Davide Cherubini - PhD Thesis - UniCA Eprints

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6.3 Multicommodity Flow ProblemsAlgorithm ComplexityDijkstra [32] O(N 2 )Bellman-Ford [27] O(N · A)A ∗ Search [1] PolynomialFloyd-Warshall [41] O(N 3 )Johnson [50] O(N 2 log(N) + N · A)Table 6.2: Algorithms to solve the Shortest Path problemthe Routing Information Protocol (RIP). Moreover, the shortest path problemcan be applied in a huge number of applications [25], such as:• Robot navigation• Urban traffic planning• Optimal pipelining of microelectronic chips• And many others6.3 Multicommodity Flow ProblemsSometimes, the networks are dedicated to the transport of a single commodity(e.g., water). More often, the edge capacities are shared by different flows representingmultiple commodities, where one commodity will always remain the samewithout any transformation in a different commodity (e.g., an apple will not everbecome a pear!).In mathematical terms this means that, at each vertex, each commodity hasits own flow conservation constraint, and the total flow through each arc cannotexceed the maximum capacity.Important examples of multicommodity flow problems arise in transportation,manufacturing networks, and telecommunication, where a separate commodityper class of traffic and origin/destination pair can be identified. In a multicommodityflow problem, either a Min-Cost flow or a Max Flow problem, each49
6.3 Multicommodity Flow Problemscommodity has its own single or multiple sources s i and its own single or multipledestinations t i .In the integer multicommodity flow problem, the capacities and flows are restrictedto be integers. Unlike the single commodity flow problem, for problemswith integral capacities and demands, the existence of a feasible fractional solutionto the multicommodity flow problem does not guarantee a feasible integralsolution.In typical telecommunication systems model an extra constraint, which maybe imposed, is to restrict each commodity to be sent along a single path or to besplit in equal parts and sent along multiple equivalent paths as seen in chapter 2.Moreover, adding a further constraint representing the survivability of the networkin case of failure makes the model NP-hard.In that case, a new formulation is needed, and the present work is focusedon the development of Multicommodity Min-Cost Flow models that match theproblem statement introduced in chapter 3 and that reduce the complexity.6.3.1 Multicommodity Min-Cost FlowLet G = (N, A) be a directed graph with n nodes and m arcs, and x = [x 1 , · · · , x k ]be a vector, representing the multicommodity flow, of k distinct flow vectors [45].A linear Multicommodity Min-Cost Flow problem can be formalized as anextension of the single commodity formulation as follows:min ∑ h∈K∑x h ij −∑j:(i,j)∈A j:(j,i)∈A∑c h ijx h ij (6.24)(i,j)∈Ax h ji = b h i ∀i ∈ N, ∀h ∈ K (6.25)0 ≤ x h ij ≤ u h ij ∀(i, j) ∈ A, ∀h ∈ K (6.26)∑x h ij ≤ u ij ∀(i, j) ∈ A (6.27)h∈Kwith K = {1, · · · , k}.This representation is called node-arc formulation and describes an LP problem;equation 6.24 states that the k commodities have to be “routed” over the50
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AcknowledgementsThis dissertation w
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AbstractThis dissertation investiga
Page 9 and 10: CONTENTS7.4 The Real ISP’s Italia
Page 11 and 12: LIST OF FIGURES7.3 Default metrics
Page 13 and 14: 1.2 Objectives of the dissertation2
Page 15 and 16: Chapter 2Basic knowledgeThis chapte
Page 17 and 18: 2.1 Understanding IS-IS• A = sour
Page 19 and 20: 2.2 Understanding MPLS Technology2.
Page 21 and 22: 2.2 Understanding MPLS TechnologyTh
Page 23 and 24: 2.4 Restoration SchemesHowever, fai
Page 26 and 27: 3.1 Optimization of IS-IS/OSPF rout
Page 28: 3.3 Survivability problemhas been d
Page 31 and 32: 4.1 Heuristic Algorithmsmeta-heuris
Page 33 and 34: 4.1 Heuristic Algorithmsto keep a c
Page 35 and 36: 4.1 Heuristic Algorithmsprobability
Page 37 and 38: 4.3 Linear Programming ApproachExam
Page 39 and 40: 5.2 Traffic monitoring• 60% of th
Page 41 and 42: 5.2 Traffic monitoringExporting the
Page 43 and 44: 5.2 Traffic monitoringare included
Page 45 and 46: 5.2 Traffic monitoringthe Internet.
Page 47 and 48: 5.2 Traffic monitoring5.2.3 The net
Page 49 and 50: 5.2 Traffic monitoring5.2.4 Monitor
Page 51 and 52: 5.2 Traffic monitoringFigure 5.9: S
Page 53 and 54: Chapter 6LP Optimization ModelsThis
Page 55 and 56: 6.2 Graphs and Network Flowswith a
Page 57 and 58: 6.2 Graphs and Network FlowsWhere B
Page 59: 6.2 Graphs and Network Flows6.2.3 T
Page 63 and 64: 6.4 LP modelsThe second layer intro
Page 65 and 66: 6.4 LP modelsEquation 6.32 is the o
Page 67 and 68: 6.4 LP modelsmalized as follows:z =
Page 69 and 70: 6.4 LP models(a) Rerouting flow p
Page 71 and 72: 6.4 LP models(a) 1 st solution(b) 2
Page 73 and 74: 6.4 LP models∑f∈Fx f,lij⎛⎝i
Page 75 and 76: 6.4 LP modelsterm is the MPLS part
Page 77 and 78: 7.2 The highly meshed network7.2 Th
Page 79 and 80: 7.3 The IBCN European network7.3 Th
Page 81 and 82: 7.3 The IBCN European networkmaximu
Page 83 and 84: 7.3 The IBCN European networkSurviv
Page 85 and 86: 7.3 The IBCN European networkFigure
Page 87 and 88: 7.4 The Real ISP’s Italian Backbo
Page 93 and 94: 7.5 Conclusion and validation of re
Page 95 and 96: optimal configuration of network pa
Page 97 and 98: REFERENCES[14] Jones Lustig Format
Page 99 and 100: REFERENCES[40] A. V. Fiacco and G.
Page 101: REFERENCES[61] A. Nucci, B. Schroed
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Davide Cherubini - PhD Thesis - UniCA Eprints

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