Multicriteria Steiner Tree Problem for Communication Network

Инфор мационные процессы, Т ом 9, № 3, 2009, стр. 199–209. c  2009 L evin, Nuriakhmetov. INF ORMA TION TECHNOLOGY IN ENGINEERING SYSTEMS Multicriteria Steiner T ree Problem for Comm unication Net w ork (Information Pro cesses, 9(3), 2009, 199-209) Mark Sh. Levin*, Rustem I. Nuriakhmeto v** *Inst. for Information T r ansmission Pr oblems, Russian A c ademy of Scienc es, Mosc ow, Russia email: mslevin@acm.or g **Mosc ow Inst. of Physics and T e chnolo gy (State Univ.), Dolgoprudny, R ussia, email: rust87@mail.ru Receiv ed Septem ber 20, 2009 Abstract— This pap er addresses com binatorial optimization schemes for solving the multicri- teria Steiner tree problem for comm unication net w ork topology design (e.g., wireless mesh net w ork). The solving sc heme is based on several mo dels: multicriteria ranking, clustering, minim um spanning tree, and minimum Steiner tree problem. An illustrative n umerical example corresp onds to designing a cov ering long-distance Wi-Fi net w ork (static Ad-Ho c netw ork). The set of criteria (i.e., ob jectiv e functions) inv olv es the follo wing: total cost, total edge length, o v erall throughput (capacit y), and estimate of QoS. Obtained computing results sho w the sug- gested solving scheme provides goo d netw ork topologies which can b e compared with minimum spanning trees. 1. INTRODUCTION Man y y ears minim um spanning tree problem (MST) is a basic scien tific/telecommunication problem for design of communication/access/computer netw ork top ology , for netw ork routing ([3], [4], [5], [6], [17], [18], [21]). In this problem a traditional ob jectiv e function consists in minim um total length (or weigh t) of spanning tree edges. Minim um Steiner tree problem (STP) can pro vide decreasing of the ab o v e-men tioned total length b y usage of additional no des (vertices) ([4], [7], [8], [17], [21], [23]). This problem is studied in mathematics (e.g., [2], [7], [8]), in combinatorial optimization (e.g., [4], [21], [23]). In recent t w o decades the significance of STP was increased (e.g., activ e usage in comm unication netw orks for top ology design, routing, protocol engineering, etc.) (e.g., [17], [21]). STP belongs to class of NP-hard problems (e.g., [4]) and exact enumerativ e solving metho ds (e.g., [23]) or appro ximation algorithms (e.g., heuristics) (e.g., [20]) are used. A brief survey of well-kno wn kinds of Steiner tree problems is presented in [14]. STP with tw o criteria was inv estigated in [22]. A description of using a partitioning-syn thesis heuristic based on Hierarc hical Morphological Multicriteria Design approach for Steiner tree problem was describ ed in [10]. In this article m ulticriteria Steiner tree problem (multicriteria STP) is firstly suggested. A static Ad-Ho c comm unication netw ork is examined as an application domain. In the multicriteria spanning problem, each edge has the following attributes: (i) length (cost), (ii) throughput (capac- it y), (iii) reliability or QoS parameters. Our comp osite (four-stage) solving sc heme is targeted to building some Pareto-effectiv e Steiner trees (i.e., alternativ e solutions). The solving scheme con- sists of stages: (a) building a spanning tree, (b) clustering of netw ork nodes (b y a mo dification of agglomerative algorithm), (c) building a Steiner tree for each obtained no de cluster, and (d) rev elation of Pareto-effectiv e solutions and their analysis. 200 LEVIN, NURIAKHMETO V Presen ted n umerical examples for a wireless communication net work illustrate the suggested de- sign approac h. Computing w as based on authors MatLab programs (h ttp://www.mathw orks.com/). A preliminary material was presented as a conference pap er [15]. 2. SP ANNING PR OBLEMS A static multihop wireless netw ork is under examination. Multihop no de paths going through a set of no des are used for t wo-node communications. Altitude map is introduced and four main criteria are assigned for each P2P connection. The altitude is the one of significant parameters b ecause it affects not only netw ork productivity (wireless links require line-of-sight clearance) but also link costs. The examined net work is considered as undirected graph G ( V , E ) where V is the set of no des (vertices) and E is the set of edges. It is assumed that the net work is tw o dimensional one, though no de stations are at different height. This fact has an affect on altitude criterion for eac h P2P connection. 2.1. Par ameters The parameters under consideration are follo wing: 1. A distance b et ween t wo vertices v i , v j ∈ V : l i,j . 2. QoS characteristics for tw o vertices v i , v j ∈ V : q ij . 3. An altitude b et ween t wo vertices v i , v j ∈ V : δ ij . 4. A cost of connection b et ween v i , v j ∈ V depends on δ ij and q ij : c ij = F ( δ ij , q ij ). Here it is assumed that F is prop ortional to linear aggregate δ 3 ij and q ij . 2.2. Basic engine ering pr oblem A set of transmission stations is considered in the netw ork that is represen ted by graph G . Each connection/edge of G is ev aluated up on four characteristics ab o ve. The examined problem is: Find Par eto-effe ctive Steiner tr e e for gr aph G while taking into ac c ount the fol lowing criteria : (i) over al l c ost: C st = P e ij ⊂ G st c ij ; (ii) total network length: L st = P e ij ⊂ G st l ij ; (iii) over al l QoS: Q st = P e ij ⊂ G st q ij ; and (iv) summarize d altitude: ∆ st = P e ij ⊂ G st δ ij wher e G st ( V 0 , E 0 ) is a Steiner tr e e, e ij ⊂ E 0 : V 0 ⊇ V and E 0 is the extende d set of e dges. 2.3. Pr oblem formulations The basic problem (Minimum Spanning T ree MST) is: min X e ij ⊂ G span l ij where G span ( V 0 , E 0 ) : V 0 ≡ V , E 0 ≡ E . Here there are w ell-known solving metho ds such as Prim’s algorithm and Krusk al’s algorithm (e.g., [1]). ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 MUL TICRITERIA STEINER TREE PR OBLEM FOR COMMUNICA TION NETWORK 201 If extra v ertices can b e added to minimize o verall length, Steiner tree problem (STP) can b e examined: min X e ij ⊂ G st l ij where G st ( V 0 , E 0 ) : V 0 ⊇ V . This problem is NP-hard. There are man y heuristics prop osed during the recen t y ears (e.g., [7], [20]). An extension of MST problem is targeted to finding an efficient set of ov erall characteristics (i.e., optimization by vector function) (m ulticriteria MST, i.e., MMST): min C span ( G span ) , min L span ( G span ) , max Q span ( G span ) , min ∆ span ( G span ) where: G span ( V 0 , E 0 ) : V 0 ≡ V , E 0 ≡ E Here a p ossible solving metho d consists in m ulticriteria ranking of graph edges (by estimates) and using standard approaches for MST. Figure 1. Interconnection of problem formulations Multicriteria STP (MSTP) is an extension of STP . Here optimization is based on v ector function: min C st ( G st ) , min L st ( G st ) , max Q st ( G st ) , min ∆ st ( G st ) where: G st ( V 0 , E 0 ) : V 0 ⊇ V Figure 1 depicts interconnection for the problems ab o ve. 3. SOL VING SCHEME, EXAMPLE Our four-stage comp osite solving scheme (macro-heuristic) is: Stage 1. Building a multicriteria spanning tree for the initial netw ork (a mo dified Prim’s algo- rithm [1]). ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 202 LEVIN, NURIAKHMETO V Figure 2. Initial comm unication net w ork (example) ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 MUL TICRITERIA STEINER TREE PR OBLEM FOR COMMUNICA TION NETWORK 203 Stage 2. Clustering of netw ork no des (by a mo dification of agglomerative algorithm [12]). Stage 3. Building a spanning Steiner tree for each obtained no de cluster (a mo dified Melzak’s algorithm [7]). Stage 4. Revelation of Pareto-effectiv e solutions and their analysis. Figure 3. Spanning tree (example) Let us consider our solving scheme. Concurrently , our numerical example is describ ed. Figure 2 depicts the initial communication netw ork. Thus stages of the solving schemes are considered. Stage 1 (building a multicriteria spanning tree) is based in the following initial data: the set of netw ork elemen ts (graph v ertices, access p oin ts). Here a mo dified Prim’s algorithm is used [1]: addition to an existing built tree a ”most close subtree”(or a vertex). Multicriteria ranking is based on quadratic utilit y function. At the first step an initial set of ro ots is selected and, as a result, ”spanning forest”is obtained. A t the end step several v ertices are selected to extend the solution set. ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 204 LEVIN, NURIAKHMETO V A solution of minimum spanning tree (MST) is depicted in Figure 3. Figure 4 depicts the solution for multicriteria spanning tree MST. Figure 4. Multicriteria spanning tree (example) A t the stage of clustering (Stage 2), clusters as groups of close v ertices in the spanning structure are defined. Here a cluster includes ab out 5...6 vertices, this cardinality of a cluster elements set is useful to decrease complexity of the solving pro cess for Steiner tree problem (Stage 3). The mo dified agglomerativ e algorithm for hierarchical clustering is used [12]. The result of clustering is depicted in Figure 5. A mo dified Melzak’s algorithm [7] (decreased complexit y) is used for building spanning Steiner tree (MSTP) for eac h cluster (Stage 3). The resultan t spanning Steiner tree is depicted in Figure 6. ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 MUL TICRITERIA STEINER TREE PR OBLEM FOR COMMUNICA TION NETWORK 205 Figure 5. Clustering (example) ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 206 LEVIN, NURIAKHMETO V A t the Stage 4 rev elation of P areto-effective spanning structures is executed. F our ab o ve- men tioned ob jectiv e functions are used. Figure 7 illustrates six P areto-effective multicriteria span- ning Steiner trees (netw ork top ology solutions). Basic algorithms for top ology design are shown in T able 1. Figure 6. Steiner tree (example) ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 MUL TICRITERIA STEINER TREE PR OBLEM FOR COMMUNICA TION NETWORK 207 T able 1. Basic approac hes for net work top ology design T op ology design problems Basic algorithms Used algorithms 1.Minim um spanning tree problem MST 1.Krusk al’s algorithm [1] 2.Prim’s algorithm [1] Prim’s algorithm [1] 2.Steiner tree problem STP 1.Melzak’s algorithm [7] 2.Win ter’s algorithm [7] 3.Sim ulated annealing [16] 4.Ev olutionary methods [9] Melzak’s algorithm [7] 3.Mulicriteria spanning tree MMST 1.W eighted sum function [19] 2.Maximin metho d [19] W eighted sum function [19] 4.Mulicriteria Steiner tree MSTP Metho ds ab o ve Figure 7. Pareto-effectiv e solutions for Steiner tree problem 4. COMP ARISON OF APPR OA CHES T able 2 integrates computing results of multicriteria comparison of three approaches: (a) min- im um spanning tree problem (MST, Figure 3), (b) multicriteria spanning tree problem (MMST, Figure 4), and (c) multicriteria Steiner tree problem (MSTP , Figure 7). F or the comparison, total estimates for four considered ob jective functions are computed for each approach results and rev ela- tion of P areto-effective solutions (i.e., estimates v ectors) is executed: tw o series steps with selection of the 1st P areto-effective solution set and the 2nd P areto-effective solution set (after deletion of the 1st Pareto-effectiv e solution set). ИНФОРМАЦИОННЫЕ ПРОЦЕССЫ ТОМ 9 № 3 2009 208 LEVIN, NURIAKHMETO V 5. CONCLUSION In the pap er, multicriteria spanning Steiner tree problem is firstly suggested for communication net work (a case of wireless communication netw ork). The solving scheme (a comp osite macro- heuristic) consists of the following stages: (i) spanning an initial netw ork b y a spanning tree, (ii) clustering of netw ork nodes, (iii) building of spanning Steiner tree for each obtained cluster (a subnet work), and (iv) revelation and analysis of alternative spanning Pareto-effectiv e Steiner trees. Eviden tly , it can b e reasonable to examine mo difications of the used solving scheme and its stages, for example: (a) impro vemen t of clustering metho ds, (b) increasing of clusters (i.e., cluster node sets), (c) usage and comparison of v arious algorithms for Steiner tree problem. In addition, a sp ecial research computing exp erimen ts may b e carried out to study the suggested solving scheme, a mo dified its v ersions, and ev olutionary optimization heuristics. The draft material for the article w as prepared within framew ork of course ”Design of Systems”in Mosco w Institute of Ph ysics and T echnology (State Univ ersity) (creator and lecturer: M.Sh. Levin) ([11], [13]) as lab oratory w ork 12 (studen t: R.I. Nuriakhmetov) and BS-thesis of R.I. Nuriakhmeto v (2008, advisor: M.Sh. Levin). T able 2. 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