A Parallel DFA Minimization Algorithm
|
|
- Felicity Sutton
- 5 years ago
- Views:
Transcription
1 A Parallel DFA Miimizatio Algorithm Ambuj Tewari, Utkarsh Srivastava, ad P. Gupta Departmet of Computer Sciece & Egieerig Idia Istitute of Techology Kapur Kapur ,INDIA Abstract. I this paper,we have cosidered the state miimizatio problem for Determiistic Fiite Automata (DFA). A efficiet parallel algorithm for solvig the problem o a arbitrary CRCW PRAM has bee proposed. For umber of states ad k umber of iputs i Σ of the DFA to be miimized,the algorithm rus i O(k ) time ad uses O( ) processors. 1 Itroductio The problem of miimizig a give DFA (Determiistic F iite Automata) has a log history datig back to the begiigs of automata theory. Cosider a determiistic fiite automato M as a tuple (Q, Σ, q 0,F,δ)whereQ, Σ, q 0 Q, F Q ad δ : Q Σ Q are a fiite set of states, a fiite iput alphabet, the start state, the set of acceptig (or fial) states ad the trasitio fuctio, respectively. A iput strig x is a sequece of symbols over Σ. O a iput strig x = x 1 x 2...x m, the DFA visits a sequece of states q 0 q 1...q m startig with the start state by successively applyig the trasitio fuctio δ. Thus q i+1 = δ(q i,x i+1 )for0 i m 1. The laguage L(M) accepted by a DFA is defied as the set of strigs x that takes the DFA to a acceptig state, i.e. x is i L(M) if ad oly if q m F. Two DFAs are said to be equivalet if they accept the same set of strigs. The problem of DFA miimizatio is to fid a DFA with the miimum umber of states which is equivalet to the give DFA. A fudametal result i automata theory states that such a miimal DFA is uique up to reamig of states. The umber of states i the miimal DFA is give by the umber of equivalece classes i the partitio o the set of all strigs i Σ defied as follows: Two strigs x ad y are equivalet if ad oly if for all strigs z, xz L(M) if ad oly if yz L(M) [6]. Besides beig widely studied, DFA has may applicatios i the diverse fields like patter matchig, optimizatio of logic programs, protocol verificatio ad specificatio ad modelig of fiite state systems [14]. It is kow that odetermiistic fiite automata (NFA) are equivalet to determiistic oes as far as the laguages recogized by them are cocered. Huffma [4] admoore[10] have preseted O( 2 ) algorithms for DFA miimizatio ad are sufficietly fast for most of the classical applicatios. However, there exist umerous algorithms S. Sahi et al. (Eds.) HiPC 2002, LNCS 2552, pp , c Spriger-Verlag Berli Heidelberg 2002
2 A Parallel DFA Miimizatio Algorithm 35 which are variatios of the same basic idea. A efficiet O( ) algorithm is due to Hopcroft [5]. Blum [2] has also proposed a simpler algorithm with same time complexity. Traditioal applicatios of the DFA miimizatio algorithm ivolve a few thousad states ad the sequetial algorithms available geerally perform well i these settigs. But if the umber of states i a DFA is of the order of a few millios, the the efficiet sequetial algorithms may take a sigificat amout of time ad may eed much more tha the available physical memory. Oe way to achieve the speed-up is the use of multiple processors. DFA miimizatio has bee extesively studied o may parallel computatio models. Jaja ad Kosaraju [7] have preseted a efficiet algorithm o mesh coected computer for the case whe Σ = 1. A simple NC algorithm has also bee outlied i their work. Cho ad Huyh [3] have proved the problem to be NLOGSPACE-complete. Efficiet ad close to cost-optimal algorithms are kow oly for the case of the alphabet cosistig of a sigle iput symbol. A very simple algorithm is proposed by Srikat [13] but the best algorithm for this problem is due to Jaja ad Ryu [8]. It is a CRCW algorithm with time complexity O() ad cost O( log ). Ufortuately, o efficiet algorithm, which is also ecoomical with respect to cost, is kow for the geeral case of multiple iput symbols. The stadard NC algorithm for this problem requires O( 6 ) processors. This is due the use of trasitive closure computatio o a cross product graph. I [12], a simple parallel algorithm for DFA miimizatio alog with its implemetatio is preseted. I this paper we have proposed a efficiet parallel algorithm for DFA miimizatio havig O(klog) time complexity o a arbitrary CRCW PRAM with ( ) processors. The rest of the paper is orgaized as follows. Sectio 2 cotais some prelimiaries alog with the aive sequetial algorithm. I Sectio 3, log we discuss a fast parallel algorithm for the problem which uses a large umber of processors. A efficiet parallel algorithm has bee proposed i Sectio 4. Coclusio is give i the last sectio. 2 Review The DFA miimizatio problem is closely related to the coarsest partitioig problem which ca be stated as follows. We are give a set Q ad its iitial partitio ito m disjoit sets {B 0,...,B m 1 }, ad a collectio of fuctios, f i : Q Q. We have to fid a coarsest partitio of Q, say{e 1,...,E q },such that: (1) each E i is a subset of some B j, ad (2) the partitio respects the give fuctios, i.e. j, ifa ad b both belog to the same E i the f j (a) adf j (b) also belog to the same E k for some k. Assume, Q is the set of states, f i is the restrictio of the trasitio fuctio δ to the ith iput symbol, i.e. f i (q) =δ(q, a i ), the iitial partitio cotais two sets, amely F ad Q F. The size of the miimal DFA is the umber of equivalece classes i the coarsest partitio. For the geeral case of multiple fuctio coarsest partitio problem, a O( ) solutio is give i [1]. Later, a liear
3 36 Ambuj Tewari et al. Sequetial Algorithm 1. for all fial states q i do block o[q i]=1 2. for all o-fial states q i do block o[q i]=2 3. do 4. for i= 0to k-1 do 5. for j=0to -1 do 6. b 1 =blocko[q j] 7. b 2 =blocko[δ(q j,x i)] 8. label state q j with (b 1,b 2) 9. edfor 10. Assig same block umber to states havig same labels 11. edfor 12. while <umber of blocks is chagig> Algorithm 1: The sequetial algorithm for DFA miimizatio time solutio has bee proposed i [11] for the case of the sigle fuctio coarsest partitio problem. The sigle fuctio versio of the problem correspods to havig oly a sigle symbol i the iput alphabet Σ. But the simplest sequetial algorithm for solvig the problem, give i Algorithm 1, rus i O(k 2 ) time, where Σ = k [1]. It performs as follows. Iitially there are oly two blocks: oe cotaiig all the fial states ad the other cotaiig all the o-fial states. If two states q ad q are foud i the same block such that for some iput symbol a i,thestatesδ(q, a i )adδ(q,a i )arei differet blocks, q ad q are placed i differet blocks for the ext iteratio. The algorithm is iterated for at most times because, i the worst case, each state will be i a block cotaiig just itself. I each iteratio ew block umbers are assiged i O(k) time. Therefore, the total time take is O(k 2 ). 3 A Fast Parallel Algorithm The fastest kow algorithm for the multiple iput symbol case is due to Cho ad Huyh [3]. The DFA miimizatio problem is iitially traslated to a istace of the multiple fuctio coarsest partitio problem to yield the set S of states, the iitial partitio B cotaiig sets of fial ad o-fial states ad the fuctios f i which are restrictios of the trasitio fuctio to sigle iput symbols. A graph G =< V,E>may be geerated as follows: V = {(a, b) a, b S} ad E = {((a, b), (c, d)) c = f i (a), d = f i (b) forsomei}. For ay pair x, y S, x ad y get differet labels i the coarsest partitio if ad oly if there is a path from ode (x, y) tosomeode(a, b) such that a ad b have differet B-labels. The algorithm is give below. It ca be show that the algorithm ca be implemeted o a EREW i time O(log 2 ) with total cost O( 6 ). The boud O( 6 ) arises because of the trasitive closure computatio of agraphwith 2 odes. So far, it has ot bee possible to fid a algorithm with a reasoable cost, say O( 2 ), ad a small ruig time.
4 A Parallel DFA Miimizatio Algorithm 37 Parallel Algorithm 1. Costruct graph G as defied above 2. Mark all odes (p, q) p ad q belog to differet sets of the B-partitio 3. Uses trasitive closure to mark pairs reachable from marked pairs 4. commet Note that all umarked pairs are equivalet Algorithm 2: A fast parallel algorithm for DFA miimizatio 4 A Efficiet Algorithm I this sectio we have proposed a parallel versio of the simple O( 2 )time sequetial algorithm outlied i Sectio 3. We use O( ) processors to achieve a expected ruig time of O(). Usig a arbitrary-crcw PRAM, we have parallelized the ier for-loop which iterates over the set of states. The the ew labels obtaied i this for-loop are hashed usig parallel hashig algorithm of Matias ad Vishki [9] to get ew block umbers for the states. The algorithm is give below. Lies 1-8 are the same as i the sequetial algorithm except that Lies 5-8 of Algorithm 3 are ow doe i parallel. The labels obtaied i Lie 8 are hashed to [1..O()] usig the hashig techique due to Matias ad Vishki [9]. Theorem 1 (Parallel Hashig Theorem). LetWbeamultisetof umbers from the rage [1..m], where m +1 = p is a prime. Suppose we have New Parallel Algorithm 1. Iitialize block o array 2. do 3. for i =0to k-1 do 4. for j =1to do i parallel 5. for m =(j 1) to j 1 do 6. b 1 =blocko[q m] 7. b 2 =blocko[δ(q m,a i)] 8. label state q m with (b 1,b 2) 9. edfor 10. edfor 11. Use parallel hashig to map the labels to [1..O()] 12. aumbertowhichastate slabelgetsmappedto its ew block o 13. Reduce the rage of block o from O() to 14. edfor 15. while umber of blocks is chagig Algorithm 3: New parallel algorithm for DFA miimizatio
5 38 Ambuj Tewari et al. processors o a arbitrary-crcw PRAM. A oe-to-oe fuctio F : W [1..O()] ca be foud i O() expected time. The evaluatio of F (x) for each x W, takes O(1) arithmetic operatios (usig umbers from [1..m]). We have to assig ew block umbers to states such that states with differet labels get differet block umbers ad states with same label get same block umber. Also we do ot wat the block umbers to become too large. The labels are pairs of the form (b 1,b 2 )whereb 1,b 2. Therefore we ca map a label (b 1,b 2 )tob 1 ( +1)+b 2 ad we ca treat these labels as umbers i the rage [ ]. A umber m such that 2 2 <m 4 2 ad m +1 is a prime ca be foud i O() time(see[9]) ad this has to be doe just oce. For istace, it ca be doe after the iitializatio phase of step 1. We wat the block umbers to remai i the rage [1..]. However, after hashig, we get umbers i the rage [1..K] somefixedk. This rage shrikig ca be implemeted o a arbitrary-crcw PRAM i time O(). The procedure is give as Algorithm 4. First, each processor hashes labels ad sets PRESENT[x] to1ifsome label got hashed to the value x, wherepresent[1..k] is a array. Several processors might try to write i the same locatio i the array but sice we have assumed a arbitrary-crcw PRAM, oe of them will succeed arbitrarily. Each of the processors ow cosiders a rage of K idices of the PRESENT array ad computes the umber of locatios which are set to 1 usig O(log( )) prefix sum algorithm. New block umber for a give locatio is simply the umber of 1 s occurrig before that locatio. A processor ca easily compute the ew block umber for a locatio i its rage by addig the umber of 1 s occurrig before that locatio withi the processor s rage to the umber of 1 s occurrig i earlier rages (this has already bee computed by prefix-sum). To fid the time complexity, let us cosider first Algorithm 4. From the parallel hashig theorem we kow that evaluatio of the hashig fuctio i lie 3 takes O(1) time. Lie 4 is a assigmet ad so the loop i Lies 2-5 takes O() time. Similarly the loop i Lies 9-11 takes O(). Prefix sum computatio i Lies also takes O() time. The loop i Lies takes O() time. Evaluatio of the hash fuctio at lie 25 takes O(1) time. Therefore, the last loop (Lies 24-27) too takes O() time. The outermost for-loop (Lies 3-14) of Algorithm 3 rus exactly k times where k is the size of the iput alphabet ad the outer do-while-loop (Lies 2-15) of our algorithm ca ru for at most times sice the miimal DFA does othavemoretha states. Therefore, the expected time complexity of our algorithm is O(k ). Sice we use O( ) processors, the cost is O(k2 ) which is the cost optimal parallel adaptatio of the O(k 2 ) sequetial method. 5Coclusio I this paper, we have cosidered a well-kow problem from classical automata theory ad have preseted a parallel algorithm for the problem. We have essetially adapted the aive O(k 2 ) sequetial algorithm ad have show that our
6 A Parallel DFA Miimizatio Algorithm 39 algorithm requires O( ) timeusigo( ) processors. Thus it is a cost optimal parallelizatio o a arbitrary-crcw PRAM. Fially, it will be of immese theoretical ad practical importace to come up with impossibility results about the limited parallelizability of the sequetial DFA miimizatio algorithms. // Iitialize the PRESENT[1..K] array 1. for i =1to do i parallel 2. for j =(i 1) K +1to i K do 3. Let x be the value to which the label of q j hashes 4. PRESENT[x] =1 5. edfor 6. edfor // Compute umber of 1 s i each processor s rage 7. for i =1to do i parallel 8. a i =0 9. for j=(i 1) K +1to i K do 10. a i = a i + PRESENT[j] 11. edfor 12. edfor //Compute partial sums 13. s 0 =0 14. Compute s i = i k=1 a i for 1 i usig prefix sum //Compute ew block umbers 15. for i =1to do i parallel 16. a i = s i for j =(i 1) K +1to i K do 18. a i = a i + PRESENT[j] 19. if PRESENT[j] =1the 20. ew block o[j] = a i 21. edfor 22. edfor // Update the block o array with the ew block umbers 23. for i =1to do i parallel 24. for j =(i 1) to i 1 do 25. Let x be the value to which the label of q j hashes 26. block o[q j ]=ewblock o[x] 27. edfor 28. edfor Algorithm 4: Reducig the rage of block umbers
7 40 Ambuj Tewari et al. Refereces [1] Aho A. V.,Hopcroft J. E. ad Ullma J. D.: The desig ad aalysis of computer algorithms. Addiso-Wesley,Readig,Massachusetts (1974) 35, 36 [2] Blum N.: A O( ) implemetatio of the stadard method of miimizig -state fiite automata. Iformatio Processig Letters 57 (1996) [3] Cho S. ad Huyh D. T.: The parallel complexity of coarsest set partitio problems. Iformatio Processig Letters 42 (1992) , 36 [4] Huffma D. A.: The Sythesis of Sequetial Switchig Circuits. Joural of Frakli Istitute 257 (1954) [5] Hopcroft J. E.: A algorithm for miimizig states i a fiite automata. Theory of Machies ad Computatio,Academic Press (1971) [6] Hopcroft J. E. ad Ullma J. D.: Itroductio to automata theory,laguages,ad computatio. Addiso-Wesley,Readig,Massachusetts (1979) 34 [7] Jaja J. ad Kosaraju S. R.: Parallel algorithms for plaar graph isomorphism ad related problems. IEEE Trasactios o Circuits ad Systems 35 (1988) [8] Jaja J. ad Ryu K. W.: A Efficiet Parallel Algorithm for the Sigle Fuctio Coarsest Partitio Problem. Theoretical Computer Sciece 129 (1994) [9] Matias Y. ad Vishki U.: O parallel hashig ad iteger sortig. Joural of Algorithms 4 (1991) , 38 [10] Moore E. F.: Gedake-experimets o sequetial circuits. Automata Studies, Priceto Uiversity Press (1956) [11] Paige R.,Tarja R. E. ad Boic R.: A liear time solutio to the sigle fuctio coarsest partitio problem. Theoretical Computer Sciece 40 (1985) [12] Ravikumar B. ad Xiog X.: A parallel algorithm for miimizatio of fiite automata. Proceedigs of the 10th Iteratioal Parallel Processig Symposium, Hoululu,Hawaii (1996) [13] Srikat Y. N.: A parallel algorithm for the miimizatio of fiite state automata. Iteratioal Joural Computer Math. 32 (1990) [14] Vardi M.: Notraditioal applicatios of automata theory. Lecture Notes i Computer Sciece,Spriger-Verlag 789 (1994)
Ones Assignment Method for Solving Traveling Salesman Problem
Joural of mathematics ad computer sciece 0 (0), 58-65 Oes Assigmet Method for Solvig Travelig Salesma Problem Hadi Basirzadeh Departmet of Mathematics, Shahid Chamra Uiversity, Ahvaz, Ira Article history:
More informationCounting the Number of Minimum Roman Dominating Functions of a Graph
Coutig the Number of Miimum Roma Domiatig Fuctios of a Graph SHI ZHENG ad KOH KHEE MENG, Natioal Uiversity of Sigapore We provide two algorithms coutig the umber of miimum Roma domiatig fuctios of a graph
More informationLecture Notes 6 Introduction to algorithm analysis CSS 501 Data Structures and Object-Oriented Programming
Lecture Notes 6 Itroductio to algorithm aalysis CSS 501 Data Structures ad Object-Orieted Programmig Readig for this lecture: Carrao, Chapter 10 To be covered i this lecture: Itroductio to algorithm aalysis
More informationA SOFTWARE MODEL FOR THE MULTILAYER PERCEPTRON
A SOFTWARE MODEL FOR THE MULTILAYER PERCEPTRON Roberto Lopez ad Eugeio Oñate Iteratioal Ceter for Numerical Methods i Egieerig (CIMNE) Edificio C1, Gra Capitá s/, 08034 Barceloa, Spai ABSTRACT I this work
More informationCSC165H1 Worksheet: Tutorial 8 Algorithm analysis (SOLUTIONS)
CSC165H1, Witer 018 Learig Objectives By the ed of this worksheet, you will: Aalyse the ruig time of fuctios cotaiig ested loops. 1. Nested loop variatios. Each of the followig fuctios takes as iput a
More informationLecture 1: Introduction and Strassen s Algorithm
5-750: Graduate Algorithms Jauary 7, 08 Lecture : Itroductio ad Strasse s Algorithm Lecturer: Gary Miller Scribe: Robert Parker Itroductio Machie models I this class, we will primarily use the Radom Access
More informationChapter 1. Introduction to Computers and C++ Programming. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 1 Itroductio to Computers ad C++ Programmig Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 1.1 Computer Systems 1.2 Programmig ad Problem Solvig 1.3 Itroductio to C++ 1.4 Testig
More information. Written in factored form it is easy to see that the roots are 2, 2, i,
CMPS A Itroductio to Programmig Programmig Assigmet 4 I this assigmet you will write a java program that determies the real roots of a polyomial that lie withi a specified rage. Recall that the roots (or
More informationLecture 18. Optimization in n dimensions
Lecture 8 Optimizatio i dimesios Itroductio We ow cosider the problem of miimizig a sigle scalar fuctio of variables, f x, where x=[ x, x,, x ]T. The D case ca be visualized as fidig the lowest poit of
More informationPseudocode ( 1.1) Analysis of Algorithms. Primitive Operations. Pseudocode Details. Running Time ( 1.1) Estimating performance
Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Pseudocode ( 1.1) High-level descriptio of a algorithm More structured
More informationImproving Information Retrieval System Security via an Optimal Maximal Coding Scheme
Improvig Iformatio Retrieval System Security via a Optimal Maximal Codig Scheme Dogyag Log Departmet of Computer Sciece, City Uiversity of Hog Kog, 8 Tat Chee Aveue Kowloo, Hog Kog SAR, PRC dylog@cs.cityu.edu.hk
More informationSolving Fuzzy Assignment Problem Using Fourier Elimination Method
Global Joural of Pure ad Applied Mathematics. ISSN 0973-768 Volume 3, Number 2 (207), pp. 453-462 Research Idia Publicatios http://www.ripublicatio.com Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio
More informationAn Efficient Algorithm for Graph Bisection of Triangularizations
A Efficiet Algorithm for Graph Bisectio of Triagularizatios Gerold Jäger Departmet of Computer Sciece Washigto Uiversity Campus Box 1045 Oe Brookigs Drive St. Louis, Missouri 63130-4899, USA jaegerg@cse.wustl.edu
More informationLecture 5. Counting Sort / Radix Sort
Lecture 5. Coutig Sort / Radix Sort T. H. Corme, C. E. Leiserso ad R. L. Rivest Itroductio to Algorithms, 3rd Editio, MIT Press, 2009 Sugkyukwa Uiversity Hyuseug Choo choo@skku.edu Copyright 2000-2018
More information1 Graph Sparsfication
CME 305: Discrete Mathematics ad Algorithms 1 Graph Sparsficatio I this sectio we discuss the approximatio of a graph G(V, E) by a sparse graph H(V, F ) o the same vertex set. I particular, we cosider
More informationA New Morphological 3D Shape Decomposition: Grayscale Interframe Interpolation Method
A ew Morphological 3D Shape Decompositio: Grayscale Iterframe Iterpolatio Method D.. Vizireau Politehica Uiversity Bucharest, Romaia ae@comm.pub.ro R. M. Udrea Politehica Uiversity Bucharest, Romaia mihea@comm.pub.ro
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpeCourseWare http://ocw.mit.edu 6.854J / 18.415J Advaced Algorithms Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.415/6.854 Advaced Algorithms
More informationCS200: Hash Tables. Prichard Ch CS200 - Hash Tables 1
CS200: Hash Tables Prichard Ch. 13.2 CS200 - Hash Tables 1 Table Implemetatios: average cases Search Add Remove Sorted array-based Usorted array-based Balaced Search Trees O(log ) O() O() O() O(1) O()
More informationPETRI NETS GENERATING KOLAM PATTERNS
. Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) PETRI NETS GENERATING KOLAM PATTERNS. Lalitha epartmet of Mathematics Sathyabama Uiversity, heai-119, Idia lalkrish_24@yahoo.co.i K. Ragaraja
More informationCIS 121 Data Structures and Algorithms with Java Spring Stacks and Queues Monday, February 12 / Tuesday, February 13
CIS Data Structures ad Algorithms with Java Sprig 08 Stacks ad Queues Moday, February / Tuesday, February Learig Goals Durig this lab, you will: Review stacks ad queues. Lear amortized ruig time aalysis
More informationRunning Time. Analysis of Algorithms. Experimental Studies. Limitations of Experiments
Ruig Time Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects. The
More informationAnalysis Metrics. Intro to Algorithm Analysis. Slides. 12. Alg Analysis. 12. Alg Analysis
Itro to Algorithm Aalysis Aalysis Metrics Slides. Table of Cotets. Aalysis Metrics 3. Exact Aalysis Rules 4. Simple Summatio 5. Summatio Formulas 6. Order of Magitude 7. Big-O otatio 8. Big-O Theorems
More informationData Structures and Algorithms. Analysis of Algorithms
Data Structures ad Algorithms Aalysis of Algorithms Outlie Ruig time Pseudo-code Big-oh otatio Big-theta otatio Big-omega otatio Asymptotic algorithm aalysis Aalysis of Algorithms Iput Algorithm Output
More informationECE4050 Data Structures and Algorithms. Lecture 6: Searching
ECE4050 Data Structures ad Algorithms Lecture 6: Searchig 1 Search Give: Distict keys k 1, k 2,, k ad collectio L of records of the form (k 1, I 1 ), (k 2, I 2 ),, (k, I ) where I j is the iformatio associated
More informationSorting in Linear Time. Data Structures and Algorithms Andrei Bulatov
Sortig i Liear Time Data Structures ad Algorithms Adrei Bulatov Algorithms Sortig i Liear Time 7-2 Compariso Sorts The oly test that all the algorithms we have cosidered so far is compariso The oly iformatio
More informationBASED ON ITERATIVE ERROR-CORRECTION
A COHPARISO OF CRYPTAALYTIC PRICIPLES BASED O ITERATIVE ERROR-CORRECTIO Miodrag J. MihaljeviC ad Jova Dj. GoliC Istitute of Applied Mathematics ad Electroics. Belgrade School of Electrical Egieerig. Uiversity
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies. Limitations of Experiments
Ruig Time ( 3.1) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step- by- step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationAnalysis of Algorithms
Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Ruig Time Most algorithms trasform iput objects ito output objects. The
More informationHow do we evaluate algorithms?
F2 Readig referece: chapter 2 + slides Algorithm complexity Big O ad big Ω To calculate ruig time Aalysis of recursive Algorithms Next time: Litterature: slides mostly The first Algorithm desig methods:
More informationAn Efficient Algorithm for Graph Bisection of Triangularizations
Applied Mathematical Scieces, Vol. 1, 2007, o. 25, 1203-1215 A Efficiet Algorithm for Graph Bisectio of Triagularizatios Gerold Jäger Departmet of Computer Sciece Washigto Uiversity Campus Box 1045, Oe
More informationBOOLEAN MATHEMATICS: GENERAL THEORY
CHAPTER 3 BOOLEAN MATHEMATICS: GENERAL THEORY 3.1 ISOMORPHIC PROPERTIES The ame Boolea Arithmetic was chose because it was discovered that literal Boolea Algebra could have a isomorphic umerical aspect.
More informationAnalysis of Algorithms
Aalysis of Algorithms Ruig Time of a algorithm Ruig Time Upper Bouds Lower Bouds Examples Mathematical facts Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite
More informationCopyright 2016 Ramez Elmasri and Shamkant B. Navathe
Copyright 2016 Ramez Elmasri ad Shamkat B. Navathe CHAPTER 18 Strategies for Query Processig Copyright 2016 Ramez Elmasri ad Shamkat B. Navathe Itroductio DBMS techiques to process a query Scaer idetifies
More informationAn Algorithm to Solve Multi-Objective Assignment. Problem Using Interactive Fuzzy. Goal Programming Approach
It. J. Cotemp. Math. Scieces, Vol. 6, 0, o. 34, 65-66 A Algorm to Solve Multi-Objective Assigmet Problem Usig Iteractive Fuzzy Goal Programmig Approach P. K. De ad Bharti Yadav Departmet of Mathematics
More informationCIS 121 Data Structures and Algorithms with Java Spring Stacks, Queues, and Heaps Monday, February 18 / Tuesday, February 19
CIS Data Structures ad Algorithms with Java Sprig 09 Stacks, Queues, ad Heaps Moday, February 8 / Tuesday, February 9 Stacks ad Queues Recall the stack ad queue ADTs (abstract data types from lecture.
More informationA New Bit Wise Technique for 3-Partitioning Algorithm
Special Issue of Iteratioal Joural of Computer Applicatios (0975 8887) o Optimizatio ad O-chip Commuicatio, No.1. Feb.2012, ww.ijcaolie.org A New Bit Wise Techique for 3-Partitioig Algorithm Rajumar Jai
More informationMinimum Spanning Trees
Miimum Spaig Trees Miimum Spaig Trees Spaig subgraph Subgraph of a graph G cotaiig all the vertices of G Spaig tree Spaig subgraph that is itself a (free) tree Miimum spaig tree (MST) Spaig tree of a weighted
More informationAdministrative UNSUPERVISED LEARNING. Unsupervised learning. Supervised learning 11/25/13. Final project. No office hours today
Admiistrative Fial project No office hours today UNSUPERVISED LEARNING David Kauchak CS 451 Fall 2013 Supervised learig Usupervised learig label label 1 label 3 model/ predictor label 4 label 5 Supervised
More informationarxiv: v2 [cs.ds] 24 Mar 2018
Similar Elemets ad Metric Labelig o Complete Graphs arxiv:1803.08037v [cs.ds] 4 Mar 018 Pedro F. Felzeszwalb Brow Uiversity Providece, RI, USA pff@brow.edu March 8, 018 We cosider a problem that ivolves
More informationcondition w i B i S maximum u i
ecture 10 Dyamic Programmig 10.1 Kapsack Problem November 1, 2004 ecturer: Kamal Jai Notes: Tobias Holgers We are give a set of items U = {a 1, a 2,..., a }. Each item has a weight w i Z + ad a utility
More informationWhat are we going to learn? CSC Data Structures Analysis of Algorithms. Overview. Algorithm, and Inputs
What are we goig to lear? CSC316-003 Data Structures Aalysis of Algorithms Computer Sciece North Carolia State Uiversity Need to say that some algorithms are better tha others Criteria for evaluatio Structure
More informationPython Programming: An Introduction to Computer Science
Pytho Programmig: A Itroductio to Computer Sciece Chapter 1 Computers ad Programs 1 Objectives To uderstad the respective roles of hardware ad software i a computig system. To lear what computer scietists
More informationComputational Geometry
Computatioal Geometry Chapter 4 Liear programmig Duality Smallest eclosig disk O the Ageda Liear Programmig Slides courtesy of Craig Gotsma 4. 4. Liear Programmig - Example Defie: (amout amout cosumed
More informationOn Infinite Groups that are Isomorphic to its Proper Infinite Subgroup. Jaymar Talledo Balihon. Abstract
O Ifiite Groups that are Isomorphic to its Proper Ifiite Subgroup Jaymar Talledo Baliho Abstract Two groups are isomorphic if there exists a isomorphism betwee them Lagrage Theorem states that the order
More informationParallel Polygon Approximation Algorithm Targeted at Reconfigurable Multi-Ring Hardware
Parallel Polygo Approximatio Algorithm Targeted at Recofigurable Multi-Rig Hardware M. Arif Wai* ad Hamid R. Arabia** *Califoria State Uiversity Bakersfield, Califoria, USA **Uiversity of Georgia, Georgia,
More informationRandom Graphs and Complex Networks T
Radom Graphs ad Complex Networks T-79.7003 Charalampos E. Tsourakakis Aalto Uiversity Lecture 3 7 September 013 Aoucemet Homework 1 is out, due i two weeks from ow. Exercises: Probabilistic iequalities
More informationBig-O Analysis. Asymptotics
Big-O Aalysis 1 Defiitio: Suppose that f() ad g() are oegative fuctios of. The we say that f() is O(g()) provided that there are costats C > 0 ad N > 0 such that for all > N, f() Cg(). Big-O expresses
More information9.1. Sequences and Series. Sequences. What you should learn. Why you should learn it. Definition of Sequence
_9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability 9. Sequeces ad Series What you should lear Use sequece otatio to write the terms of sequeces. Use factorial otatio. Use summatio otatio to
More informationCIS 121 Data Structures and Algorithms with Java Fall Big-Oh Notation Tuesday, September 5 (Make-up Friday, September 8)
CIS 11 Data Structures ad Algorithms with Java Fall 017 Big-Oh Notatio Tuesday, September 5 (Make-up Friday, September 8) Learig Goals Review Big-Oh ad lear big/small omega/theta otatios Practice solvig
More informationHash Tables. Presentation for use with the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015.
Presetatio for use with the textbook Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 2015 Hash Tables xkcd. http://xkcd.com/221/. Radom Number. Used with permissio uder Creative
More informationn n B. How many subsets of C are there of cardinality n. We are selecting elements for such a
4. [10] Usig a combiatorial argumet, prove that for 1: = 0 = Let A ad B be disjoit sets of cardiality each ad C = A B. How may subsets of C are there of cardiality. We are selectig elemets for such a subset
More informationImproved Random Graph Isomorphism
Improved Radom Graph Isomorphism Tomek Czajka Gopal Paduraga Abstract Caoical labelig of a graph cosists of assigig a uique label to each vertex such that the labels are ivariat uder isomorphism. Such
More informationChapter 10. Defining Classes. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 10 Defiig Classes Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 10.1 Structures 10.2 Classes 10.3 Abstract Data Types 10.4 Itroductio to Iheritace Copyright 2015 Pearso Educatio,
More information15-859E: Advanced Algorithms CMU, Spring 2015 Lecture #2: Randomized MST and MST Verification January 14, 2015
15-859E: Advaced Algorithms CMU, Sprig 2015 Lecture #2: Radomized MST ad MST Verificatio Jauary 14, 2015 Lecturer: Aupam Gupta Scribe: Yu Zhao 1 Prelimiaries I this lecture we are talkig about two cotets:
More informationCombination Labelings Of Graphs
Applied Mathematics E-Notes, (0), - c ISSN 0-0 Available free at mirror sites of http://wwwmaththuedutw/ame/ Combiatio Labeligs Of Graphs Pak Chig Li y Received February 0 Abstract Suppose G = (V; E) is
More informationLower Bounds for Sorting
Liear Sortig Topics Covered: Lower Bouds for Sortig Coutig Sort Radix Sort Bucket Sort Lower Bouds for Sortig Compariso vs. o-compariso sortig Decisio tree model Worst case lower boud Compariso Sortig
More informationRedundancy Allocation for Series Parallel Systems with Multiple Constraints and Sensitivity Analysis
IOSR Joural of Egieerig Redudacy Allocatio for Series Parallel Systems with Multiple Costraits ad Sesitivity Aalysis S. V. Suresh Babu, D.Maheswar 2, G. Ragaath 3 Y.Viaya Kumar d G.Sakaraiah e (Mechaical
More informationComputers and Scientific Thinking
Computers ad Scietific Thikig David Reed, Creighto Uiversity Chapter 15 JavaScript Strigs 1 Strigs as Objects so far, your iteractive Web pages have maipulated strigs i simple ways use text box to iput
More informationExact Minimum Lower Bound Algorithm for Traveling Salesman Problem
Exact Miimum Lower Boud Algorithm for Travelig Salesma Problem Mohamed Eleiche GeoTiba Systems mohamed.eleiche@gmail.com Abstract The miimum-travel-cost algorithm is a dyamic programmig algorithm to compute
More informationCubic Polynomial Curves with a Shape Parameter
roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April -8 00 (pp5-70) Cubic olyomial Curves with a Shape arameter MO GUOLIANG ZHAO YANAN Iformatio ad
More information5.3 Recursive definitions and structural induction
/8/05 5.3 Recursive defiitios ad structural iductio CSE03 Discrete Computatioal Structures Lecture 6 A recursively defied picture Recursive defiitios e sequece of powers of is give by a = for =0,,, Ca
More informationHashing Functions Performance in Packet Classification
Hashig Fuctios Performace i Packet Classificatio Mahmood Ahmadi ad Stepha Wog Computer Egieerig Laboratory Faculty of Electrical Egieerig, Mathematics ad Computer Sciece Delft Uiversity of Techology {mahmadi,
More informationBig-O Analysis. Asymptotics
Big-O Aalysis 1 Defiitio: Suppose that f() ad g() are oegative fuctios of. The we say that f() is O(g()) provided that there are costats C > 0 ad N > 0 such that for all > N, f() Cg(). Big-O expresses
More informationOn (K t e)-saturated Graphs
Noame mauscript No. (will be iserted by the editor O (K t e-saturated Graphs Jessica Fuller Roald J. Gould the date of receipt ad acceptace should be iserted later Abstract Give a graph H, we say a graph
More informationMAXIMUM MATCHINGS IN COMPLETE MULTIPARTITE GRAPHS
Fura Uiversity Electroic Joural of Udergraduate Matheatics Volue 00, 1996 6-16 MAXIMUM MATCHINGS IN COMPLETE MULTIPARTITE GRAPHS DAVID SITTON Abstract. How ay edges ca there be i a axiu atchig i a coplete
More informationCopyright 2016 Ramez Elmasri and Shamkant B. Navathe
Copyright 2016 Ramez Elmasri ad Shamkat B. Navathe CHAPTER 19 Query Optimizatio Copyright 2016 Ramez Elmasri ad Shamkat B. Navathe Itroductio Query optimizatio Coducted by a query optimizer i a DBMS Goal:
More informationPattern Recognition Systems Lab 1 Least Mean Squares
Patter Recogitio Systems Lab 1 Least Mea Squares 1. Objectives This laboratory work itroduces the OpeCV-based framework used throughout the course. I this assigmet a lie is fitted to a set of poits usig
More informationData Structures Week #9. Sorting
Data Structures Week #9 Sortig Outlie Motivatio Types of Sortig Elemetary (O( 2 )) Sortig Techiques Other (O(*log())) Sortig Techiques 21.Aralık.2010 Boraha Tümer, Ph.D. 2 Sortig 21.Aralık.2010 Boraha
More informationCSC 220: Computer Organization Unit 11 Basic Computer Organization and Design
College of Computer ad Iformatio Scieces Departmet of Computer Sciece CSC 220: Computer Orgaizatio Uit 11 Basic Computer Orgaizatio ad Desig 1 For the rest of the semester, we ll focus o computer architecture:
More informationStrong Complementary Acyclic Domination of a Graph
Aals of Pure ad Applied Mathematics Vol 8, No, 04, 83-89 ISSN: 79-087X (P), 79-0888(olie) Published o 7 December 04 wwwresearchmathsciorg Aals of Strog Complemetary Acyclic Domiatio of a Graph NSaradha
More informationGraphs. Minimum Spanning Trees. Slides by Rose Hoberman (CMU)
Graphs Miimum Spaig Trees Slides by Rose Hoberma (CMU) Problem: Layig Telephoe Wire Cetral office 2 Wirig: Naïve Approach Cetral office Expesive! 3 Wirig: Better Approach Cetral office Miimize the total
More informationA study on Interior Domination in Graphs
IOSR Joural of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 219-765X. Volume 12, Issue 2 Ver. VI (Mar. - Apr. 2016), PP 55-59 www.iosrjourals.org A study o Iterior Domiatio i Graphs A. Ato Kisley 1,
More informationFast Fourier Transform (FFT) Algorithms
Fast Fourier Trasform FFT Algorithms Relatio to the z-trasform elsewhere, ozero, z x z X x [ ] 2 ~ elsewhere,, ~ e j x X x x π j e z z X X π 2 ~ The DFS X represets evely spaced samples of the z- trasform
More informationNew HSL Distance Based Colour Clustering Algorithm
The 4th Midwest Artificial Itelligece ad Cogitive Scieces Coferece (MAICS 03 pp 85-9 New Albay Idiaa USA April 3-4 03 New HSL Distace Based Colour Clusterig Algorithm Vasile Patrascu Departemet of Iformatics
More informationRecursion. Computer Science S-111 Harvard University David G. Sullivan, Ph.D. Review: Method Frames
Uit 4, Part 3 Recursio Computer Sciece S-111 Harvard Uiversity David G. Sulliva, Ph.D. Review: Method Frames Whe you make a method call, the Java rutime sets aside a block of memory kow as the frame of
More informationLecturers: Sanjam Garg and Prasad Raghavendra Feb 21, Midterm 1 Solutions
U.C. Berkeley CS170 : Algorithms Midterm 1 Solutios Lecturers: Sajam Garg ad Prasad Raghavedra Feb 1, 017 Midterm 1 Solutios 1. (4 poits) For the directed graph below, fid all the strogly coected compoets
More informationA Note on Chromatic Transversal Weak Domination in Graphs
Iteratioal Joural of Mathematics Treds ad Techology Volume 17 Number 2 Ja 2015 A Note o Chromatic Trasversal Weak Domiatio i Graphs S Balamuruga 1, P Selvalakshmi 2 ad A Arivalaga 1 Assistat Professor,
More informationANN WHICH COVERS MLP AND RBF
ANN WHICH COVERS MLP AND RBF Josef Boští, Jaromír Kual Faculty of Nuclear Scieces ad Physical Egieerig, CTU i Prague Departmet of Software Egieerig Abstract Two basic types of artificial eural etwors Multi
More informationChapter 11. Friends, Overloaded Operators, and Arrays in Classes. Copyright 2014 Pearson Addison-Wesley. All rights reserved.
Chapter 11 Frieds, Overloaded Operators, ad Arrays i Classes Copyright 2014 Pearso Addiso-Wesley. All rights reserved. Overview 11.1 Fried Fuctios 11.2 Overloadig Operators 11.3 Arrays ad Classes 11.4
More informationPerhaps the method will give that for every e > U f() > p - 3/+e There is o o-trivial upper boud for f() ad ot eve f() < Z - e. seems to be kow, where
ON MAXIMUM CHORDAL SUBGRAPH * Paul Erdos Mathematical Istitute of the Hugaria Academy of Scieces ad Reu Laskar Clemso Uiversity 1. Let G() deote a udirected graph, with vertices ad V(G) deote the vertex
More informationFREQUENCY ESTIMATION OF INTERNET PACKET STREAMS WITH LIMITED SPACE: UPPER AND LOWER BOUNDS
FREQUENCY ESTIMATION OF INTERNET PACKET STREAMS WITH LIMITED SPACE: UPPER AND LOWER BOUNDS Prosejit Bose Evagelos Kraakis Pat Mori Yihui Tag School of Computer Sciece, Carleto Uiversity {jit,kraakis,mori,y
More informationAn Improved Shuffled Frog-Leaping Algorithm for Knapsack Problem
A Improved Shuffled Frog-Leapig Algorithm for Kapsack Problem Zhoufag Li, Ya Zhou, ad Peg Cheg School of Iformatio Sciece ad Egieerig Hea Uiversity of Techology ZhegZhou, Chia lzhf1978@126.com Abstract.
More informationCS 683: Advanced Design and Analysis of Algorithms
CS 683: Advaced Desig ad Aalysis of Algorithms Lecture 6, February 1, 2008 Lecturer: Joh Hopcroft Scribes: Shaomei Wu, Etha Feldma February 7, 2008 1 Threshold for k CNF Satisfiability I the previous lecture,
More informationINTERSECTION CORDIAL LABELING OF GRAPHS
INTERSECTION CORDIAL LABELING OF GRAPHS G Meea, K Nagaraja Departmet of Mathematics, PSR Egieerig College, Sivakasi- 66 4, Virudhuagar(Dist) Tamil Nadu, INDIA meeag9@yahoocoi Departmet of Mathematics,
More informationCSE 417: Algorithms and Computational Complexity
Time CSE 47: Algorithms ad Computatioal Readig assigmet Read Chapter of The ALGORITHM Desig Maual Aalysis & Sortig Autum 00 Paul Beame aalysis Problem size Worst-case complexity: max # steps algorithm
More informationReversible Realization of Quaternary Decoder, Multiplexer, and Demultiplexer Circuits
Egieerig Letters, :, EL Reversible Realizatio of Quaterary Decoder, Multiplexer, ad Demultiplexer Circuits Mozammel H.. Kha, Member, ENG bstract quaterary reversible circuit is more compact tha the correspodig
More informationXiaozhou (Steve) Li, Atri Rudra, Ram Swaminathan. HP Laboratories HPL Keyword(s): graph coloring; hardness of approximation
Flexible Colorig Xiaozhou (Steve) Li, Atri Rudra, Ram Swamiatha HP Laboratories HPL-2010-177 Keyword(s): graph colorig; hardess of approximatio Abstract: Motivated b y reliability cosideratios i data deduplicatio
More informationthe beginning of the program in order for it to work correctly. Similarly, a Confirm
I our sytax, a Assume statemet will be used to record what must be true at the begiig of the program i order for it to work correctly. Similarly, a Cofirm statemet is used to record what should be true
More informationHigher-order iterative methods free from second derivative for solving nonlinear equations
Iteratioal Joural of the Phsical Scieces Vol 6(8, pp 887-89, 8 April, Available olie at http://wwwacademicjouralsorg/ijps DOI: 5897/IJPS45 ISSN 99-95 Academic Jourals Full Legth Research Paper Higher-order
More informationOutline and Reading. Analysis of Algorithms. Running Time. Experimental Studies. Limitations of Experiments. Theoretical Analysis
Outlie ad Readig Aalysis of Algorithms Iput Algorithm Output Ruig time ( 3.) Pseudo-code ( 3.2) Coutig primitive operatios ( 3.3-3.) Asymptotic otatio ( 3.6) Asymptotic aalysis ( 3.7) Case study Aalysis
More information3. b. Present a combinatorial argument that for all positive integers n : : 2 n
. b. Preset a combiatorial argumet that for all positive itegers : : Cosider two distict sets A ad B each of size. Sice they are distict, the cardiality of A B is. The umber of ways of choosig a pair of
More information3D Model Retrieval Method Based on Sample Prediction
20 Iteratioal Coferece o Computer Commuicatio ad Maagemet Proc.of CSIT vol.5 (20) (20) IACSIT Press, Sigapore 3D Model Retrieval Method Based o Sample Predictio Qigche Zhag, Ya Tag* School of Computer
More informationDynamic Programming and Curve Fitting Based Road Boundary Detection
Dyamic Programmig ad Curve Fittig Based Road Boudary Detectio SHYAM PRASAD ADHIKARI, HYONGSUK KIM, Divisio of Electroics ad Iformatio Egieerig Chobuk Natioal Uiversity 664-4 Ga Deokji-Dog Jeoju-City Jeobuk
More informationRelationship between augmented eccentric connectivity index and some other graph invariants
Iteratioal Joural of Advaced Mathematical Scieces, () (03) 6-3 Sciece Publishig Corporatio wwwsciecepubcocom/idexphp/ijams Relatioship betwee augmeted eccetric coectivity idex ad some other graph ivariats
More informationAdaptive Resource Allocation for Electric Environmental Pollution through the Control Network
Available olie at www.sciecedirect.com Eergy Procedia 6 (202) 60 64 202 Iteratioal Coferece o Future Eergy, Eviromet, ad Materials Adaptive Resource Allocatio for Electric Evirometal Pollutio through the
More informationSECURITY PROOF FOR SHENGBAO WANG S IDENTITY-BASED ENCRYPTION SCHEME
SCURITY PROOF FOR SNGBAO WANG S IDNTITY-BASD NCRYPTION SCM Suder Lal ad Priyam Sharma Derpartmet of Mathematics, Dr. B.R.A.(Agra), Uiversity, Agra-800(UP), Idia. -mail- suder_lal@rediffmail.com, priyam_sharma.ibs@rediffmail.com
More informationAccuracy Improvement in Camera Calibration
Accuracy Improvemet i Camera Calibratio FaJie L Qi Zag ad Reihard Klette CITR, Computer Sciece Departmet The Uiversity of Aucklad Tamaki Campus, Aucklad, New Zealad fli006, qza001@ec.aucklad.ac.z r.klette@aucklad.ac.z
More informationAppendix D. Controller Implementation
COMPUTER ORGANIZATION AND DESIGN The Hardware/Software Iterface 5 th Editio Appedix D Cotroller Implemetatio Cotroller Implemetatios Combiatioal logic (sigle-cycle); Fiite state machie (multi-cycle, pipelied);
More informationForce Network Analysis using Complementary Energy
orce Network Aalysis usig Complemetary Eergy Adrew BORGART Assistat Professor Delft Uiversity of Techology Delft, The Netherlads A.Borgart@tudelft.l Yaick LIEM Studet Delft Uiversity of Techology Delft,
More informationMinimum Spanning Trees
Presetatio for use with the textbook, lgorithm esig ad pplicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 0 Miimum Spaig Trees 0 Goodrich ad Tamassia Miimum Spaig Trees pplicatio: oectig a Network Suppose
More information