Clustering on antimatroids and convex geometries
|
|
- Gilbert Thornton
- 5 years ago
- Views:
Transcription
1 Clusterng on antmatrods and convex geometres YULIA KEMPNER 1, ILYA MUCNIK 2 1 Department of Computer cence olon Academc Insttute of Technology 52 Golomb tr., P.O. Box 305, olon IRAEL 2 Department of Computer cence Rutgers Unversty, NJ, DIMAC 96 Frelnghuysen Road, Pscataway, NJ U Abstract: - The clusterng problem as a problem of set functon optmzaton wth constrants s consdered. The behavor of quas-concave functons on antmatrods and on convex geometres s nvestgated. The dualty of these two set functon optmzatons s proved. The greedy type Chan algorthm, whch allows to fnd an optmal cluster, both as the most dstant group on antmatrods and as a dense cluster on convex geometres, s descrbed. Key-Words: - Quas-concave functon, antmatrod, convex geometry, cluster, greedy algorthm 1 Introducton In ths paper we consder the problem of clusterng on antmatrods and on convex geometres. There are two approaches to clusterng. In the frst one we try to fnd extreme dense clusters (ether as a partton of the consdered set of obects nto homogeneous groups, or as a sngle tght set of obects). The second approach fnds a cluster whch s the most dstant from ts complementary subset n the whole set (as n the sngle lnkage clusterng method) wthout a control how close are obects wthn a cluster. The problem of densty (n the frst approach) or solaton (n the second approach) of a subset may be formalzed n terms of a set functon - the functon that ts values score the subsets accordng to ther tghtness or to dstant relatons wth ther complementary subsets. Usually, n cluster analyss each subset may be referred as a cluster. owever, n last years the problem to fnd clusters as optmzaton problem wth constrants became actual, partcularly n scentfc database mnng, where obects have complex structure. In those cases ther smlarty measurement requres to consder the problem from mult-crtera perspectves. For nstance, f one consders transportaton problems n a large cty and wants to apply clusterng technque, she or he has to take nto account at least two crtera: a populaton flow densty along transportaton lnes and a capacty of all types of transports along the lnes. In ths paper we also consder restrcted stuatons for clusterng. We found that such stuatons can be characterzed as constructvely defned famles of subsets, such that only elements from these famles can be consdered as feasble clusters. For example, let us consder a boolean matrx of data n whch varables (columns of matrx) are dvded nto several groups. Varables, belongng to one group, are ordered. In ths case every group of varables can be nterpreted as a complex varable whch represent an ordnal scale for some property: value 1 of the frst (accordng to the order) boolean varable means that an obect represents the correspondng property on very general level, 1 for the second varable means that the obect represents the property on more specfc level, etc. Thus the value 1 of a boolean varable wth rank k n ts group means that all prevous varables of the group also have ths value 1. In other words, t s an ordnal data matrx. The problem s to fnd an extreme n ths ordnal data whch represents a group of ponts, that are smaller (accordng to the ordnal vector comparson) than the choosen extreme. It s obvously that ths case can be generalzed on a common ordnal data. If one nterprets ordnal scales as crtera for a partcular multcrtera evaluaton then he can nterpret the above cluster as the best choce soluton. Thus the cluster problem can be formulated as follows: for a gven set system over E,.e. for a par (E,) where 2 E s a famly of feasble subsets of fnte E, and for a gven functon F : R, fnd
2 elements of for whch the value of the functon F s maxmal. In general, ths optmzaton problem s NP-hard, but for some specfc functon and for some specfc set systems polynomal algorthms are known. The well-known examples are modular cost functons that can be optmzed over matrods by usng polynomal greedy algorthm [2] and bottleneck functon that can be maxmzed over greedods [5]. The other example s set functon defned as the mnmum value of monotone lnkage functons. uch set functon can be maxmzed by a greedy type algorthm over a famly of all subsets of E [4,7] and over antmatrods [3]. In ths paper we refne our results obtaned for antmatrods and extend them to convex geometres. In ecton 2 some basc nformaton about convex geometres and antmatrods are lsted. ecton 3 gves a bref ntroducton to the optmzaton of quas-concave functons and nvestgates the optmzaton problem on antmatrods. In ecton 4 the optmzaton problem s consdered on convex geometres and dualty of two problems are establshed. 2 Prelmnares Let E be a fnte set. A set system over E s a par (E,) where 2 E s a famly of feasble subsets of E, called feasble sets. We wll use X x for X {x} and X x for X {x}. Defnton 2.1 A closure operator, τ : 2 E 2 E, s a map satsfyng the closure axoms: C1: X τ(x) C2: X Y τ(x) τ(y) C3: τ(τ(x)) = τ(x) A set X s sad to be closed f τ(x) = X. The par (E,τ) s called a closure space. To obtan an equvalent defnton consder a set system (E,) such that () E () X,Y X Y Then the operator (1) τ(a) = {X : A X and X } s a closure operator. A set system ( E, ) satsfyng () and () wth closure operator τ defned n (1) s an equvalent axomatzaton of closure space [5]. Defnton 2.2 A closure space s called a convex geometry f the followng ant-exchange property s satsfed: f y,z τ(x) then z τ(x y) mples y τ(x z) We call an element x A an extreme pont of A f x τ(a x). For a convex set X,.e. for a set from a convex geometry (E,), ths s equvalent to X x. The set of extreme ponts of A s denoted ex(a). Consder another set system. Defnton 2.3 A nonempty set system (E,) s an antmatrod f (A1) for each nonempty X there s an x X such that X x (A2) for all X,Y, and X Y, there exsts an x X Y such that Y x Any set system satsfyng (A1) s called accessble. Another antmatrod defnton s based on the followng property: Defnton 2.4 A set system (E,) has the nterval property wthout upper bounds f for all X,Y wth X Y and for all x E Y, X x mples Y x. Theorem 2.1 [1,5] For an accessble set system (E,) the followng statements are equvalent: () (E,) s an antmatrod () s closed under unon () (E,) satsfes the nterval property wthout upper bounds A maxmal feasble subset of set X E s called a bass of X, and s denoted by B X. Clearly, (by ()), there s only one bass for each set. Now consder the Theorem that establshes dualty between two structures. Theorem 2.2 [5] A set system (E,) s an antmatrod f and only f (E,) s a convex geometry, where = { E X : X }. As an mmedate consequence we have (2) τ(e X) = E B X For a set X, let Γ(X) ={x E X : X x } be the set of feasble contnuatons of X, then (3) Γ(X) = ex(e
3 3 Quas-concave functons and antmatrods In ths secton we consder set functons defned as the mnmum value of monotone lnkage functons. uch set functons can be maxmzed by a greedy type algorthm over a famly of all subsets of E [10] and over antmatrods [3]. ere we detal the behavor of these functons defned on antmatrods n order to use them for clusterng. We consder the problem to fnd a cluster whch s most dstant from ts complementary subset n the whole consdered set. To defne a dstance between the subset and rest obects, the concept of an element-to-set lnkage functon can be used. A lnkage functon π(x,x) measures a dstance between subset X and element x. The well-known example s the sngle lnkage functon π ( x, = mn d y X whch s defned n terms of par-wse dstances d xy. As another example consder a data table A = (x k ) where x k s a value of varable k K for entty E. A lnkage functon can be defned as π (, = mn xk x k X k K An mportant feature of these examples s the monotoncty of lnkage functons. The monotone lnkage functons were ntroduced by Mullat [9]. Defnton 3.1 A functon π : E 2 E R s a monotone lnkage functon f for all X,Y E and x E (4) X Y mples π(x,x) π(x,y) Consder F : 2 E R defned for each X E F( = mn π ( x, x E X Ths functon may be consdered as the functon that measures solaton of the subset X,.e. t measures how ths subset s dstant from rest obects. ence a subset maxmzng ths set functon can be referred as a cluster. It was shown [6], that the functon F satsfes the next condton: for each X,Y E, F(X Y) mn {F(X), F(Y)} uch functons are called quas-concave set functons. These functons were studed n [10,4]. In partcular, the smple polynomal algorthm whch fnds the mnmal set X E such that F(X) = max{f(y) : Y E} was developed and used for ncomplete clusterng constructor [10]. In ths secton we extend our results to antmatrods. For ths purpose we defne a new functon specfed for antmatrods: xy F ( = mn π ( x, ( and f Γ(X) =, then F (X) = -. It should be mentoned, that an antmatrod (E,) has one and only one maxmal feasble set, E = X X. Thus, for each feasble set X - E the contnuaton Γ(X) s not-empty,.e., F s well defned on - E for any antmatrod (E,). Addtonal notaton s that the feasble sets of an antmatrod (E,), ordered by ncluson, form a lattce L, wth lattce operatons: X Y = X Y, X Y = B X Y Theorem 3.1. If a set system (E,) s an antmatrod, then the functon F s a quas-concave functon on L,.e., for each X,Y, F (X Y) mn {F (X), F (Y)} Proof. The case X Y = E s trval, then assume that X Y - E,.e., F ( X Y) = mn π ( x, X Y ). ( X Y ) Let F (X Y) = π(x 0, X Y), where x 0 Γ(X Y). ence, x 0 E (X Y), snce Γ (B E X. Indeed, suppose that x Γ (B and x X, then B X x X and B X x. Thus B X x s also a bass, a contradcton. Assume, wthout loss of generalty, that x 0 E X. ence, we have x 0 E X, and X Y X, and x 0 Γ (X Y ), so from the nterval property wthout upper bounds follows that x 0 Γ (. Then 0 0 F ( X Y) = π ( x, X Y) π ( x, mn π ( x, = F ( mn{ F (, F ( Y )} ( Consder the followng optmzaton problem - gven a monotone lnkage functon π and a set system (E,), fnd the feasble set X such that F (X) = max{f (Y) : Y }. From Theorem 3.1 t follows that the set of the optmzaton problem solutons s a meet-semlattce wth unque mnmal element. To fnd ths mnmal element we use the followng algorthm [3]: The Chan Algorthm ( E,, π ) 1. X := 2. X 0 := 3. Whle Γ (X) do 3.1 f F (X) > F (X 0 ), X 0 :=X 3.2 choose (X) such that π (x, π (y, for all y Γ (X) 3.3 X := X x 4. Return X 0
4 Thus, the algorthm generates a chan of sets = X 0 X 1 X k, where X = X -1 x and x Γ (X -1 ), and returns the mnmal set X 0 of the chan on whch the value F (X 0 ) s maxmal. Theorem 3.2 Let a set system (E,) be an antmatrod, then for all monotone lnkage functon π the Chan Algorthm fnds a mnmal feasble set that maxmzes the functon F. Proof. Let X 0 be a set obtaned by the Chan Algorthm. To prove that X 0 s a mnmal feasble set that maxmzes F, we have to prove two statements: () F ( < F ( X 0 ) for each X and X 0 X Let = X 0 X 1 X k be the chan generated by the Chan Algorthm. Let be the least nteger for whch X X. Then X -1 X, x X and X -1 x, that mples (from the nterval property wthout upper bounds) x Γ (. ence (5) F ( π(x, π(x, X -1 ) = F ( X -1 ) nce X 0 X we have X -1 X 0, so F (X -1 )<F ( X 0 ) () F ( F ( X 0 ) for each X and X 0 X If X = E or X = X k the statement s obvously, otherwse, by analogy wth prevous case, we obtan (5), but now F (X -1 ) F (X 0 ), because t s possble that X 0 X -1. The Chan Algorthm s a greedy type algorthm snce t based on the best choce prncple: t chooses on each step the extreme element (n sense of lnkage functon) and thus approaches the optmal soluton. Let P s the maxmum complexty of π computaton, then the Chan Algorthm fnds the mnmal set that maxmzes the functon F n O(P E 2 ) tme. Notce, that a quas-concave functon determnes on an antmatrod (E,) the specal structure. It has been already noted that the famly of feasble sets maxmzng the functon F s a meet-semlattce wth unque mnmal element. Denote ths famly by T 0 and let a 0 wll be the value of the functon F on the sets from T 0. The famly of sets, that maxmze the functon F over - T 0, we wll denote by T 1 and by a 1 wll be denoted the value of the functon F on these sets. Contnung ths process we have = T. It s easy to see that U L = T s a U t =0 =0 subsemlattce of, where L = { X : F ( a }. We wll call these subsemlattces by level semlattces. Denote by K the mnmal element (null) of level semlattce L. nce L L +1, we obtan K 0... K t. Theorem 3.3. Let = A 0... A p be a chan of sets that were stored as local optmal subsets X 0 n the Chan Algorthm runnng on an antmatrod (E,), then ths chan concdes wth the chan of level semlattce nulls K 0... K t. Proof. From the algorthm constructon follows that f = X 0 X 1 X k s a sequence of sets generated by the Chan Algorthm, then for each X such that Al 1 X Al, we have F ( X ) F 1) < F ). ence, (6) X Al F ( X ) F 1) To prove the Theorem, we frst prove the followng clam: Clam. For each l = 0,1,...,p and for each X, f A l X, then F ( F 1) < F ) By analogy wth the proof of Theorem 3.2 (case ()) we obtan that there exst the set X belongng to the chan generated by the Chan Algorthm, such that X A l and F (X) F (X ), then the Clam follows from (6). Now prove that for each l = 0,1,...,p, A l s a null of some level semlattce L,.e., A l = K. Let a =F (A l ). From the Clam mmedately follows that A l s a null of L. It remans to show that for each null K exsts A l = K. nce F (A p ) s a maxmal value of the functon F, for each null K exsts 1 l p, such that F 1) < F ( K ) F ) or F ( K ) F ) where A l =. how that K A l. In the second case (A l = ) t s obvously. uppose, that n the frst case A l K, then from the Clam we have F ( K ) F 1), a contradcton. On the other hand, A l L,.e., K A l because K s a null of L. Then A l = K. The obtaned chan-nested structure, referred n [7,8] as a layered cluster, can be consdered an abstract mplementaton of the dea of multresolutonal vew on a system of nterrelated obects.
5 4 Quas-concave functons and convex geometry In ths secton we consder convex geometres. As set functons, defned on convex geometres, we nvestgate functons whch are dual to quasconcave functons ntroduced n the prevous sectons. Let (E,) be a convex geometry and let F ( = F ( E for each X. From Theorem 2.2 follows that the set system = {E X : X } s an antmatrod, then the defnton s correct. From the defnton follows that F ( = mn π ( x, E. Then, by usng (3) we F ( = ( E mn π ( x, E = x ex( obtan * mn π ( x, x ex( It s easy to see that the functon π * s a monotone ncreasng lnkage functon where f X,Y and X Y then π * (x,x) π * (x,y). It should be menton, that we not assume that the orgnal gven functon s ust a monotone lnkage functon π. If a gven functon s a monotone ncreasng lnkage functon π *, then we straghtforwardly obtan the set functon F defned on the convex geometry. Consder, for example, * π ( x, = max s, defned n terms of par-wse y X xy smlarty s xy. Obvously, thus defned π * s a monotone ncreasng functon. Then the functon F may be referred as the tghtness functon [8], and a subset maxmzng ths functon can be consdered as a dense cluster. Notce that the feasble sets of a convex geometry ordered by ncluson form a lattce L, wth lattce operatons: X Y = τ ( X Y), X Y = X Y then we obtan: Lemma 4.1 If a set system (E,) s a convex geometry, then for two lattces L and L we have F ( X Y ) = F ( X Y ) The proof s based on the Theorem 2.2 and on (2). Theorem 4.1 If a set system (E,) s a convex geometry, then the functon F s a quas-concave functon on L,.e., for each X,Y, F (X Y ) mn {F (X), F (Y)} Proof. From the Lemma 4.1 and from the Theorem 3.1 we have F ( X Y ) = F ( X Y ) mn{ F = mn{ F (, F ( Y )} (, F ( Y )} = Consder the followng optmzaton problem - gven a monotone lnkage functon π *, and a convex geometry (E,), fnd the feasble set X such that F (X) = max{f (Y) : Y }. From Theorem 4.1 t follows that the set of the optmzaton problem solutons s a on-semlattce wth unque maxmal element. To fnd ths maxmal element we use the algorthm that s a mrror verson of the Chan Algorthm: The Mrror Chan Algorthm ( E,, π * ) 1. X := E 2. X 0 := E 3. Whle ex (X) do 3.1 f F (X) > F (X 0 ), X 0 :=X 3.2 choose x ex (X) such that π * (x, π * (y, for all y ex (X) 3.3 X := X - x 4. Return X 0 Thus, the algorthm generates a chan of sets E = X 0 X 1 X k, where X = X -1 - x and x ex(x -1 ), and returns the maxmal set X 0 of the chan on whch the value F (X 0 ) s maxmal. Theorem 3.2 Let a set system (E,) be a convex geometry, then for all monotone lnkage functon π * the Mrror Chan Algorthm fnds a maxmal feasble set that maxmzes the functon F. The proof of ths Theorem mmedately follows from Theorem 3.2, snce each step X X x of the Mrror Chan algorthm correspondences to the step X X x of the orgnal Chan Algorthm, and the result of the Mrror Chan Algorthm s a complement of the mnmal set obtaned by the Chan Algorthm. By analogy wth a quas-concave functon F a quas-concave functon F determnes on a convex geometry (E,) the chan-nested structure of onsemlattces L 0 L t, where 0 t L = { X : F ( a } and a >... > a are the all values of functon F on a convex geometry (E,). We also call these subsemlattces by level semlattces. Denote by I the maxmal element (one) of level semlattce L. nce L L +1, we obtan 0 1 t I I... I.
6 Theorem 4.3 Let E = A 0 A 1 A p be a chan of sets that were stored as a local maxmal subsets X 0 n the Mrror Chan Algorthm runnng on a convex geometry (E,), then ths chan concdes wth the chan of level semlattce ones 0 1 t I I... I, such that the maxmal optmal subset A p =I 0 and E=A 0 =I t. [9] J.Mullat, Extremal subsystems of monotone systems: I, II, Automaton and Remote Control 37, (1976) ; [10] Y.Zaks (Kempner), and I.Muchnk, Incomplete classfcatons of a fnte set of obects usng monotone systems, Automaton and Remote Control 50, (1989), The proof of the Theorem 4.3 s based on the dualty of antmatrods and convex geometres, and on Theorem Concluson In ths paper we have nvestgated the problem of clusterng wth constrants. The clusterng problem was consdered as optmzaton problem on antmatrods and on convex geometres wth quasconcave goal functon. The greedy type Chan algorthm, whch allows to fnd an optmal cluster, also as the most dstant group on antmatrods and as dense cluster on convex geometres, was descrbed. References: [1] A. Börner and G.M. Zegler, Introducton to greedods, n Matrod applcatons, ed. N. Whte, Cambrdge Unv.Press, Cambrdge, UK,1992 [2] J.Edmonds, Matrod and the greedy algorthm, Mathematcal Programmng 1, (1971), [3] Y. Kempner, and V.Levt, Algorthmc characterzatons of antmatrods, Thrd afa Workshop on Interdscplnary Applcatons of Graph Theory, Combnatorcs and Algorthms, afa, Israel,2003 [4] Y.Kempner, B.Mrkn, and I.Muchnk, Monotone lnkage clusterng and quas-concave functons, Appl.Math.Lett. 10,No.4 (1997) [5] B.Korte, L.Lovász, and R.chrader, ''Greedods'', prnger-verlag, New York/Berln, 1991 [6] A.Malshevsk, Propertes of ordnal set functons, n A.Malshevsk,''Qualtatve Models n the Theory of Complex ystems'', Nauka, Moscow,1998 (n Russan) [7] B.Mrkn, and I.Muchnk, Layered clusters of tghtness set functons, Appl.Math.Lett,15 (2002), [8] B. Mrkn, and I.Muchnk, Induced layered clusters, heredtary mappngs and convex geometres, Appl.Math.Lett,15 (2002),
Parallelism for Nested Loops with Non-uniform and Flow Dependences
Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr
More informationGSLM Operations Research II Fall 13/14
GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are
More informationFor instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)
Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A
More informationAn Optimal Algorithm for Prufer Codes *
J. Software Engneerng & Applcatons, 2009, 2: 111-115 do:10.4236/jsea.2009.22016 Publshed Onlne July 2009 (www.scrp.org/journal/jsea) An Optmal Algorthm for Prufer Codes * Xaodong Wang 1, 2, Le Wang 3,
More informationA NOTE ON FUZZY CLOSURE OF A FUZZY SET
(JPMNT) Journal of Process Management New Technologes, Internatonal A NOTE ON FUZZY CLOSURE OF A FUZZY SET Bhmraj Basumatary Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, Assam, Inda,
More informationSupport Vector Machines
/9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.
More information6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour
6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the
More informationMachine Learning: Algorithms and Applications
14/05/1 Machne Learnng: Algorthms and Applcatons Florano Zn Free Unversty of Bozen-Bolzano Faculty of Computer Scence Academc Year 011-01 Lecture 10: 14 May 01 Unsupervsed Learnng cont Sldes courtesy of
More informationALEKSANDROV URYSOHN COMPACTNESS CRITERION ON INTUITIONISTIC FUZZY S * STRUCTURE SPACE
mercan Journal of Mathematcs and cences Vol. 5, No., (January-December, 206) Copyrght Mnd Reader Publcatons IN No: 2250-302 www.journalshub.com LEKNDROV URYOHN COMPCTNE CRITERION ON INTUITIONITIC FUZZY
More informationb * -Open Sets in Bispaces
Internatonal Journal of Mathematcs and Statstcs Inventon (IJMSI) E-ISSN: 2321 4767 P-ISSN: 2321-4759 wwwjmsorg Volume 4 Issue 6 August 2016 PP- 39-43 b * -Open Sets n Bspaces Amar Kumar Banerjee 1 and
More informationHierarchical clustering for gene expression data analysis
Herarchcal clusterng for gene expresson data analyss Gorgo Valentn e-mal: valentn@ds.unm.t Clusterng of Mcroarray Data. Clusterng of gene expresson profles (rows) => dscovery of co-regulated and functonally
More informationA New Approach For the Ranking of Fuzzy Sets With Different Heights
New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts Pushpnder Sngh School of Mathematcs Computer pplcatons Thapar Unversty, Patala-7 00 Inda pushpndersnl@gmalcom STCT ankng of fuzzy sets plays
More information5 The Primal-Dual Method
5 The Prmal-Dual Method Orgnally desgned as a method for solvng lnear programs, where t reduces weghted optmzaton problems to smpler combnatoral ones, the prmal-dual method (PDM) has receved much attenton
More informationSum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints
Australan Journal of Basc and Appled Scences, 2(4): 1204-1208, 2008 ISSN 1991-8178 Sum of Lnear and Fractonal Multobjectve Programmng Problem under Fuzzy Rules Constrants 1 2 Sanjay Jan and Kalash Lachhwan
More informationThe Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique
//00 :0 AM Outlne and Readng The Greedy Method The Greedy Method Technque (secton.) Fractonal Knapsack Problem (secton..) Task Schedulng (secton..) Mnmum Spannng Trees (secton.) Change Money Problem Greedy
More informationModule Management Tool in Software Development Organizations
Journal of Computer Scence (5): 8-, 7 ISSN 59-66 7 Scence Publcatons Management Tool n Software Development Organzatons Ahmad A. Al-Rababah and Mohammad A. Al-Rababah Faculty of IT, Al-Ahlyyah Amman Unversty,
More informationIntra-Parametric Analysis of a Fuzzy MOLP
Intra-Parametrc Analyss of a Fuzzy MOLP a MIAO-LING WANG a Department of Industral Engneerng and Management a Mnghsn Insttute of Technology and Hsnchu Tawan, ROC b HSIAO-FAN WANG b Insttute of Industral
More informationProblem Definitions and Evaluation Criteria for Computational Expensive Optimization
Problem efntons and Evaluaton Crtera for Computatonal Expensve Optmzaton B. Lu 1, Q. Chen and Q. Zhang 3, J. J. Lang 4, P. N. Suganthan, B. Y. Qu 6 1 epartment of Computng, Glyndwr Unversty, UK Faclty
More informationSolving two-person zero-sum game by Matlab
Appled Mechancs and Materals Onlne: 2011-02-02 ISSN: 1662-7482, Vols. 50-51, pp 262-265 do:10.4028/www.scentfc.net/amm.50-51.262 2011 Trans Tech Publcatons, Swtzerland Solvng two-person zero-sum game by
More informationHermite Splines in Lie Groups as Products of Geodesics
Hermte Splnes n Le Groups as Products of Geodescs Ethan Eade Updated May 28, 2017 1 Introducton 1.1 Goal Ths document defnes a curve n the Le group G parametrzed by tme and by structural parameters n the
More informationKent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming
CS 4/560 Desgn and Analyss of Algorthms Kent State Unversty Dept. of Math & Computer Scence LECT-6 Dynamc Programmng 2 Dynamc Programmng Dynamc Programmng, lke the dvde-and-conquer method, solves problems
More informationSmoothing Spline ANOVA for variable screening
Smoothng Splne ANOVA for varable screenng a useful tool for metamodels tranng and mult-objectve optmzaton L. Rcco, E. Rgon, A. Turco Outlne RSM Introducton Possble couplng Test case MOO MOO wth Game Theory
More informationcos(a, b) = at b a b. To get a distance measure, subtract the cosine similarity from one. dist(a, b) =1 cos(a, b)
8 Clusterng 8.1 Some Clusterng Examples Clusterng comes up n many contexts. For example, one mght want to cluster journal artcles nto clusters of artcles on related topcs. In dong ths, one frst represents
More informationSemi - - Connectedness in Bitopological Spaces
Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second صفحة 45-53 Sem - - Connectedness n Btopologcal Spaces By Qays Hatem
More informationCourse Introduction. Algorithm 8/31/2017. COSC 320 Advanced Data Structures and Algorithms. COSC 320 Advanced Data Structures and Algorithms
Course Introducton Course Topcs Exams, abs, Proects A quc loo at a few algorthms 1 Advanced Data Structures and Algorthms Descrpton: We are gong to dscuss algorthm complexty analyss, algorthm desgn technques
More informationLearning the Kernel Parameters in Kernel Minimum Distance Classifier
Learnng the Kernel Parameters n Kernel Mnmum Dstance Classfer Daoqang Zhang 1,, Songcan Chen and Zh-Hua Zhou 1* 1 Natonal Laboratory for Novel Software Technology Nanjng Unversty, Nanjng 193, Chna Department
More informationCMPS 10 Introduction to Computer Science Lecture Notes
CPS 0 Introducton to Computer Scence Lecture Notes Chapter : Algorthm Desgn How should we present algorthms? Natural languages lke Englsh, Spansh, or French whch are rch n nterpretaton and meanng are not
More informationCompiler Design. Spring Register Allocation. Sample Exercises and Solutions. Prof. Pedro C. Diniz
Compler Desgn Sprng 2014 Regster Allocaton Sample Exercses and Solutons Prof. Pedro C. Dnz USC / Informaton Scences Insttute 4676 Admralty Way, Sute 1001 Marna del Rey, Calforna 90292 pedro@s.edu Regster
More informationNon-Split Restrained Dominating Set of an Interval Graph Using an Algorithm
Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - on-splt Restraned Domnatng Set of an Interval Graph Usng an Algorthm ABSTRACT Dr.A.Sudhakaraah *, E. Gnana Deepka,
More informationSupport Vector Machines
Support Vector Machnes Decson surface s a hyperplane (lne n 2D) n feature space (smlar to the Perceptron) Arguably, the most mportant recent dscovery n machne learnng In a nutshell: map the data to a predetermned
More informationDetermining the Optimal Bandwidth Based on Multi-criterion Fusion
Proceedngs of 01 4th Internatonal Conference on Machne Learnng and Computng IPCSIT vol. 5 (01) (01) IACSIT Press, Sngapore Determnng the Optmal Bandwdth Based on Mult-crteron Fuson Ha-L Lang 1+, Xan-Mn
More informationRamsey numbers of cubes versus cliques
Ramsey numbers of cubes versus clques Davd Conlon Jacob Fox Choongbum Lee Benny Sudakov Abstract The cube graph Q n s the skeleton of the n-dmensonal cube. It s an n-regular graph on 2 n vertces. The Ramsey
More informationUNIT 2 : INEQUALITIES AND CONVEX SETS
UNT 2 : NEQUALTES AND CONVEX SETS ' Structure 2. ntroducton Objectves, nequaltes and ther Graphs Convex Sets and ther Geometry Noton of Convex Sets Extreme Ponts of Convex Set Hyper Planes and Half Spaces
More informationLobachevsky State University of Nizhni Novgorod. Polyhedron. Quick Start Guide
Lobachevsky State Unversty of Nzhn Novgorod Polyhedron Quck Start Gude Nzhn Novgorod 2016 Contents Specfcaton of Polyhedron software... 3 Theoretcal background... 4 1. Interface of Polyhedron... 6 1.1.
More informationOutline. Type of Machine Learning. Examples of Application. Unsupervised Learning
Outlne Artfcal Intellgence and ts applcatons Lecture 8 Unsupervsed Learnng Professor Danel Yeung danyeung@eee.org Dr. Patrck Chan patrckchan@eee.org South Chna Unversty of Technology, Chna Introducton
More informationCHAPTER 2 DECOMPOSITION OF GRAPHS
CHAPTER DECOMPOSITION OF GRAPHS. INTRODUCTION A graph H s called a Supersubdvson of a graph G f H s obtaned from G by replacng every edge uv of G by a bpartte graph,m (m may vary for each edge by dentfyng
More informationCordial and 3-Equitable Labeling for Some Star Related Graphs
Internatonal Mathematcal Forum, 4, 009, no. 31, 1543-1553 Cordal and 3-Equtable Labelng for Some Star Related Graphs S. K. Vadya Department of Mathematcs, Saurashtra Unversty Rajkot - 360005, Gujarat,
More informationMachine Learning. Support Vector Machines. (contains material adapted from talks by Constantin F. Aliferis & Ioannis Tsamardinos, and Martin Law)
Machne Learnng Support Vector Machnes (contans materal adapted from talks by Constantn F. Alfers & Ioanns Tsamardnos, and Martn Law) Bryan Pardo, Machne Learnng: EECS 349 Fall 2014 Support Vector Machnes
More information11. APPROXIMATION ALGORITHMS
Copng wth NP-completeness 11. APPROXIMATION ALGORITHMS load balancng center selecton prcng method: vertex cover LP roundng: vertex cover generalzed load balancng knapsack problem Q. Suppose I need to solve
More informationProblem Set 3 Solutions
Introducton to Algorthms October 4, 2002 Massachusetts Insttute of Technology 6046J/18410J Professors Erk Demane and Shaf Goldwasser Handout 14 Problem Set 3 Solutons (Exercses were not to be turned n,
More informationClassifier Selection Based on Data Complexity Measures *
Classfer Selecton Based on Data Complexty Measures * Edth Hernández-Reyes, J.A. Carrasco-Ochoa, and J.Fco. Martínez-Trndad Natonal Insttute for Astrophyscs, Optcs and Electroncs, Lus Enrque Erro No.1 Sta.
More informationHelsinki University Of Technology, Systems Analysis Laboratory Mat Independent research projects in applied mathematics (3 cr)
Helsnk Unversty Of Technology, Systems Analyss Laboratory Mat-2.08 Independent research projects n appled mathematcs (3 cr) "! #$&% Antt Laukkanen 506 R ajlaukka@cc.hut.f 2 Introducton...3 2 Multattrbute
More informationMath Homotopy Theory Additional notes
Math 527 - Homotopy Theory Addtonal notes Martn Frankland February 4, 2013 The category Top s not Cartesan closed. problem. In these notes, we explan how to remedy that 1 Compactly generated spaces Ths
More informationReport on On-line Graph Coloring
2003 Fall Semester Comp 670K Onlne Algorthm Report on LO Yuet Me (00086365) cndylo@ust.hk Abstract Onlne algorthm deals wth data that has no future nformaton. Lots of examples demonstrate that onlne algorthm
More informationUnsupervised Learning
Pattern Recognton Lecture 8 Outlne Introducton Unsupervsed Learnng Parametrc VS Non-Parametrc Approach Mxture of Denstes Maxmum-Lkelhood Estmates Clusterng Prof. Danel Yeung School of Computer Scence and
More informationCluster Analysis of Electrical Behavior
Journal of Computer and Communcatons, 205, 3, 88-93 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/cc http://dx.do.org/0.4236/cc.205.350 Cluster Analyss of Electrcal Behavor Ln Lu Ln Lu, School
More informationAn Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices
Internatonal Mathematcal Forum, Vol 7, 2012, no 52, 2549-2554 An Applcaton of the Dulmage-Mendelsohn Decomposton to Sparse Null Space Bases of Full Row Rank Matrces Mostafa Khorramzadeh Department of Mathematcal
More informationthe nber of vertces n the graph. spannng tree T beng part of a par of maxmally dstant trees s called extremal. Extremal trees are useful n the mxed an
On Central Spannng Trees of a Graph S. Bezrukov Unverstat-GH Paderborn FB Mathematk/Informatk Furstenallee 11 D{33102 Paderborn F. Kaderal, W. Poguntke FernUnverstat Hagen LG Kommunkatonssysteme Bergscher
More informationON SOME ENTERTAINING APPLICATIONS OF THE CONCEPT OF SET IN COMPUTER SCIENCE COURSE
Yordzhev K., Kostadnova H. Інформаційні технології в освіті ON SOME ENTERTAINING APPLICATIONS OF THE CONCEPT OF SET IN COMPUTER SCIENCE COURSE Yordzhev K., Kostadnova H. Some aspects of programmng educaton
More informationA Binarization Algorithm specialized on Document Images and Photos
A Bnarzaton Algorthm specalzed on Document mages and Photos Ergna Kavalleratou Dept. of nformaton and Communcaton Systems Engneerng Unversty of the Aegean kavalleratou@aegean.gr Abstract n ths paper, a
More informationThe Erdős Pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces
Dscrete Mathematcs 307 (2007) 764 768 www.elsever.com/locate/dsc Note The Erdős Pósa property for vertex- and edge-dsjont odd cycles n graphs on orentable surfaces Ken-Ich Kawarabayash a, Atsuhro Nakamoto
More informationK-means and Hierarchical Clustering
Note to other teachers and users of these sldes. Andrew would be delghted f you found ths source materal useful n gvng your own lectures. Feel free to use these sldes verbatm, or to modfy them to ft your
More informationMinimization of the Expected Total Net Loss in a Stationary Multistate Flow Network System
Appled Mathematcs, 6, 7, 793-87 Publshed Onlne May 6 n ScRes. http://www.scrp.org/journal/am http://dx.do.org/.436/am.6.787 Mnmzaton of the Expected Total Net Loss n a Statonary Multstate Flow Networ System
More informationGraph-based Clustering
Graphbased Clusterng Transform the data nto a graph representaton ertces are the data ponts to be clustered Edges are eghted based on smlarty beteen data ponts Graph parttonng Þ Each connected component
More informationOn Some Entertaining Applications of the Concept of Set in Computer Science Course
On Some Entertanng Applcatons of the Concept of Set n Computer Scence Course Krasmr Yordzhev *, Hrstna Kostadnova ** * Assocate Professor Krasmr Yordzhev, Ph.D., Faculty of Mathematcs and Natural Scences,
More informationS1 Note. Basis functions.
S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 5 Luca Trevisan September 7, 2017
U.C. Bereley CS294: Beyond Worst-Case Analyss Handout 5 Luca Trevsan September 7, 207 Scrbed by Haars Khan Last modfed 0/3/207 Lecture 5 In whch we study the SDP relaxaton of Max Cut n random graphs. Quc
More informationProgramming in Fortran 90 : 2017/2018
Programmng n Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Exercse 1 : Evaluaton of functon dependng on nput Wrte a program who evaluate the functon f (x,y) for any two user specfed values
More informationF Geometric Mean Graphs
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 2 (December 2015), pp. 937-952 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) F Geometrc Mean Graphs A.
More informationBridges and cut-vertices of Intuitionistic Fuzzy Graph Structure
Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: http://www.jesm.co.n, Emal: jesmj@gmal.com Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal -
More informationSubspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;
Subspace clusterng Clusterng Fundamental to all clusterng technques s the choce of dstance measure between data ponts; D q ( ) ( ) 2 x x = x x, j k = 1 k jk Squared Eucldean dstance Assumpton: All features
More informationNAG Fortran Library Chapter Introduction. G10 Smoothing in Statistics
Introducton G10 NAG Fortran Lbrary Chapter Introducton G10 Smoothng n Statstcs Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Smoothng Methods... 2 2.2 Smoothng Splnes and Regresson
More informationOptimization Methods: Integer Programming Integer Linear Programming 1. Module 7 Lecture Notes 1. Integer Linear Programming
Optzaton Methods: Integer Prograng Integer Lnear Prograng Module Lecture Notes Integer Lnear Prograng Introducton In all the prevous lectures n lnear prograng dscussed so far, the desgn varables consdered
More informationLecture 4: Principal components
/3/6 Lecture 4: Prncpal components 3..6 Multvarate lnear regresson MLR s optmal for the estmaton data...but poor for handlng collnear data Covarance matrx s not nvertble (large condton number) Robustness
More informationTsinghua University at TAC 2009: Summarizing Multi-documents by Information Distance
Tsnghua Unversty at TAC 2009: Summarzng Mult-documents by Informaton Dstance Chong Long, Mnle Huang, Xaoyan Zhu State Key Laboratory of Intellgent Technology and Systems, Tsnghua Natonal Laboratory for
More informationType-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data
Malaysan Journal of Mathematcal Scences 11(S) Aprl : 35 46 (2017) Specal Issue: The 2nd Internatonal Conference and Workshop on Mathematcal Analyss (ICWOMA 2016) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES
More informationA mathematical programming approach to the analysis, design and scheduling of offshore oilfields
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 A mathematcal programmng approach to the analyss, desgn and
More informationAssignment # 2. Farrukh Jabeen Algorithms 510 Assignment #2 Due Date: June 15, 2009.
Farrukh Jabeen Algorthms 51 Assgnment #2 Due Date: June 15, 29. Assgnment # 2 Chapter 3 Dscrete Fourer Transforms Implement the FFT for the DFT. Descrbed n sectons 3.1 and 3.2. Delverables: 1. Concse descrpton
More informationAn Adjusted Recursive Operator Allocation Optimization Algorithm for Line Balancing Control
IAENG Internatonal Journal of Appled Maematcs, 36:1, IJAM_36_1_5 An Adusted ecursve Operator Allocaton Optmzaton Algorm for Lne Balancng Control B.L. ong, W.K. Wong, J. Fan, and.f. Chan 1 Abstract Ths
More informationOntology Generator from Relational Database Based on Jena
Computer and Informaton Scence Vol. 3, No. 2; May 2010 Ontology Generator from Relatonal Database Based on Jena Shufeng Zhou (Correspondng author) College of Mathematcs Scence, Laocheng Unversty No.34
More informationSupport Vector Machines. CS534 - Machine Learning
Support Vector Machnes CS534 - Machne Learnng Perceptron Revsted: Lnear Separators Bnar classfcaton can be veed as the task of separatng classes n feature space: b > 0 b 0 b < 0 f() sgn( b) Lnear Separators
More informationFrom Comparing Clusterings to Combining Clusterings
Proceedngs of the Twenty-Thrd AAAI Conference on Artfcal Intellgence (008 From Comparng Clusterngs to Combnng Clusterngs Zhwu Lu and Yuxn Peng and Janguo Xao Insttute of Computer Scence and Technology,
More information1 Introducton Gven a graph G = (V; E), a non-negatve cost on each edge n E, and a set of vertces Z V, the mnmum Stener problem s to nd a mnmum cost su
Stener Problems on Drected Acyclc Graphs Tsan-sheng Hsu y, Kuo-Hu Tsa yz, Da-We Wang yz and D. T. Lee? September 1, 1995 Abstract In ths paper, we consder two varatons of the mnmum-cost Stener problem
More informationA Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme
Mathematcal and Computatonal Applcatons Artcle A Fve-Pont Subdvson Scheme wth Two Parameters and a Four-Pont Shape-Preservng Scheme Jeqng Tan,2, Bo Wang, * and Jun Sh School of Mathematcs, Hefe Unversty
More information12/2/2009. Announcements. Parametric / Non-parametric. Case-Based Reasoning. Nearest-Neighbor on Images. Nearest-Neighbor Classification
Introducton to Artfcal Intellgence V22.0472-001 Fall 2009 Lecture 24: Nearest-Neghbors & Support Vector Machnes Rob Fergus Dept of Computer Scence, Courant Insttute, NYU Sldes from Danel Yeung, John DeNero
More informationOutline. Self-Organizing Maps (SOM) US Hebbian Learning, Cntd. The learning rule is Hebbian like:
Self-Organzng Maps (SOM) Turgay İBRİKÇİ, PhD. Outlne Introducton Structures of SOM SOM Archtecture Neghborhoods SOM Algorthm Examples Summary 1 2 Unsupervsed Hebban Learnng US Hebban Learnng, Cntd 3 A
More informationComputation of a Minimum Average Distance Tree on Permutation Graphs*
Annals of Pure and Appled Mathematcs Vol, No, 0, 74-85 ISSN: 79-087X (P), 79-0888(onlne) Publshed on 8 December 0 wwwresearchmathscorg Annals of Computaton of a Mnmum Average Dstance Tree on Permutaton
More informationChapter 6 Programmng the fnte element method Inow turn to the man subject of ths book: The mplementaton of the fnte element algorthm n computer programs. In order to make my dscusson as straghtforward
More informationLECTURE NOTES Duality Theory, Sensitivity Analysis, and Parametric Programming
CEE 60 Davd Rosenberg p. LECTURE NOTES Dualty Theory, Senstvty Analyss, and Parametrc Programmng Learnng Objectves. Revew the prmal LP model formulaton 2. Formulate the Dual Problem of an LP problem (TUES)
More informationAn Approach in Coloring Semi-Regular Tilings on the Hyperbolic Plane
An Approach n Colorng Sem-Regular Tlngs on the Hyperbolc Plane Ma Louse Antonette N De Las Peñas, mlp@mathscmathadmueduph Glenn R Lago, glago@yahoocom Math Department, Ateneo de Manla Unversty, Loyola
More informationAnalysis of Continuous Beams in General
Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support,
More informationCSE 326: Data Structures Quicksort Comparison Sorting Bound
CSE 326: Data Structures Qucksort Comparson Sortng Bound Steve Setz Wnter 2009 Qucksort Qucksort uses a dvde and conquer strategy, but does not requre the O(N) extra space that MergeSort does. Here s the
More informationRough Neutrosophic Multisets Relation with Application in Marketing Strategy
Neutrosophc Sets and Systems, Vol. 21, 2018 Unversty of New Mexco 36 Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy Surana Alas 1, Daud Mohamad 2, and Adbah Shub 3 1 Faculty of Computer
More informationClassification / Regression Support Vector Machines
Classfcaton / Regresson Support Vector Machnes Jeff Howbert Introducton to Machne Learnng Wnter 04 Topcs SVM classfers for lnearly separable classes SVM classfers for non-lnearly separable classes SVM
More informationMathematics 256 a course in differential equations for engineering students
Mathematcs 56 a course n dfferental equatons for engneerng students Chapter 5. More effcent methods of numercal soluton Euler s method s qute neffcent. Because the error s essentally proportonal to the
More informationComplex Numbers. Now we also saw that if a and b were both positive then ab = a b. For a second let s forget that restriction and do the following.
Complex Numbers The last topc n ths secton s not really related to most of what we ve done n ths chapter, although t s somewhat related to the radcals secton as we wll see. We also won t need the materal
More informationCombinatorial Auctions with Structured Item Graphs
Combnatoral Auctons wth Structured Item Graphs Vncent Contzer and Jonathan Derryberry and Tuomas Sandholm Carnege Mellon Unversty 5000 Forbes Avenue Pttsburgh, PA 15213 {contzer, jonderry, sandholm}@cs.cmu.edu
More informationConstructing Minimum Connected Dominating Set: Algorithmic approach
Constructng Mnmum Connected Domnatng Set: Algorthmc approach G.N. Puroht and Usha Sharma Centre for Mathematcal Scences, Banasthal Unversty, Rajasthan 304022 usha.sharma94@yahoo.com Abstract: Connected
More informationWavefront Reconstructor
A Dstrbuted Smplex B-Splne Based Wavefront Reconstructor Coen de Vsser and Mchel Verhaegen 14-12-201212 2012 Delft Unversty of Technology Contents Introducton Wavefront reconstructon usng Smplex B-Splnes
More informationR s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges
More informationHarvard University CS 101 Fall 2005, Shimon Schocken. Assembler. Elements of Computing Systems 1 Assembler (Ch. 6)
Harvard Unversty CS 101 Fall 2005, Shmon Schocken Assembler Elements of Computng Systems 1 Assembler (Ch. 6) Why care about assemblers? Because Assemblers employ some nfty trcks Assemblers are the frst
More informationMeta-heuristics for Multidimensional Knapsack Problems
2012 4th Internatonal Conference on Computer Research and Development IPCSIT vol.39 (2012) (2012) IACSIT Press, Sngapore Meta-heurstcs for Multdmensonal Knapsack Problems Zhbao Man + Computer Scence Department,
More informationNUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS
ARPN Journal of Engneerng and Appled Scences 006-017 Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana
More informationApproximating Clique and Biclique Problems*
Ž. JOURNAL OF ALGORITHMS 9, 17400 1998 ARTICLE NO. AL980964 Approxmatng Clque and Bclque Problems* Dort S. Hochbaum Department of Industral Engneerng and Operatons Research and Walter A. Haas School of
More informationMULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION
MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION Paulo Quntlano 1 & Antono Santa-Rosa 1 Federal Polce Department, Brasla, Brazl. E-mals: quntlano.pqs@dpf.gov.br and
More informationAn Efficient Genetic Algorithm with Fuzzy c-means Clustering for Traveling Salesman Problem
An Effcent Genetc Algorthm wth Fuzzy c-means Clusterng for Travelng Salesman Problem Jong-Won Yoon and Sung-Bae Cho Dept. of Computer Scence Yonse Unversty Seoul, Korea jwyoon@sclab.yonse.ac.r, sbcho@cs.yonse.ac.r
More informationOn Two Segmentation Problems
Journal of Algorthms 33, 173184 Ž 1999. Artcle ID jagm.1999.1024, avalable onlne at http:www.dealbrary.com on On Two Segmentaton Problems Noga Alon* Department of Mathematcs, Raymond and Beerly Sacler
More informationCOPS AND ROBBER WITH CONSTRAINTS
COPS AND ROBBER WITH CONSTRAINTS FEDOR V. FOMIN, PETR A. GOLOVACH, AND PAWE L PRA LAT Abstract. Cops & Robber s a classcal pursut-evason game on undrected graphs, where the task s to dentfy the mnmum number
More informationPolyhedral Compilation Foundations
Polyhedral Complaton Foundatons Lous-Noël Pouchet pouchet@cse.oho-state.edu Dept. of Computer Scence and Engneerng, the Oho State Unversty Feb 8, 200 888., Class # Introducton: Polyhedral Complaton Foundatons
More informationA new paradigm of fuzzy control point in space curve
MATEMATIKA, 2016, Volume 32, Number 2, 153 159 c Penerbt UTM Press All rghts reserved A new paradgm of fuzzy control pont n space curve 1 Abd Fatah Wahab, 2 Mohd Sallehuddn Husan and 3 Mohammad Izat Emr
More information