, Mrch 2006
2 Outline of Lecture 10 - Minute Introduction to Grph Theory nd Complexity Introduction to the MLST Prolem A GA for the MLST Prolem Four Modified Versions of the Benchmrk Heuristic A Modified Genetic Algorithm Results nd Conclusions 2
3 Defining Trees A grph with no cycles is cyclic A tree is connected cyclic grph Some exmples of trees A spnning tree of grph G contins ll the nodes of G 3
4 Spnning Trees Grph G A spnning tree of G Another spnning tree of G 4
5 Miniml Spnning Trees A network prolem for which there is simple solution method is the selection of minimum spnning tree from n undirected network over n cities The cost of instlling communiction link etween cities i nd j is c ij = c ji 0 Ech city must e connected, directly or indirectly, to ll others, nd this is to e done t minimum totl cost Attention cn e confined to trees, ecuse if the network contins cycle, removing one link of the cycle leves the network connected nd reduces cost 5
6 A Miniml Spnning Tree Originl Network Minimum Spnning Tree 6
7 The Trveling Slesmn Prolem Imgine suurn college cmpus with 140 seprte uildings scttered over 800 cres of lnd To promote sfety, security gurd must inspect ech uilding every evening The gol is to sequence the 140 uildings so tht the totl time (trvel time plus inspection time) is minimized This is n exmple of the well-known TSP Originl prolem Possile solution 7
8 Definitions Anlysis of Algorithms Algorithm- method for solving clss of prolems on computer Optiml lgorithm verifile optiml solution Heuristic lgorithm fesile solution Performnce Mesures Numer of sic computtions / Running time Computtionl effort --- Prolem size --- Plyer one --- Plyer two 8
9 Computtionl Effort s Function of Prolem Size Computtionl effort 10,000,000 1,000,000 2 n n nlog2n n2 n3 2^n n 3 100,000 10,000 1, n 2 nlog 2 n n n Prolem size 9
10 Terminology Good vs. Bd Algorithms Reserchers hve emphsized the importnce of finding polynomil time lgorithms, y referring to ll such polynomil lgorithms s inherently good Algorithms tht re not polynomilly ounded, re leled inherently d Good Optiml Algorithms Exist for these Prolems Trnsporttion prolem Miniml spnning tree prolem Shortest pth prolem Liner progrmming 10
11 High Qulity Heuristic Algorithms Good Optiml Algorithms Don t Exist for these Prolems Trveling slesmn prolem (TSP) Minimum lel spnning tree prolem (MLST) Why Focus on Heuristic Algorithms? For the ove prolems, optiml lgorithms re not prcticl Efficient, ner optiml heuristics re needed to solve rel-world prolems The key is to find fst, high-qulity heuristic lgorithms 11
12 One More Concept from Grph Theory A disconnected grph consists of two or more connected grphs Ech of these connected sugrphs is clled component A disconnected grph with two components 12
13 Introduction The Minimum Lel Spnning Tree (MLST) Prolem Communictions network design Edges my e of different types or medi (e.g., fier optics, cle, microwve, telephone lines, etc.) Ech edge type is denoted y unique letter or color Construct spnning tree tht minimizes the numer of colors 13
14 A Smll Exmple Introduction Input 1 c e 6 e d Solution 1 6 e e 2 5 d
15 Where did we strt? Literture Review Proposed y Chng & Leu (1997) The MLST Prolem is NP-hrd Severl heuristics hd een proposed One of these, MVCA (mximum vertex covering lgorithm), ws very fst nd effective Worst-cse ounds for MVCA hd een otined 15
16 Literture Review An optiml lgorithm (using cktrck serch) hd een proposed On smll prolems, MVCA consistently otined nerly optiml solutions A description of MVCA follows 16
17 Description of MVCA 0. Input: G (V, E, L). 1. Let C { } e the set of used lels. 2. repet 3. Let H e the sugrph of G restricted to V nd edges with lels from C. 4. for ll i L C do 5. Determine the numer of connected components when inserting ll edges with lel i in H. 6. end for 7. Choose lel i with the smllest resulting numer of components nd do: C C {i}. 8. Until H is connected. 17
18 How MVCA Works Input 1 c e 6 e d 2 5 d 3 4 Intermedite Solution Solution 1 6 e e
19 Worst-Cse Results 1. Krumke, Wirth (1998): MVCA OPT 1+ 2ln n 2. Wn, Chen, Xu (2002): MVCA OPT 1+ ln ( n 1) 3. Xiong, Golden, Wsil (2005): MVCA OPT H = 1 < 1+ ln i i= 1 where = mx lel frequency, nd H = th hrmonic numer 19
20 Some Oservtions The Xiong, Golden, Wsil worst-cse ound is tight Unlike the MST, where we focus on the edges, here it mkes sense to focus on the lels or colors Next, we present genetic lgorithm (GA) for the MLST prolem 20
21 Genetic Algorithm: Overview Rndomly choose p solutions to serve s the initil popultion Suppose s [0], s [1],, s [p 1] re the individuls (solutions) in genertion 0 Build genertion k from genertion k 1 s elow For ech j etween 0 nd p 1, do: End For t [ j ] = crossover { s [ j ], s [ (j + k) mod p ] } t [ j ] = muttion { t [ j ] } s [ j ] = the etter solution of s [ j ] nd t [ j ] Run until genertion p 1 nd output the est solution from the finl 21 genertion
22 Crossover Schemtic (p = 4) Genertion 0 S[0] S[1] S[2] S[3] Genertion 1 S[0] S[1] S[2] S[3] Genertion 2 S[0] S[1] S[2] S[3] Genertion 3 S[0] S[1] S[2] S[3] 22
23 Crossover Given two solutions s [ 1 ] nd s [ 2 ], find the child T = crossover { s [ 1 ], s [ 2 ] } Define ech solution y its lels or colors Description of Crossover. Let S = s [ 1 ] s [ 2 ] nd T e the empty set. Sort S in decresing order of the frequency of lels in G c. Add lels of S, from the first to the lst, to T until T represents fesile solution d. Output T 23
24 An Exmple of Crossover s [ 1 ] = {,, d } s [ 2 ] = {, c, d } d d d d c c c T = { } S = {,, c, d } Ordering:,, c, d 24
25 An Exmple of Crossover T = { } T = {, } T = {,, c } c c c 25
26 Muttion Given solution S, find muttion T Description of Muttion. Rndomly select c not in S nd let T = S c. Sort T in decresing order of the frequency of the lels in G c. From the lst lel on the ove list to the first, try to remove one lel from T nd keep T s fesile solution d. Repet the ove step until no lels cn e removed e. Output T 26
27 An Exmple of Muttion S = {,, c } S = {,, c, d } d d c c c Add { d } c c c Ordering:,, c, d 27
28 An Exmple of Muttion Remove { d } S = {,, c } Remove { } S = {, c } c c c c c c T = {, c } 28
29 Three Modified Versions of MVCA Voss et l. (2005) implement MVCA using their pilot method The results were quite time-consuming We dded prmeter ( % ) to improve the results Three modified versions of MVCA MVCA1 uses % = 100 MVCA2 uses % = 10 MVCA3 uses % = 30 29
30 MVCA1 We try ech lel in L (% = 100) s the first or pilot lel Run MVCA to determine the remining lels We output the est solution of the l solutions otined For lrge l, we expect MVCA1 to e very slow 30
31 MVCA2 (nd MVCA3) We sort ll lels y their frequencies in G, from highest to lowest We select ech of the top 10% (% = 10) of the lels to serve s the pilot lel Run MVCA to determine the remining lels We output the est solution of the l/10 solutions otined MVCA2 will e fster thn MVCA1, ut not s effective MVCA3 selects the top 30% (% = 30) nd exmines 3l/10 solutions MVCA3 is compromise pproch 31
32 A Rndomized Version of MVCA (RMVCA) We follow MVCA in spirit At ech step, we consider the three most promising lels s cndidtes We select one of the three lels The est lel is selected with pro. = 0.4 The second est lel is selected with pro. = 0.3 The third est lel is selected with pro. = 0.3 We run RMVCA 50 times for ech instnce nd output the est solution 32
33 A Modified Genetic Algorithm (MGA) We modify the crossover opertion descried erlier We tke the union of the prents (i.e., S = S 1^S 2 ) s efore Next, pply MVCA to the sugrph of G with lel set S (Sf L), node set V, nd the edge set E ' (E ' f E) ssocited with S The new crossover opertion is more time-consuming thn the old one The muttion opertion remins s efore 33
34 Computtionl Results 48 comintions: n = 50 to 200 / l = 12 to 250 / density = 0.2, 0.5, smple grphs for ech comintion The verge numer of lels is compred 34
35 Performnce Comprison MVCA GA MGA MVCA1 MVCA2 MVCA3 RMVCA Row Totl MVCA GA MGA MVCA MVCA MVCA RMVCA Summry of computtionl results with respect to ccurcy for seven heuristics on 48 cses. The entry (i, j) represents the numer of cses heuristic i genertes solution tht is etter thn the solution generted y heuristic j. 35
36 Running Times MVCA GA MGA MVCA1 MVCA2 MVCA3 RMVCA n = 100, l = 125, d = n = 150, l = 150, d = n = 150, l = 150, d = n = 150, l = 187, d = n = 150, l = 187, d = n = 200, l = 100, d = n = 200, l = 200, d = n = 200, l = 200, d = n = 200, l = 200, d = n = 200, l = 250, d = n = 200, l = 250, d = n = 200, l = 250, d = Averge running time Running times for 12 demnding cses (in seconds). 36
37 One Finl Experiment for Smll Grphs 240 instnces for n = 20 to 50 re solved y the seven heuristics Bcktrck serch solves ech instnce to optimlity The seven heuristics re compred sed on how often ech otins n optiml solution Procedure OPT MVCA GA MGA MVCA1 MVCA2 MVCA3 RMVCA % optiml
38 Conclusions We presented three modified (deterministic) versions of MVCA, rndomized version of MVCA, nd modified GA All five of the modified procedures generted etter results thn MVCA nd GA, ut were more time-consuming With respect to running time nd performnce, MGA seems to e the est 38
39 Relted Work The Lel-Constrined Minimum Spnning Tree (LCMST) Prolem We show the LCMST prolem is NP-hrd We introduce two locl serch methods We present n effective genetic lgorithm We formulte the LCMST s MIP nd solve for smll cses We introduce dul prolem 39
Genetic Programming. Outline. Evolutionary Strategies. Evolutionary strategies Genetic programming Summary
Outline Genetic Progrmming Evolutionry strtegies Genetic progrmming Summry Bsed on the mteril provided y Professor Michel Negnevitsky Evolutionry Strtegies An pproch simulting nturl evolution ws proposed
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationRegular expressions, Finite Automata, transition graphs are all the same!!
CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104-W11 1
More informationConnected-components. Summary of lecture 9. Algorithms and Data Structures Disjoint sets. Example: connected components in graphs
Prm University, Mth. Deprtment Summry of lecture 9 Algorithms nd Dt Structures Disjoint sets Summry of this lecture: (CLR.1-3) Dt Structures for Disjoint sets: Union opertion Find opertion Mrco Pellegrini
More informationSurface maps into free groups
Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The
More information12.1 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 informationFirst 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 informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationFast 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 informationResources. 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 informationEvolutionary Computation
Topic 9 Evolutionry Computtion Introduction, or cn evolution e intelligent? Simultion of nturl evolution Genetic lgorithms Evolution strtegies Genetic progrmming Summry Cn evolution e intelligent? Intelligence
More information1 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 informationCS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University
CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationFirst 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 informationCS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata
CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationBayesian Networks: Approximate Inference
pproches to inference yesin Networks: pproximte Inference xct inference Vrillimintion Join tree lgorithm pproximte inference Simplify the structure of the network to mkxct inferencfficient (vritionl methods,
More information1. 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 informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationLecture 2: January 27
CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full
More informationCompiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz
University of Southern Cliforni Computer Science Deprtment Compiler Design Fll Lexicl Anlysis Smple Exercises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sciences Institute 4676 Admirlty Wy, Suite
More information19 Optimal behavior: Game theory
Intro. to Artificil Intelligence: Dle Schuurmns, Relu Ptrscu 1 19 Optiml behvior: Gme theory Adversril stte dynmics hve to ccount for worst cse Compute policy π : S A tht mximizes minimum rewrd Let S (,
More informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
More informationRandom 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
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationThe size of subsequence automaton
Theoreticl Computer Science 4 (005) 79 84 www.elsevier.com/locte/tcs Note The size of susequence utomton Zdeněk Troníček,, Ayumi Shinohr,c Deprtment of Computer Science nd Engineering, FEE CTU in Prgue,
More informationLecture 08: Feb. 08, 2019
4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationCS 188: Artificial Intelligence Spring 2007
CS 188: Artificil Intelligence Spring 2007 Lecture 3: Queue-Bsed Serch 1/23/2007 Srini Nrynn UC Berkeley Mny slides over the course dpted from Dn Klein, Sturt Russell or Andrew Moore Announcements Assignment
More informationFormal languages, automata, and theory of computation
Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm
More information5.1 How do we Measure Distance Traveled given Velocity? Student Notes
. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis
More informationCS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2014
CS125 Lecture 12 Fll 2014 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 informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should
More informationDesigning 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 informationCS 275 Automata and Formal Language Theory
CS 275 utomt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Prolem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) nton Setzer (Bsed on ook drft y J. V. Tucker nd K. Stephenson)
More informationCMSC 330: Organization of Programming Languages
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 CMSC 330 1 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic
More informationThe Quest for Perfect and Compact Symmetry Breaking for Graph Problems
The Quest for Perfect nd Compct Symmetry Breking for Grph Prolems Mrijn J.H. Heule SYNASC Septemer 25, 2016 1/19 Stisfiility (SAT) solving hs mny pplictions... forml verifiction grph theory ioinformtics
More informationModel 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 informationTypes of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More informationTypes of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More informationThe Knapsack Problem. COSC 3101A - Design and Analysis of Algorithms 9. Fractional Knapsack Problem. Fractional Knapsack Problem
The Knpsck Prolem COSC A - Design nd Anlsis of Algorithms Knpsck Prolem Huffmn Codes Introduction to Grphs Mn of these slides re tken from Monic Nicolescu, Univ. of Nevd, Reno, monic@cs.unr.edu The - knpsck
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationDesigning Information Devices and Systems I Anant Sahai, Ali Niknejad. This homework is due October 19, 2015, at Noon.
EECS 16A Designing Informtion Devices nd Systems I Fll 2015 Annt Shi, Ali Niknejd Homework 7 This homework is due Octoer 19, 2015, t Noon. 1. Circuits with cpcitors nd resistors () Find the voltges cross
More informationChapter 5 Plan-Space Planning
Lecture slides for Automted Plnning: Theory nd Prctice Chpter 5 Pln-Spce Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Stte-Spce Plnning Motivtion g 1 1 g 4 4 s 0 g 5 5 g 2
More informationDesigning Information Devices and Systems I Spring 2018 Homework 8
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission
More informationTransportation of handicapped persons
Trnsporttion of hndicpped s Andres Reinholz Mrí Soto Mrc Sevux André Rossi DLR Institute of Solr Reserch Cologne GERMANY Lb-STICC Université de Bretgne-Sud Lorient FRANCE LOT September 2014 1/25 A collbortion
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationC Dutch System Version as agreed by the 83rd FIDE Congress in Istanbul 2012
04.3.1. Dutch System Version s greed y the 83rd FIDE Congress in Istnul 2012 A Introductory Remrks nd Definitions A.1 Initil rnking list A.2 Order See 04.2.B (Generl Hndling Rules - Initil order) For pirings
More informationSection 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 informationAssignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages
Deprtment of Computer Science, Austrlin Ntionl University COMP2600 Forml Methods for Softwre Engineering Semester 2, 206 Assignment Automt, Lnguges, nd Computility Smple Solutions Finite Stte Automt nd
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: 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
More informationCSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science
CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSignal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:
3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: 3 4 5 where n is the scttering mtri of ech device, nd is the overll scttering
More informationPreview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms
Preview Greed Algorithms Greed Algorithms Coin Chnge Huffmn Code Greed lgorithms end to e simple nd strightforwrd. Are often used to solve optimiztion prolems. Alws mke the choice tht looks est t the moment,
More informationHomework 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 informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More informationChapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis
Chpter 4: Techniques of Circuit Anlysis Terminology Node-Voltge Method Introduction Dependent Sources Specil Cses Mesh-Current Method Introduction Dependent Sources Specil Cses Comprison of Methods Source
More informationPhysics 1402: Lecture 7 Today s Agenda
1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:
More informationexpression 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 informationClassification: Rules. Prof. Pier Luca Lanzi Laurea in Ingegneria Informatica Politecnico di Milano Polo regionale di Como
Metodologie per Sistemi Intelligenti Clssifiction: Prof. Pier Luc Lnzi Lure in Ingegneri Informtic Politecnico di Milno Polo regionle di Como Rules Lecture outline Why rules? Wht re clssifiction rules?
More informationCHAPTER 1 Regular Languages. Contents
Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr
More informationBases 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 informationHamiltonian Cycle in Complete Multipartite Graphs
Annls of Pure nd Applied Mthemtics Vol 13, No 2, 2017, 223-228 ISSN: 2279-087X (P), 2279-0888(online) Pulished on 18 April 2017 wwwreserchmthsciorg DOI: http://dxdoiorg/1022457/pmv13n28 Annls of Hmiltonin
More information1 From NFA to regular expression
Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work
More informationFinite Automata-cont d
Automt Theory nd Forml Lnguges Professor Leslie Lnder Lecture # 6 Finite Automt-cont d The Pumping Lemm WEB SITE: http://ingwe.inghmton.edu/ ~lnder/cs573.html Septemer 18, 2000 Exmple 1 Consider L = {ww
More informationImproper 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 informationCMPSCI 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 informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More information1 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 informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS
The University of Nottinghm SCHOOL OF COMPUTER SCIENCE LEVEL 2 MODULE, SPRING SEMESTER 2016 2017 LNGUGES ND COMPUTTION NSWERS Time llowed TWO hours Cndidtes my complete the front cover of their nswer ook
More informationCS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018
CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA
More informationBalanced binary search trees
02110 Inge Li Gørtz Overview Blnced binry serch trees: Red-blck trees nd 2-3-4 trees Amortized nlysis Dynmic progrmming Network flows String mtching String indexing Computtionl geometry Introduction to
More informationDesigning Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.
EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work
More informationChapter 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 information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationLexical Analysis Finite Automate
Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition
More information1. 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 informationLecture 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 informationSatellite Retrieval Data Assimilation
tellite etrievl Dt Assimiltion odgers C. D. Inverse Methods for Atmospheric ounding: Theor nd Prctice World cientific Pu. Co. Hckensck N.J. 2000 Chpter 3 nd Chpter 8 Dve uhl Artist depiction of NAA terr
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationFABER Formal Languages, Automata and Models of Computation
DVA337 FABER Forml Lnguges, Automt nd Models of Computtion Lecture 5 chool of Innovtion, Design nd Engineering Mälrdlen University 2015 1 Recp of lecture 4 y definition suset construction DFA NFA stte
More informationCS 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 information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationEE273 Lecture 15 Asynchronous Design November 16, Today s Assignment
EE273 Lecture 15 Asynchronous Design Novemer 16, 199 Willim J. Dlly Computer Systems Lortory Stnford University illd@csl.stnford.edu 1 Tody s Assignment Term Project see project updte hndout on we checkpoint
More informationKleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15
Models of Comput:on Lecture #8 Chpter 7 con:nued Any lnguge tht e defined y regulr expression, finite utomton, or trnsi:on grph cn e defined y ll three methods We prove this y showing tht ny lnguge defined
More informationCounting 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 informationConverting Regular Expressions to Discrete Finite Automata: A Tutorial
Converting Regulr Expressions to Discrete Finite Automt: A Tutoril Dvid Christinsen 2013-01-03 This is tutoril on how to convert regulr expressions to nondeterministic finite utomt (NFA) nd how to convert
More informationFree groups, Lecture 2, part 1
Free groups, Lecture 2, prt 1 Olg Khrlmpovich NYC, Sep. 2 1 / 22 Theorem Every sugroup H F of free group F is free. Given finite numer of genertors of H we cn compute its sis. 2 / 22 Schreir s grph The
More informationAlignment of Long Sequences. BMI/CS Spring 2016 Anthony Gitter
Alignment of Long Sequences BMI/CS 776 www.biostt.wisc.edu/bmi776/ Spring 2016 Anthony Gitter gitter@biostt.wisc.edu Gols for Lecture Key concepts how lrge-scle lignment differs from the simple cse the
More informationA Bounded Incremental Test Generation Algorithm for Finite State Machines
A Bounded Incrementl Test Genertion Algorithm for Finite Stte Mchines Zoltán Pp 1, Mhdevn Surmnim 2, Gáor Kovács 3, Gáor Árpád Németh3 1 Ericsson Telecomm. Hungry, H-1117 Budpest, Irinyi J. u. 4-20, Hungry
More information