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#### Algorithms_Part5

Course: IT 367, Spring 2011

School: King Abdulaziz University

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Dasgupta, S. C.H. Papadimitriou, and U.V. Vazirani 81 5. And nally, notice that the FFT circuit is a natural for parallel computation and direct implementation in hardware. Figure 2.10 The fast Fourier transform circuit. a2 A( 1 ) 001 A( 2 ) 010 A( 3 ) 011 A( 4 ) 100 A( 5 ) 101 6 A( 6 ) 110 7 A( 7 ) 111 () 4 4 67 &' 010 000 45 a4 100 A( 0 ) 01 a0 23 000 2 a1 6 4...

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King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani61Figure 2.4 The sequence of merge operations in mergesort.Input: 10 21022333352 10575213135661617 135 1017 13 167 10 1367 13.function merge(x[1 . . . k ], y [1 . . . l])
King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani41An application of number theory?The renowned mathematician G. H. Hardy once declared of his work: I have never doneanything useful. Hardy was an expert in the theory of numbers, which has long been r
King Abdulaziz University - IT - 367
Chapter 1Algorithms with numbersOne of the main themes of this chapter is the dramatic contrast between two ancient problemsthat at rst seem very similar:Factoring: Given a number N , express it as a product of its prime factors.Primality: Given a nu
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AlgorithmsCopyright c 2006 S. Dasgupta, C. H. Papadimitriou, and U. V. VaziraniJuly 18, 20062AlgorithmsContentsPreface0 Prologue0.1 Books and algorithms0.2 Enter Fibonacci . . .0.3 Big-O notation . . . .Exercises . . . . . . . . . .9.....
King Abdulaziz University - IT - 367
Chapter 7Linear programming andreductionsMany of the problems for which we want algorithms are optimization tasks: the shortest path,the cheapest spanning tree, the longest increasing subsequence, and so on. In such cases, weseek a solution that (1)
King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani221Lets investigate the issue by describing what we expect of these three multipliers, callthem y1 , y2 , y3 .MultiplierInequalityy1x1 200y2x2 300y3x1 + x2 400To start with, these yi s must be
King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani2417.12. For the linear programmax x1 2x3x1 x 2 12x2 x3 1x1 , x 2 , x 3 0prove that the solution (x1 , x2 , x3 ) = (3/2, 1/2, 0) is optimal.7.13. Matching pennies. In this simple two-player game, t
King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani261FactoringOne last point: we started off this book by introducing another famously hard search problem:FACTORING, the task of nding all prime factors of a given integer. But the difculty ofFACTORING
King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani281In this case, ingredients 2 and 3 go together pretty well whereas 1 and 5 clash badly. Notice thatthis matrix is necessarily symmetric; and that the diagonal entries are always 0.0. Any set ofingred
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S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani301Figure 9.8 Local search.Figure 9.7 shows a specic example of local search at work. Figure 9.8 is a more abstract,stylized depiction of local search. The solutions crowd the unshaded area, and cost d
King Abdulaziz University - IT - 367
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani321Lets make this more precise.Lemma Suppose s independent samples are drawn uniformly from0,M 2M(k 1)M,,.,.kkkThen with probability at least 1 k/2 s , the greatest common divisor of these samp
King Abdulaziz University - IT - 367
Algorithms and Data StructuresAlfred Strohmeier alfred.strohmeier@epfl.ch http:/lglwww.epfl.chMarch 2000Swiss Federal Institute of Technology in Lausanne Department of Computer Science Software Engineering LaboratoryAlgorithms and Data Structures 1995
King Abdulaziz University - IT - 367
Unit GTBasic Concepts in Graph TheorySection 1: What is a Graph?There are various types of graphs, each with its own definition. Unfortunately, some people apply the term &quot;graph&quot; rather loosely, so you can't be sure what type of graph they're talking a
King Abdulaziz University - IT - 367
Computational Modelingand Complexity ScienceVersion 0.0.10Computational Modelingand Complexity ScienceVersion 0.0.10Allen DowneyGreen Tea PressNeedham, MassachusettsCopyright 2008 Allen Downey.Printing history:Fall 2008: First edition.Green Te
King Abdulaziz University - IT - 367
Chapter 5Cellular AutomataA cellular automaton is a model of a world with very simple physics. Cellular means that thespace is divided into discrete chunks, called cells. An automaton is a machine that performscomputationsit could be a real machine, b
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Jrgen Bang-Jensen, Gregory GutinDigraphsTheory, Algorithms andApplications15th August 2007Springer-VerlagBerlin Heidelberg NewYorkLondon Paris TokyoHong Kong BarcelonaBudapestWe dedicate this book to our parents, especially to our fathers, Brge
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1.2 Digraphs, Subdigraphs, Neighbours, Degreesx3vzuywFigure 1.1 A digraph Di.e. u is adjacent to1 v and v is adjacent to u. If (u, v ) is an arc, we also saythat u dominates v (or v is dominated by u) and denote it by uv . Wesay that a vertex u
King Abdulaziz University - IT - 367
1.7 Mixed Graphs and Hypergraphs23A mixed graph M = (V, A, E ) contains both arcs (ordered pairs ofvertices in A) and edges (unordered pairs of vertices in E ). We do not allowloops or parallel arcs and edges, but M may have an edge and an arc with th
King Abdulaziz University - IT - 367
1.11 Exercises431.67. 4-kings in bipartite tournaments. A vertex v in a digraph D is a k-king,if for every u V (D) cfw_v there is a (v, u)-path of length at most k. Provethat a vertex of maximum out-degree in a strong bipartite tournament is a4-king
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2.6 Minimum Diameter of Orientations of Multigraphs632.6 Minimum Diameter of Orientations of MultigraphsThe same complexity result holds for the following problem: nd a minimumdiameter orientation of a graph. Indeed, the following assertion holds.The
King Abdulaziz University - IT - 367
2.12 Application: Exponential Neighbourhood Local Search for the TSP83is normally far from optimal but is much better than a random solution7 .(Some construction heuristics for the TSP are described later in this book.)In the second phase, a local sea
King Abdulaziz University - IT - 367
3.2 Reductions Among Dierent Flow Models103mean the obvious thing when b(v ) = 0 and if b(v ) &gt; 0 (b(v ) &lt; 0) we think ofl (v ), u (v ), c (v ) as bounds and costs per unit on the total amount of owout of (in to) v .Let DST be the digraph obtained fr
King Abdulaziz University - IT - 367
3.7 Unit Capacity Networks and Simple Networks123Lemma 3.7.2 Let L = (V = V0 V1 . . . Vk , A, l 0, u 1) be a layeredunit capacity network with V0 = cfw_s and Vk = cfw_t. One can nd a blocking(s, t)-ow in L in time O(m).Proof: It suces to see that the
King Abdulaziz University - IT - 367
3.11 Applications of Flows143of value M in N : xsvi = ai , xvi t = bi , for each i = 1, 2, . . . , n and xvi vjequals one if vi vj A and zero otherwise.Suppose now that x is an integer (s, t)-ow of value M in N and letA = cfw_vi vj : xvi vj = 1. Then
King Abdulaziz University - IT - 367
3.13 Exercises163do augment x along P ;6.C := C/27. return xProve that the algorithm correctly determines a maximum ow in theinput network N .(d) Argue that every time Step 4 is performed the residual capacity of everyminimum (s, t)-cut is at mos
King Abdulaziz University - IT - 367
4.5 Line Digraphs2331243445125H1832554QFigure 4.4 A digraph H and its line digraph Q = L(H ).[419] by Beineke and Hemminger. The proof presented here is adapted from[419]. For an n n-matrix M = [mik ], a row i is orthogonal to a row j if
King Abdulaziz University - IT - 367
4.11 Locally Semicomplete Digraphs203digraphs rst proved by Bang-Jensen, Guo, Gutin and Volkmann [55]. Inthe process of deriving this classication, we will show several importantproperties of locally semicomplete digraphs. We start our consideration f
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4.15 Application: Gaussian Elimination223V (Di ) = cfw_v(ni +1) , v(ni +2) , . . . , v(ni+1 ) .It is easy to see that B has (n1 , . . . , np )-block-triangular structure. This implies that A has block-triangular structure. The above observation suggest
King Abdulaziz University - IT - 367
5.6 Hamilton Cycles and Paths in Degree-Constrained Digraphs243Short proofs of Meyniels theorem were given by Overbeck-Larisch [597]and Bondy and Thomassen [128]. The second proof is slightly simpler thanthe rst one and can also be found in the book [
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5.7 Longest Paths and Cycles in Semicomplete Multipartite Digraphs263Corollary 5.7.25 [744] If a k -strong semicomplete multipartite digraph Dhas at most k vertices in each partite set, then D contains a Hamilton cycle.Corollary 5.7.26 [744] A k -stro
King Abdulaziz University - IT - 367
6.1 Hamiltonian Paths with a Prescribed End-Vertex283Theorem 6.1.2 [66] Let D = (V, A) be a digraph which is either semicomplete bipartite or extended locally out-semicomplete and let x V . Then Dhas a hamiltonian path starting at x if and only if D co
King Abdulaziz University - IT - 367
6.5 Pancyclicity of Digraphs303(i = 0, 1, 2) contains a directed path of length 2, then D contains no cycle oflength 5.Now we prove the suciency of the condition in (a). According to Theorem 4.8.5, there exists a semicomplete digraph T on k vertices f
King Abdulaziz University - IT - 367
6.9 Oriented Hamiltonian Paths and Cycles123456789101132312Figure 6.6 An oriented path with intervals [1, 3], [3, 6], [6, 7], [7, 8], [8, 10], [10, 11],[11, 12].For every set X V in a tournament T = (V, A), we dene the setsR+ (X ) (R (X )
King Abdulaziz University - IT - 367
6.13 Exercises3436.53. Let Dr be the digraph which is dened in the end of Subsection 6.10.2.Show that every strong spanning subdigraph of Dr has cyclomatic numberat least 2r 1. Next show that every cyclic spanning subdigraph of Dr withcyclomatic numb
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7.6 Increasing the Arc-Strong Connectivity Optimally363arcs. First let us make the simple observation that such a set F indeed exists,since we may just add k parallel arcs in both directions between a xed vertexv V and all other vertices in V (it is e
King Abdulaziz University - IT - 367
7.10 Minimally k-(Arc)-Strong Directed Multigraphs383precisely one minimal k -out-critical set Xu and enters precisely one minimalk -in-critical set Yu . Here minimal means with respect to inclusion.Lemma 7.10.2 If X, Y are crossing k -in-critical set
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7.15 Packing Cuts403Let C = cfw_Z1 , Z2 , . . . , ZM and let Ai = (Zi , V Zi ), i = 1, 2, . . . , M bethe corresponding arc sets. Construct an undirected graph G(C ) = (V, E ) asfollows: V = cfw_v1 , v2 , . . . , vM and there is an edge between vi a
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8.1 Underlying Graphs of Various Classes of DigraphsAiAjApBiBjBpCj423CpCiFigure 8.3 Implication classes for orientations of a graph G as a local tournamentdigraph.The sets Ci , Cj , Cp denote distinct connected components of G. For eachcompon
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8.6 Orientations Achieving High Arc-Strong Connectivity4438.6 Orientations Achieving High Arc-StrongConnectivityLet us recall that an orientation D of a multigraph G = (V, E ) is obtained byassigning one of the two possible orientations to each edge
King Abdulaziz University - IT - 367
8.9 Orientations of Mixed Graphs463It is not dicult to see that one can formulate the problem of orienting amixed graph so as to get a k -arc-strong directed multigraph as a submodularow problem. We can use the same approach as in Subsection 8.8.4. Th
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9.2 Disjoint Path Problems483Theorem 9.2.7 [545] Let D = (V, A) be a digraph of order n and let k be aninteger such that n 2k 2. If |A| n(n 2) + 2k then D is k -linked.The proof of Theorem 9.2.7 in [545] is based on the following lemma.Lemma 9.2.8 [5
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9.5 Arc-Disjoint Branchings503above was two. Bang-Jensen, Frank and Jackson proved that, if (z, x) kholds for those vertices x V (D) for which d+ (x) &gt; d (x) (that is, the valueof k is restricted by the local arc-connectivities from z to these vertice
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9.9 Arc-Disjoint In- and Out-Branchings523It is easy to reduce (in polynomial time) Problem 9.9.1 for the case whenu = v to the case when u = v for arbitrary digraphs (Exercise 9.49). Hencethe problem remains N P -complete when we ask for an out-branc
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9.12 Exercises5439.59. Suppose D is a digraph which has k but not k + 1 arc-disjoint out-branchingsrooted at s and let F = cfw_X V s : d (X ) = k. Explain how to nd aDminimal member of F (that is, no Y X belongs to F ). Hint: rst show howto nd a mem
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10.4 Disjoint Cycles Versus Feedback Sets56310.4.2 Solution of Youngers ConjectureThe vertex and arc versions of Youngers conjecture were proved for variousfamilies of digraphs including the families mentioned above. McCuaig [559]proved the existence
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10.10 Additional Topics on Cycles583Proposition 10.9.7 If n = gs, g 2, then there exists an s-regular rounddigraph of order n which is s-strong and has girth g .10.10 Additional Topics on Cycles10.10.1 Chords of CyclesThe existence of chords of cycl
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11.1 Properly Coloured Trails in Edge-Coloured Multigraphs603alternating cycles P = cfw_C1 , . . . , Cp such that x and y belong to some cyclesin P and (P ) is a connected graph.We formulate the following trivial but useful observation as a propositi
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11.2 Arc-Coloured Directed Multigraphs623some u k=1 (V (Pi ) V (Qi ). Let be the literal of cj corresponding to ui(that is, if u Pi , then by (d) and the denition of D, = xi and if u Qi ,then = xi ). If u V (Pi ), then C uses the path Qi and the trut
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12.2 Ordering the Vertices of a Digraph of Paired Comparisons643pair x, y of vertices (i.e. objects) in D, the arc xy is in D if and only ifsome experts prefer y to x. The weight of xy is the fraction of the expertsthat favour x over y . Formally, fol
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12.5 Homomorphisms A Generalization of Colourings663Lemma 12.5.15 [412] Let H be a core. If the H -colouring problem is N P complete, then so is the H -colouring problem.hjv(a)(b)(c)Figure 12.5 Illustrating the sub-indicator construction; (a) a d
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Referencesa1. A. Adm. Problem. In Theory Graphs Applications, Proc. Coll. Smolenice,pages 1218, Czech. Acad. Sci. Publ., 1964.a2. A. Adm. Bemerkungen zum graphentheoretischen Satze von I. Fidrich. ActaMath. Acad. Sci. Hungar., 16:911, 1965.3. R. Ah
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References703461. J. Jirsek. On a certain class of multidigraphs, for which reversal of no arc deacreases the number of their cycles. Comment. Math. Univ. Carolinae, 28:185189, 1987.a462. J. Jirsek. Some remarks on Adms conjecture for simple directe
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Author IndexaAdm, A., 582Aharoni, R., 501, 651Aho, A.V., 28, 178Ahuja, R.K., 95, 129Aigner, M., 652Ainouche, A., 240Aldous, J., 95Alegre, I., 59, 184, 188Alon, N., 67, 143, 145, 275, 299, 394,548550, 553, 561, 569, 580,586, 614Alspach, B., 20
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Subject Indexrelation to chromatic number,433longest path problem, 195acyclic digraph, 89weighted acyclic digraph, 53loop, 4Lovszs local lemma, 569aLovszs splitting theorem, 440, 468alower boundon an arc, 95removing from a network, 99Lucches
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Global Optimization Algorithms Theory and Application 2ndEdEvolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Genetic Pr
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1IntroductionOne of the most fundamental principles in our world is the search for an optimal state.It begins in the microcosm where atoms in physics try to form bonds1 in order to minimizethe energy of their electrons [1625]. When molecules form soli
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Fitness and heuristic values(normally) have only a meaning in thecontext of a population or a set ofsolution candidates.V RFitness Space+41fitness1.3 The Structure of Optimizationv(x)VFitness ValuesFitness Assignment Processf1(x)f1(x)Fitnes
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1.4 Problems in Optimization61A very crude and yet, sometimes eective measure is restarting the optimization process at randomly chosen points in time. One example for this method is GRASP s, GreedyRandomized Adaptive Search Procedures [663, 652] (see
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1.5 Formae and Search Space/Operator Design811 for the input 0, and false otherwise. Assume that the formulas were decoded from abinary search space G = Bn to the space of trees that represent mathematical expression bya genotype-phenotype mapping. A
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2.1 Introduction1014. Learning Classier Systems (LCS), discussed in Chapter 7 on page 233, are onlinelearning approaches that assign output values to given input values. They internally usea genetic algorithm to nd new rules for this mapping.5. Evolu
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2.4 Selection121Algorithm 2.5: v assignFitnessTournamentq,r (Pop, cmpF )Input: q : the number of tournaments per individualsInput: r: the number of other contestants per tournament, normally 1Input: Pop: the population to assign tness values toInput