Programming topology
1. IntroductionTelecommunication networks have become strategic resources for private- and state-owned institutions, and its economic importance continuously increases. There are series of recent tendencies that have a considerable impact on the economy evolution such as growing integration of networks in the productive system, integration of different services in the same communication system, and important modification in the telephone network structure. Such evolutions accompany a significant growth of the design complexity of these systems. The integration of different sorts of traffics and services and the necessity of a more accurate management of the service quality are factors that make this type of systems very hard to design, to dimension, and therefore to optimize. This situation is aggravated with a very high competitiveness context in an area of critical strategic importance. Show The conception of a WAN is a process in which dozens of sites with different characteristics require to be connected in order to satisfy certain reliability and performance restrictions with minimal costs. This design process involves the terminal site location, the concentrator location, the backbone (central network or kernel) design, the routing procedures, as well as the lines and nodes dimensioning. A key aspect on WAN design is the high complexity of the problem, as much in its globality as in the principal subproblems in which it is necessary to decompose it. Due to the high investment levels, a cost decrease of very few percentage points while preserving the service quality results in high economic benefits. Typically, a WAN network global topology can be decomposed into two main components: the access network and the backbone network. These components have very different properties, and consequently they introduce specific design problems (although they are strongly interdependent). On the one hand, this causes complicated problems (particularly algorithmic ones); on the other hand, it leads to stimulating and difficult research problems. A WAN access network is composed of a certain number of access subnetworks, having treelike topologies; and the flow concentration nodes allow to diminish the costs. These integrated flows reach the backbone which has a meshed topology, in order to satisfy security, reliability, vulnerability, survivability, and performance criteria. Consequently, the backbone is usually formed by high-capacity communication lines such as optic fiber links. Modeling a WAN design by means of the formulation of a single mathematical optimization problem is very intricate due to the interdependence of its large amount of parameters. Therefore the design of a WAN is usually divided into different subproblems [1, 2, 3, 4]. A good example of a possible decomposition approach for the WAN design process is the following [5]:
This work focuses on phase (1) of the decomposition of a WAN design process. More precisely, it deals with the topology planning process concerning the access network. Due to the NP-hard nature of the problem and even though there exist some results, there is still room for improving industrial practices applied today. In this sense, the authors believe it is of strategic importance to design powerful quantitative analysis techniques, potentially easy to integrate into tools. Combinatorial optimization models are introduced that formally define the topological design of the access networks. Moreover, different results related to the topological structure are introduced. Finally, different algorithms are proposed for the topological design which are based on Dynamic Programming and Dynamic Programming with State-Space Relaxation methodology. Advertisement 2. A model for a WAN designIn this section, a model for the design of a WAN is introduced. The model tries to show the most essential aspects which are considered when designing access and backbone networks. In this model, some parameters are not considered: the operation probability of the lines and equipment, the number of equipment ports, and the memory capacity of the equipment. The objective is to design a WAN with the smallest possible installation cost, so that the constraints are satisfied. In what follows, the data of the model are presented as well as its formalization as a combinatorial optimization problem on weighted graphs. The goal is to find the optimal topology that satisfies the imposed constraints to the access and backbone networks. Figure 1 shows an example of a wide area network. The information available for each type of equipment (switch and concentrator) and each type of connection line, as well as the line laying, is the following:
Figure 1.WAN example. In terms of graph theory, a model for the design of a WAN, based on the problem, is presented as follows. Some notation is introduced next, that is then used to formally define the problem.
Definition 1(WANDPwide area network design problem). LetG=SEbe the graph of feasible connections on the WAN. The wide area network design problemSEKWEaCDSTVSTconsists in finding a subnetwork ofGof minimum cost which satisfies the following points:
Given the complexity of the WANDP, to facilitate its solution, the topological design problem is divided into three subproblems:
The remainder of this work concentrates only in the first problem (ANDP). Advertisement 3. Access Network Design ProblemThe Access Network Design Problem is defined as follows. Definition 2(ANDPAccess Network Design Problem). LetGA=SE1E2be the graph of feasible connections on the access network andCthe matrix of connection costs defined previously. The Access Network Design ProblemSE1E2Cconsists in finding a subgraph ofGAof minimum cost such thatiST;there exists a path fromito some sitejSDof the backbone network. Notation 1.ΓANDP denotes the space of feasible solutions of ANDPSE1E2Cthat do not have any cycle and with an output only toward the backbone networktST. These have forest topology as we illustrate in Figure 2. Figure 2.A feasible solution of ANDP. In order to define these problems in terms of graph theory, the following notation is introduced:
The General Access Network Design Problem (GANDP) consists of finding a minimum-cost subgraphHGsuch that all the sites ofSTare communicated with some node of the backbone. This connection can be direct or through intermediate concentrators. The use of terminal sites as intermediate nodes is not allowed; this implies that they must have degree one in the solution. The GANDP is here simplified by collapsing the backbone into a fictitious node and given the name of Access Network Design Problem. The equivalence between both problems, GANDP and ANDP, as well as the NP-hardness of the ANDP, is proved in [7]. This work concentrates on the ANDP with the objective of proposing a new approach for solving this problem. We study different results related to the topological structure of the ANDP solutions. In particular we present results that characterize the topologies of the feasible solutions of an ANDP instance. The following proposition shows the topological form of the feasible solutions of ΓANDP for a given ANDP instance. Proposition 1.Given an ANDP with associated graphGA=SE1E2and matrix of connection costsC. If the subnetworkH=STS¯E¯(withS¯SCSDandE¯E1E2) is an optimal solution of ΓANDP, it is composed of a set of disjoint treesH=H1Hmthat satisfy:
Proof. Trivial. The following propositions present results that characterize the structure of the global optimal solution. Proposition 2.Let ANDPSE1E2Cbe a problem wherescSC,s¯SCSDandsSTSCsuch thatsscscs¯E1E2andswSD/cs,sw Proof.Let us suppose that there existsTAΓANDP global optimal solution such thatscTAa concentrator site withgsc<3inTA. Ifgsc=1;thenscis a pendant inTA; therefore, eliminating this, a feasible solution of smaller cost would be obtained. This is a contradiction; hence,gsc1. Ifgsc=2, lets¯SCSDbe the site adjacent toscinTAwhich its output site is toward the backbone network. LetsSTSCbe the other adjacent site inTA. Considering the networkH=TAscssw, whereswSDsatisfiescs,sw COSTH=COSTTAcs,sccsc,s¯+cs,sw Furthermore, it is easy to see thatHΓANDP. Hence, this implies thatHis a better feasible solution compared withTA. This is a contradiction, entailing thatgsc3inTA, as required and completing the proof. Proposition 3.Given an ANDPSE1E2Csuch that for any three sitess1s2s3, withs1STSC,s2SCands3SCSD, the strict triangular inequality is satisfied, i.e.,cs1,sk Proof.As in the previous proposition, let us suppose that there existsTAΓANDP global optimal solution such thatscTA, a concentrator site withgsc<3inTA. Clearlygssmust be different to 1. Now, let us consider the casegsc=2inTA. Lets1,s2be the adjacent sites toscinTA. By hypothesiscs1,s2 COSTT¯A=COSTTAcs1,sccsc,s2+cs1,s2 This is a contradiction; therefore,gsc3inTA, hence completing the proof. The next section presents algorithms applied to the ANDP(k) withk12. A way of computing the global optimal solution cost of it using the Dynamic Programming approach is obtained. Considering that the ANDP(1) is a NP-hard problem, we obtain lower bounds to the global optimal solution cost by Dynamic Programming with State-Space Relaxation in polynomial time. Advertisement 4. Algorithms applied to the ANDPThis chapter presents the Dynamic Programming approach as alternative methodology to find a global optimal solution cost for the ANDP(1) and ANDP(2). After we introduce the Dynamic Programming with State-Space Relaxation as a method to obtain lower bounds for the original problem. 4.1 Dynamic ProgrammingProposition 4.Given an ANDPSE1E2A, the cost of a global optimal solution ofΓANDP1is given byfSTZAQ, withf...defined by the following expression of Dynamic Programming: fSCSTZAQ=minstSTCOSTstZ+fSCST{st,Z,AQ),minscSCCOSTstsc+COSTscZ+fSCST{st,Z,AQscZ)ifST0otherwiseE3 whereCOSTsZ=minzSDCOSTsz,sZ=argminzSDCOSTszand the matrix of connection costsAQ=ai,ji,jE1E2is defined by ai,j=COSTijifijQ0otherwiseE4 Proposition 5.Given an ANDPSE1E2A, the cost of a global optimal solution ofΓANDP2is given byfSTZAQ, withf...defined by the following expression of Dynamic Programming fSCSTZAQ=minstSCCOSTstZ+fSCST{st,Z,A^Q),minscSCCOSTstsc+COSTscZ+fSCST{st,Z,AQscZ),minscuscvE2COSTstscu+COSTscuscv+COSTscvZ+fSCST{st,Z,AQscuscvscvZ)ifST0otherwiseE5 whereCOSTsZ=minzSDCOSTsz,sZ=argminzSDCOSTszand the matrix of connection costsAQ=ai,ji,jE1E2is defined by ai,j=COSTijifijQ0otherwiseE6 4.2 Dynamic programming with state-space relaxationIn order to find a lower bound offSCSTZAQ, the Dynamic Programming with State-Space Relaxation is now applied. It is a general relaxation procedure applied to a number of routing problems [8]. The motivation for this methodology stems from the fact that very few combinatorial optimization problems can be solved by Dynamic Programming alone due to the dimensionality of their state-space. To overcome this difficulty, the number of states is reduced by mapping the state-space associated with a given Dynamic Programming recursion to a smaller cardinality space. This mapping, denoted by g, must associate to every transition from a stateS1to a stateS2in the original state-space, a transitiongS1togS2in the new state-space. To be effective, the function g must give rise to a transformed recursion over the relaxed state-space which can be computed in polynomial time. Furthermore, this relaxation must generate a good lower bound for the original problem. With the aim of illustrating this methodology, we present this approach in the context of the minimization of the total schedule time for the Traveling Salesman Problem with Time Window (TSPTW), after we apply it to the Dynamic Programming recursion presented in Proposition 5. The objective of the TSPTW is to find an optimal tour where a single vehicle is required to visit each of a given set of locations (customers) exactly once and then return to its starting location. The vehicle must visit each location within a specified time window, defined by an earliest service start time and latest service start time. If the vehicle arrives at a service location before the earliest service start time, it is permitted to wait until the earliest service start time is reached. The vehicle conducts its service for a known period of time and immediately departs for the location of the next scheduled customer. Assume that the time constrained path starts at fixed time valueao. DefineFSias the shortest time it takes for a feasible path starting at nodeo, passing through every node ofSNexactly once, to end at nodeiS. Note that optimization of the total arc cost would involve an additional dimension to account for the arrival time at a node. The functionFSican be computed by solving the following recurrence equations: FSj=minijEFSji+tijiSj}SN,jSE7 The recursion formula is initialized by Fjj=maxajao+tojifojE+otherwiseE8 The optimal solution to the TSPTW is given by Note that Eq. (7) is valid ifajFSjbj.If howeverFSj Several alternatives for the mappingghave been suggested [9]. Here is presented the shortest r-path relaxation, i.e.,gS=r=iSri, whereri1is an integer associated with nodeiN; thengSi=gSri. DefineR=iSri. Hence the transformed recursion equations are Frj=minijEFrrji+tijrrjri},r1R,jNE10 Recursion (10) holds ifajFrjbj. Otherwise, ifFrj Fjj=maxajao+tojifojEandq=qj+\inftyotherwise,forq1Q,jNE11 The lower bound is given by The complexity of the bounding procedure isOn2×Qfor an-node problem. Now, we present this approach in the context of finding a good lower bound for the solution of ANDP(2). The following proposition gives a lower bound for thefSCSTZAQpresented in Proposition 5 (the optimum value of the ANDP(2)). Proposition 6.Given an ANDPSE1E2C, a lower bound offSCSTZAQis derived from the following expression of Dynamic Programming with State-Space Relaxation gSCrZAQ=minstiSTCOSTstiZ+gSCrriZAQ,minscjSCCOSTstiscj+COSTscjZ+gSCrriZAQscjZrR̂rirj,minscjusckE2COSTstiscj+COSTscjsck+COSTsckZ+gSCrriZAQscjscksckZrR̂rirj+rkifST0otherwiseE13 where1riRis an integer associated with the siteiSTSC,R=iSTSCri,R̂=jSCrjand the matrix of connection costsAQ=ai,ji,jE1E2is defined by ai,j=COSTijifijQ0otherwiseE14 The lower bound is given bygRZA. Advertisement 5. Computational resultsThis section presents the experimental results obtained with the recursions of above. The algorithms were implemented in ANSI C. The experimental results were obtained in an Intel Core i7, 2.4 GHz, and 8 GB of RAM running under a home PC. The recursions presented in Propositions 4 and 5 were applied to the ANDP(1) and the ANDP(2), respectively, whereas the recursion presented in Proposition 6 was applied to ANDP(2). They were tested using a large test set, by modifying the Steiner Problem in Graphs (SPG) instances from SteinLib [10]. This library contains many problem classes of widely different graph topologies. Most of the problems were extracted from these classes: C, MC, X, PUC, I080, I160, P6E, P6Z, and WRP3. The SPG problems were customized, transforming them into ANDP instances by means of the following changes. For each considered problem:
Moreover, if the resulting topology was unconnected, the problem instance was discarded. Let us notice that since in the ANDP the terminals cannot be used as intermediate nodes (which implies also that edges between pairs of terminals are not allowed), the cost of a SPG optimum is a lower bound for the optimum of the corresponding ANDP. Therefore they are for ANDP(k) withk12. Table 1 shows the results obtained by applying the recurrences presented in Propositions 4 and 5. In each one of them, the first column contains the names of the original SteinLib classes with the name of the customized instance. The entries from left to right are:
Table 1.Results obtained by applying Dynamic Programming tocopt1andcopt2. TheLB_GAPSPGkis computed as LB_GAPSPGk=100×coptkLBSPGLBSPG.E15 Feasible solutions were obtained here only for i080-112, i080-115, and i160-015 withk=1because, as can be seen, the cost is finite. The optimal values of the SPG instances (LBSP G) provided lower bounds for the optimal values of the ANDP (therefore to ANDP(k) withk0), considering that in the ANDP generation process, all the connections between terminal nodes were deleted and further that ANDPs feasible solution space is more restrictive than of SPG. The experimental results obtained forcopt1have an average gap with respect to the lower bound of 20.72%. Increasing k to 2 (applying the recursion presented in Proposition 5), feasible solutions were obtained for all the testing networks, and the experimental results obtained have an average gap with respect to the lower bound of 7.01%. It can be proved that (it is out of the scope of this chapter) increasingk, the following inequality is fulfilled: coptk1coptk1+floornCk·1k+nT·cmaxcmin1E16 Table 2 shows the results obtained. Despite the bound was not good in these cases (due the heterogeneity of costs of the lines), it can help us in some cases to answer the following question: how much can be saved with a higherk?
Table 2.Relation between optimal solutions ofANDP1andANDP2. Table 3 shows the results obtained by applying the recursion presented in Proposition 6. As before the first column contains the names of the original SteinLib classes with the name of the customized instance. The entries from left to right are:
Table 3.Lower bounds obtained toANDP2by applying Dynamic Programming with State-Space Relaxation. TheLB_GAPSSR2is computed as LB_GAPSSR2=100×copt2LB2SSRLB2SSRE17 In general, the gaps related to the lower bounds were low. Therito each terminal site and concentrator site were distinct integers chosen from1STSC. This lower bound can be increased by modifying the state-space through the application of subgradient optimization tori. As future work, it is possible to incorporate the method for a better choice ofri. It can be noticed that the execution times of computing global optimal solution costs were much longer than using Dynamic Programming with State-Space Relaxation. Advertisement
6. ConclusionsThe implementation of the algorithms was tested on a number of different problems with heterogeneous characteristics. In particular, a set of ANDP instances transforming 18 SPG instances extracted from SteinLib was built. The optimal values for the selected SPG instances are lower bound for the corresponding ANDP. The solutions found by the algorithm were, in average, 21% and 7% lower than the mentioned bounds in ANDP(1) and ANDP(2), respectively. It is reasonable supposing that the gaps related to the global optimum of the ANDP instances be even lower since the feasible solutions of the ANDP that are also feasible solutions of the original SPG, but not reciprocally. In this sense, remember that in any ANDP instance generated, all the edges between pairs of terminal nodes were deleted (because in our ANDP such connections are not allowed) having the additional constraint that the terminal nodes must have degree one in the solution. Besides, a Dynamic Programming with State-Space Relaxation algorithm was developed which can give a lower bound in polynomial time. The average gaps with respect to the global optimal solution costs were lower than 20%. Notice that, as expected, the execution times of the proposed algorithms are strongly dependent on the number of sites, edges, and terminal sites. To sum up, as far as the authors are concerned, the results obtained with the recurrences above are very good, considering that computing the global optimal solution of an ANDP(2) is a NP-hard problem. |