Algorithms of informatics, vol. 3 PDF
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ALGORITHMS OF INFORMATICS Volume 3 AnTonCom Budapest, 2011 This electronic book was prepared in the framework of project Eastern Hungarian Informatics Books Repository no. TÁMOP-4.1.2-08/1/A-2009-0046. This electronic book appeared with the support of European Union and with the co-financing of European Social Fund. Editor: Antal Iványi Authors of Volume 3: Béla Vizvári (Chapter 24), Antal Iványi and Shariefuddin Pirzada (Chapter 25), Zoltán Kása, and Mira-Cristiana Anisiu (Chapter 26), Ferenc Szidarovszky and László Domoszlai, (Chapter 27), László Szirmay-Kalos and László Szécsi (Chapter 28), Antal Iványi (Chapter 29), Shariefuddin Pirzada, Antal Iványi and Muhammad Ali Khan (Chapter 30) Validators of Volume 3: Gergely Kovács (Chapter 24), Zoltán Kása (Chapter 25), Antal Iványi (Chapter 26), Sándor Molnár (Chapter 27), György Antal (Chapter 28), Zoltán Kása (Chapter 29), Zoltán Kása (Chapter 30), Anna Iványi (Bibliography) c 2011 AnTonCom Infokommunikációs Kft. (cid:13) Homepage: http://www.antoncom.hu/ Contents Introduction to Volume 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 24.The Branch and Bound Method . . . . . . . . . . . . . . . . . . . . . 1208 24.1. An example: the Knapsack Problem . . . . . . . . . . . . . . . . . 1208 24.1.1. The Knapsack Problem . . . . . . . . . . . . . . . . . . . . 1209 24.1.2. A numerical example . . . . . . . . . . . . . . . . . . . . . . 1211 24.1.3. Properties in the calculation of the numerical example . . . 1214 24.1.4. How to accelerate the method . . . . . . . . . . . . . . . . . 1216 24.2. The general frame of the B&B method . . . . . . . . . . . . . . . . 1217 24.2.1. Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 24.2.2. The general frame of the B&B method . . . . . . . . . . . . 1224 24.3. Mixed integer programming with bounded variables. . . . . . . . . 1229 24.3.1. The geometric analysis of a numerical example . . . . . . . 1230 24.3.2. The linear programming background of the method . . . . . 1232 24.3.3. Fast bounds on lower and upper branches . . . . . . . . . . 1240 24.3.4. Branching strategies . . . . . . . . . . . . . . . . . . . . . . 1244 24.3.5. The selection of the branching variable . . . . . . . . . . . . 1247 24.3.6. The numerical example is revisited . . . . . . . . . . . . . . 1248 24.4. On the enumeration tree . . . . . . . . . . . . . . . . . . . . . . . . 1252 24.5. The use of information obtained from other sources . . . . . . . . . 1254 24.5.1. Application of heuristic methods . . . . . . . . . . . . . . . 1254 24.5.2. Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . 1255 24.6. Branch and Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255 24.7. Branch and Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 25. Comparison Based Ranking . . . . . . . . . . . . . . . . . . . . . . . 1262 25.1. Introduction to supertournaments . . . . . . . . . . . . . . . . . . . 1262 25.2. Introduction to (a,b)-tournaments . . . . . . . . . . . . . . . . . . 1264 25.3. Existence of (1,1)-tournaments with prescribed score sequence. . . 1265 25.4. Existence of an (a,a)-tournament with prescribed score sequence . 1267 25.5. Existence of an (a,b)-tournament with prescribed score sequence . 1269 25.5.1. Existence of a tournament with arbitrary degree sequence . 1269 25.5.2. Description of a naive reconstructing algorithm . . . . . . . 1269 25.5.3. Computation of e . . . . . . . . . . . . . . . . . . . . . . . . 1270 25.5.4. Description of a construction algorithm . . . . . . . . . . . 1271 1204 Contents 25.5.5. Computation of f and g . . . . . . . . . . . . . . . . . . . . 1271 25.5.6. Description of a testing algorithm . . . . . . . . . . . . . . . 1273 25.5.7. Description of an algorithm computing f and g . . . . . . . 1273 25.5.8. Computing of f and g in linear time . . . . . . . . . . . . . 1275 25.5.9. Tournament with f and g . . . . . . . . . . . . . . . . . . . 1276 25.5.10.Description of the score slicing algorithm. . . . . . . . . . . 1276 25.5.11.Analysis of the minimax reconstruction algorithm . . . . . . 1280 25.6. Imbalances in (0,b)-tournaments . . . . . . . . . . . . . . . . . . . 1280 25.6.1. Imbalances in (0,1)-tournaments. . . . . . . . . . . . . . . . 1281 25.6.2. Imbalances in (0,2)-tournaments . . . . . . . . . . . . . . . 1281 25.7. Supertournaments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 25.7.1. Hypertournaments . . . . . . . . . . . . . . . . . . . . . . . 1286 25.7.2. Supertournaments . . . . . . . . . . . . . . . . . . . . . . . 1293 25.8. Football tournaments . . . . . . . . . . . . . . . . . . . . . . . . . . 1294 25.8.1. Testing algorithms . . . . . . . . . . . . . . . . . . . . . . . 1294 25.8.2. Polynomial testing algorithms of the draw sequences . . . . 1303 25.9. Reconstruction of the tested sequences . . . . . . . . . . . . . . . . 1309 26.Complexity of Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 26.1. Simple complexity measures . . . . . . . . . . . . . . . . . . . . . . 1312 26.1.1. Finite words . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 26.1.2. Infinite words . . . . . . . . . . . . . . . . . . . . . . . . . . 1314 26.1.3. Word graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 1316 26.1.4. Complexity of words . . . . . . . . . . . . . . . . . . . . . . 1321 26.2. Generalized complexity measures . . . . . . . . . . . . . . . . . . . 1332 26.2.1. Rainbow words . . . . . . . . . . . . . . . . . . . . . . . . . 1332 26.2.2. General words . . . . . . . . . . . . . . . . . . . . . . . . . . 1343 26.3. Palindrome complexity . . . . . . . . . . . . . . . . . . . . . . . . . 1343 26.3.1. Palindromes in finite words . . . . . . . . . . . . . . . . . . 1343 26.3.2. Palindromes in infinite words . . . . . . . . . . . . . . . . . 1346 27. Conflict Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351 27.1. The basics of multi-objective programming . . . . . . . . . . . . . . 1351 27.1.1. Applications of utility functions . . . . . . . . . . . . . . . . 1355 27.1.2. Weighting method . . . . . . . . . . . . . . . . . . . . . . . 1357 27.1.3. Distance-dependent methods . . . . . . . . . . . . . . . . . 1358 27.1.4. Direction-dependent methods . . . . . . . . . . . . . . . . . 1361 27.2. Method of equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 1364 27.3. Methods of cooperative games . . . . . . . . . . . . . . . . . . . . . 1369 27.4. Collective decision-making . . . . . . . . . . . . . . . . . . . . . . . 1372 27.5. Applications of Pareto games . . . . . . . . . . . . . . . . . . . . . 1379 27.6. Axiomatic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382 28. General Purpose Computing on Graphics Processing Units . . . 1388 28.1. The graphics pipeline model . . . . . . . . . . . . . . . . . . . . . . 1390 28.1.1. GPU as the implementation of incremental image synthesis 1392 28.2. GPGPU with the graphics pipeline model . . . . . . . . . . . . . . 1395 28.2.1. Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395 Contents 1205 28.2.2. Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396 28.2.3. Functions and parameters . . . . . . . . . . . . . . . . . . . 1397 28.3. GPU as a vector processor . . . . . . . . . . . . . . . . . . . . . . . 1398 28.3.1. Implementing the SAXPY BLAS function . . . . . . . . . . 1400 28.3.2. Image filtering . . . . . . . . . . . . . . . . . . . . . . . . . 1401 28.4. Beyond vector processing. . . . . . . . . . . . . . . . . . . . . . . . 1402 28.4.1. SIMD or MIMD . . . . . . . . . . . . . . . . . . . . . . . . 1402 28.4.2. Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404 28.4.3. Implementing scatter . . . . . . . . . . . . . . . . . . . . . . 1405 28.4.4. Parallelism versus reuse . . . . . . . . . . . . . . . . . . . . 1407 28.5. GPGPU programming model: CUDA and OpenCL . . . . . . . . . 1409 28.6. Matrix-vector multiplication . . . . . . . . . . . . . . . . . . . . . . 1409 28.6.1. Making matrix-vector multiplication more parallel . . . . . 1411 28.7. Case study: computational fluid dynamics . . . . . . . . . . . . . . 1414 28.7.1. Eulerian solver for fluid dynamics . . . . . . . . . . . . . . . 1416 28.7.2. Lagrangian solver for differential equations. . . . . . . . . . 1421 29. Perfect Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428 29.1. Basic concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428 29.2. Necessary condition and earlier results . . . . . . . . . . . . . . . . 1431 29.3. One-dimensional perfect arrays . . . . . . . . . . . . . . . . . . . . 1431 29.3.1. Pseudocode of the algorithm Quick-Martin . . . . . . . . 1431 29.3.2. Pseudocode of the algorithm Optimal-Martin. . . . . . . 1432 29.3.3. Pseudocode of the algorithm Shift . . . . . . . . . . . . . . 1432 29.3.4. Pseudocode of the algorithm Even . . . . . . . . . . . . . . 1433 29.4. Two-dimensional perfect arrays . . . . . . . . . . . . . . . . . . . . 1433 29.4.1. Pseudocode of the algorithm Mesh . . . . . . . . . . . . . . 1434 29.4.2. Pseudocode of the algorithm Cellular . . . . . . . . . . . 1434 29.5. Three-dimensional perfect arrays . . . . . . . . . . . . . . . . . . . 1435 29.5.1. Pseudocode of the algorithm Colour . . . . . . . . . . . . 1435 29.5.2. Pseudocode of the algorithm Growing . . . . . . . . . . . 1436 29.6. Construction of growing arrays using colouring . . . . . . . . . . . 1437 29.6.1. Construction of growing sequences . . . . . . . . . . . . . . 1437 29.6.2. Construction of growing squares. . . . . . . . . . . . . . . . 1438 29.6.3. Construction of growing cubes. . . . . . . . . . . . . . . . . 1439 29.6.4. Construction of a four-dimensional double hypercube . . . . 1439 29.7. The existence theorem of perfect arrays . . . . . . . . . . . . . . . 1441 29.8. Superperfect arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443 29.9. d-complexity of one-dimensional arrays . . . . . . . . . . . . . . . . 1443 29.9.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444 29.9.2. Bounds of complexity measures . . . . . . . . . . . . . . . . 1446 29.9.3. Recurrence relations . . . . . . . . . . . . . . . . . . . . . . 1448 29.9.4. Pseudocode of the algorithm Quick-Martin . . . . . . . . 1450 29.9.5. Pseudocode of algorithm d-Complexity . . . . . . . . . . . 1450 29.9.6. Pseudocode of algorithm Super . . . . . . . . . . . . . . . 1451 29.9.7. Pseudocode of algorithm MaxSub . . . . . . . . . . . . . . 1451 29.9.8. Construction and complexity of extremal words . . . . . . . 1452 1206 Contents 29.10.Finite two-dimensional arrays with maximal complexity . . . . . . 1456 29.10.1.Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456 29.10.2.Bounds of complexity functions . . . . . . . . . . . . . . . . 1458 29.10.3.Properties of the maximal complexity function . . . . . . . 1459 29.10.4.On the existence of maximal arrays . . . . . . . . . . . . . . 1460 30. Score Sets and Kings . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463 30.1. Score sets in 1-tournaments . . . . . . . . . . . . . . . . . . . . . . 1464 30.1.1. Determining the score set . . . . . . . . . . . . . . . . . . . 1464 30.1.2. Tournaments with prescribed score set . . . . . . . . . . . . 1466 30.2. Score sets in oriented graphs . . . . . . . . . . . . . . . . . . . . . . 1473 30.2.1. Oriented graphs with prescribed scoresets . . . . . . . . . . 1475 30.3. Unicity of score sets . . . . . . . . . . . . . . . . . . . . . . . . . . 1480 30.3.1. 1-unique score sets . . . . . . . . . . . . . . . . . . . . . . . 1481 30.3.2. 2-unique score sets . . . . . . . . . . . . . . . . . . . . . . . 1482 30.4. Kings and serfs in tournaments . . . . . . . . . . . . . . . . . . . . 1484 30.5. Weak kings in oriented graphs . . . . . . . . . . . . . . . . . . . . . 1492 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518 Name Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 Introduction The third volume contains seven new chapters. Chapter 24 (The Branch and Bound Method) was written by Béla Vizvári (Eastern Mediterrean University), Chapter 25 (Comparison Based Ranking) by Antal Iványi (Eötvös Loránd University) and Shariefuddin Pirzada (University of Kashmir), Chapter 26 (Complexity of Words) by Zoltán Kása, (Sapientia Hungar- ian University of Transylvania) and Mira-Cristiana Anisiu (Tiberiu Popovici Insti- tute of Numerical Mathematics), Chapter 27 (Conflict Situations) by Ferenc Szi- darovszky(UniversityofArizona)andbyLászlóDomoszlai(EötvösLorándUniver- sity), Chapter 28 (General Purpose Computing on Graphics Processing Units) by László Szirmay-Kalos and László Szécsi (both Budapest University of Technology and Economics), Chapter 29 (Perfect Arrays) by Antal Iványi (Eötvös Loránd Uni- versity), and Chapter 30 by Shariefuddin Pirzada (University of Kashmir), Antal Iványi (Eötvös Loránd University) and Muhammad Ali Khan (King Fahd Univer- sity). The LATEX style file was written by Viktor Belényesi, Zoltán Csörnyei, László Domoszlai and Antal Iványi. The figures was drawn or corrected by Kornél Locher. and László Domoszlai. Anna Iványi transformed the bibliography into hypertext. The DOCBOOK version was made by Marton 2001. Kft. Usingthedataofthecolofonpageyoucancontactwithanyofthecreatorsofthe book. We welcome ideas for new exercises and problems, and also critical remarks or bug reports. The publication of the printed book (Volumes 1 and 2) was supported by De- partment of Mathematics of Hungarian Academy of Science. This electronic book (Volumes 1, 2, and 3) was prepared in the framework of project Eastern Hungarian Informatics Books Repository no. TÁMOP-4.1.2-08/1/A-2009-0046. This electronic book appeared with the support of European Union and with the co-financing of European Social Fund. Budapest, September 2011 Antal Iványi ([email protected]) 24. The Branch and Bound Method Ithasseriouspracticalconsequencesifitisknownthatacombinatorial problemis NP-complete. Then one can conclude according to the present state of science that no simple combinatorial algorithm can be applied and only an enumerative-type method can solve the problem in question. Enumerative methods are investigating many cases only in a non-explicit, i.e. implicit, way. It means that huge majority of the cases are dropped based on consequences obtained from the analysis of the particular numerical problem. The three most important enumerative methods are (i) implicit enumeration, (ii) dynamic programming, and (iii) branch and bound method.Thischapterisdevotedtothelatterone.Implicitenumerationanddynamic programmingcanbeappliedwithinthefamilyofoptimizationproblemsmainlyifall variableshavediscretenature.Branchandboundmethodcaneasilyhandleproblems having both discrete and continuous variables. Further on the techniques of implicit enumerationcanbeincorporatedeasilyinthebranchandboundframe.Branchand bound method can be applied even in some cases of nonlinear programming. TheBranch and Bound(abbreviatedfurtheronasB&B)methodisjustaframeofa large family of methods. Its substeps can be carried out in different ways depending on the particular problem, the available software tools and the skill of the designer of the algorithm. Boldface letters denote vectors and matrices; calligraphic letters are used for sets. Components of vectors are denoted by the same but non-boldface letter. Cap- ital letters are used for matrices and the same but lower case letters denote their elements. The columns of a matrix are denoted by the same boldface but lower case letters. Someformulaewiththeirnumbersarerepeatedseveraltimesinthischapter.The reason is that always a complete description of optimization problems is provided. Thus the fact that the number of a formula is repeated means that the formula is identical to the previous one. 24.1. An example: the Knapsack Problem In this section the branch and bound method is shown on a numerical example. The problem is a sample of the binary knapsack problem which is one of the easiest 24.1. An example: the Knapsack Problem 1209 problems of integer programming but it is still NP-complete. The calculations are carried out in a brute force way to illustrate all features of B&B. More intelligent calculations, i.e. using implicit enumeration techniques will be discussed only at the end of the section. 24.1.1. The Knapsack Problem Therearemanydifferentknapsackproblems.Thefirstandclassicaloneisthebinary knapsack problem. It has the following story. A tourist is planning a tour in the mountains.Hehasalotofobjectswhichmaybeusefulduringthetour.Forexample ice pick and can opener can be among the objects. We suppose that the following conditions are satisfied. Eachobjecthasapositivevalueandapositiveweight.(E.g.aballoonfilledwith • helium has a negative weight. See Exercises 24.1-1 and 24.1-2) The value is the degree of contribution of the object to the success of the tour. The objects are independent from each other. (E.g. can and can opener are not • independent as any of them without the other one has limited value.) The knapsack of the tourist is strong and large enough to contain all possible • objects. The strength of the tourist makes possible to bring only a limited total weight. • Butwithinthisweightlimitthetouristwanttoachievethemaximaltotalvalue. • The following notations are used to the mathematical formulation of the prob- lem: n the number of objects; j the index of the objects; w the weight of objectj; j v the value of objectj; j b the maximal weight what the tourist can bring. For each object j a so-called binary or zero-one decision variable, say x , is j introduced: 1 if object j is present on the tour x = j 0 if object j isn’t present on the tour. (cid:26) Notice that w if object j is present on the tour, w x = j j j 0 if object j isn’t present on the tour (cid:26) is the weight of the object in the knapsack. Similarly v x is the value of the object on the tour. The total weight in the j j knapsack is n w x j j j=1 X 1210 24. The Branch and Bound Method whichmaynotexceedtheweightlimit.Hencethemathematicalformoftheproblem is n max v x (24.1) j j j=1 X n w x b (24.2) j j ≤ j=1 X x =0or1, j =1,...,n. (24.3) j The difficulty of the problem is caused by the integrality requirement. If con- straint (24.3) is substituted by the relaxed constraint, i.e. by 0 x 1, j =1,...,n, (24.4) j ≤ ≤ then the Problem (29.2), (24.2), and (24.4) is a linear programming problem. (24.4) means that not only a complete object can be in the knapsack but any part of it. MoreoveritisnotnecessarytoapplythesimplexmethodoranyotherLPalgorithm to solve it as its optimal solution is described by Theorem 24.1 Suppose thatthe numbers v ,w (j =1,...,n) are allpositive and j j moreover the index order satisfies the inequality v v v 1 2 n . (24.5) w ≥ w ···≥ w 1 2 n Then there is an index p (1 p n) and an optimal solution x such that ∗ ≤ ≤ x =x = =x =1, x =x = =x =0. ∗1 ∗2 ··· ∗p−1 ∗p+1 ∗p+2 ··· ∗p+1 Noticethatthereisonlyatmostonenon-integercomponentinx .Thisproperty ∗ will be used at the numerical calculations. From the point of view of B&B the relation of the Problems (29.2), (24.2), and (24.3) and (29.2), (24.2), and (24.4) is very important. Any feasible solution of the first one is also feasible in the second one. But the opposite statement is not true. In other words the set of feasible solutions of the first problem is a proper subset of the feasible solutions of the second one. This fact has two important consequences: The optimal value of the Problem (29.2), (24.2), and (24.4) is an upper bound • of the optimal value of the Problem (29.2), (24.2), and (24.3). IftheoptimalsolutionoftheProblem(29.2),(24.2),and(24.4)isfeasibleinthe • Problem (29.2), (24.2), and (24.3) then it is the optimal solution of the latter problem as well. Thesepropertiesareusedinthecourseofthebranchandboundmethodintensively.
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