Optimal Wavelength-Routed Multicasting
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Optimal Wavelength–Routed Multicasting Bruno Beauquier, Pavol Hell, Stéphane Pérennes. N˚ 3276 Octobre 1997. THÈME 1. ISSN 0249-6399. apport de recherche.
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(4) Optimal Wavelength–Routed Multicasting Bruno Beauquier* , Pavol Hell** , Stéphane Pérennes* Thème 1 — Réseaux et systèmes Projet SLOOP Rapport de recherche n˚3276 — Octobre 1997 — 6 pages. Abstract: Motivated by wavelength division multiplexing in all-optical networks, we consider the problem of finding a set of paths from a fixed source to a multiset of destinations, which can be coloured by the fewest number of colours so that paths of the same colour do not share an arc. We prove that this minimum number of colours is equal to the maximum number of paths that share one arc, minimized over all sets of paths from the source to the destinations. We do this by modeling the problems as network flows in two different networks and relating the structure of their minimum cuts. The problem can be efficiently solved (paths found and coloured) using network flow techniques. Key-words: All–optical networks, WDM routing, multicast, graphs, flows.. (Résumé : tsvp). Submitted to Discrete Applied Mathematics. * SLOOP is a joint project with the CNRS and the University of Nice-Sophia Antipolis (I3S laboratory). This work has been partially supported by the French GDR/PRC Project PRS, by the Galileo Project. Emails : {beauquier, speren}@sophia.inria.fr ** Simon Fraser University, School of Computing Science, Burnaby, B.C., V5A 1S6, Canada. Email : pavol@cs.sfu.ca. Unité de recherche INRIA Sophia Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA ANTIPOLIS Cedex (France) Téléphone : 04 93 65 77 77 – Télécopie : 04 93 65 77 65.
(5) Algorithme optimal de multicast tout-optique Résumé : Ce rapport est motivé par l’étude du routage par multiplexage en longueur d’onde (en anglais Wavelength Division Multiplexing : WDM) dans les réseaux tout–optiques. Nous considérons le problème du multicast dans lequel un processeur fixé désire communiquer simultanément avec un certain nombre d’autres processeurs. Il s’agit de trouver dans le graphe associé des chemins reliant la source du multicast aux destinations et de colorer ces chemins, de manière à ce qu’aucun lien du réseau ne soit traversé par deux chemins de la même couleur. Nous résolvons ce problème en utilisant des techniques de flots. Il en découle un algorithme polynomial pour calculer une solution optimale. Mots-clé : Réseaux tout–optiques, multicast, graphes, routage, flots..
(6) Optimal Wavelength–Routed Multicasting. 3. 1 Motivation and Definitions Optics is emerging as a key technology in communication networks, promising very high speed local or wide area networks of the future. A single optical wavelength supports rates of gigabits per second, which in turn support multiple channels of voice, data and video [4, 5]. Multiple laser beams that are propagated over the same fiber on distinct optical wavelengths can increase this capacity even further. This is achieved through Wavelength Division Multiplexing (or WDM) [2], by partitioning the optical bandwidth into several channels and allowing the transmission of multiple data streams concurrently along the same optical fiber. All–optical (or single–hop [6]) communication networks provide all source-destination pairs with end-to-end transparent channels that are identified through a wavelength and a physical path. Wavelengths being a limited resource, solutions to the problem of efficient routing and wavelengths allocation are of importance for the future development of optical technology. The problem we consider here is motivated by switched networks with reconfigurable wavelength selective optical switches, without wavelength converters, where different signals may travel on the same communication link (but on different wavelengths) into a node, and then exit from it on different links, keeping their original wavelengths. See the recent survey [1] for an account of the theoretical problems and results obtained for this all–optical model..
(7) ! "!$#% '&( *),+-) /.0 1 1'2 3 768"1 9 142 1 ' 1 2 5 6: ; 1 %1 < 5 1 =1 < We are now to formulate our problems. Let be a digraph. in is an ordered >%ready ?9 (corresponding > toA? ).request pair of nodes to a message to be sent from An instance >% ?9 may appear more than once). A routing@ inA ofisana collection (multiset) of requests (a request CB >% ?9 'DE7>:"?F -& @;G , where 7>:"?F is a dipath from > instance ?to . @ is a collection of dipaths A JK @ A Let be a digraph and @ an instance in . For each routing A of @ , we denote by I H the minimum number of wavelengths (’colours’) that can be assigned to in A , so that no JKthe@ dipaths two dipaths of the same wavelength share an arc. The WDM problem asks for a routing A of @ which minimizes I H K @ A . We denote by I H K @ this minimum of I H K @ A over all routings A for @ . &M Let again be a digraph and an instance. For each routing A of @ , the load of an arc L @ K . in the routing A , denoted by N H the number of dipaths of A containing L . The load K @ @ A A , isL the, ismaximum JK @ A L over all arcs L &O . The of a routing A , denoted by N H of N H K. K @ congestion problemJK @ asks for a routing A of @ which minimizes its load. We denote by N H. @ A over all routings A for @ . this minimum of N H. We use the standard terminology of digraphs and flow networks [3]. A dipath (’directed path’) is a sequence of nodes , , such that in digraph for . For any sets and in a digraph, we denote by the number of arcs beginning in and ending in . In a flow network [3], we denote by the capacity of the arc , and by the capacity of the cut .. RR n˚3276.
(8) 4. B. Beauquier, P. Hell, S. Pérennes. NH. The relevance of the parameter. to our problem is shown by the following lemma:. K @ N H K @ for any instance @ Lemma 1 I H. in any digraph .. JK @ . . Proof. Indeed, to solve a given WDM problem one has to use a number of wavelengths at least equal to the maximum number of dipaths having to share an arc.. BE7>:"?F D? & G . > &M. . In this paper, we are interested in a special case of instances where the collection of requests has the form for a fixed node , called the originator, and a multiset of nodes in . Such an instance is called a multicast (or a one–to–many) instance. The particular instance where is the set , is called a broadcast.. . 2 Multicasting and Network Flows. J' . In the case of multicasting, we can model both of the above problems (the WDM problem as well as the congestion problem) by network flows. Consider a digraph with a multicast instance , with originator . In what follows we assume that is a set, i.e., that each node appears at most once in . Indeed, if is a general multiset, we can transform the problem by adding to each destination which appears in times, a set of vertices of degree one, adjacent to , each (as well as ) having multiplicity one. We begin by modeling the congestion problem (cf. Figure 1 (b)). Let be two new vertices that will play the roles of source and sink. For every positive integer , we define the network to have the vertex set , the arc set , and the capacities , for all , and for all .. @ EB ? 7>:"?F 'D? & G ?. ?. . >. ?. . 47?F. 47?F . . .
(9) ">% 6: ; 8 B 7 68G "9 '& BE 7 ? ">= 8G M? & BE?F G 5 5. 5 J K. D D Proposition 2 N H @ ) if and only if
(10) has a flow of value . Proof. From the definitions.. K @ . . . 1 1 B G 1 < B! G
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(12) D !D # = < N H 1K" @ %$3 1 1.
(13) D D D 1" D&# N H K @ 3 1 %1 < D D , for any 1(' (1) JK @ (cf. Figure 1 (c)). Let ) Next we discuss the network flow model! for the WDM problem J ; . !. . !.
(14) , and let ,+ be a be a positive integer. For )+ )*) , let be a copy of M & ! 9 ! ? & , let ?+ the copy of copy , let be the copy of in , and for each ? in ,of+ . Let. For each be two new vertices which be the source and sink.- We define the network - /% ! will - /% ! - 2 B . . 1 2 B " >. + ! ! 0 ! 0 % / ! 0 G to have the- vertex set . , the arc set 3 4 5 6! /: BE?! 7? + G 8 9 5G B77? + G , and except . F > " !. for the arcs ( K) + : ) ) ) of infinite . 1 < , > & < 1 1. For future reference we also consider the capacities of cuts in the network . Suppose is a subset of and . This gives rise to the cut in . The capacity of this cut is infinite when ; otherwise it is . According to the above proposition, is the smallest integer such that admits a flow of value . Combining this with the max-flow min-cut theorem of [3] we obtain. capacity, all other arcs having capacity one.. INRIA.
(15) Optimal Wavelength–Routed Multicasting. 5. I H K @ ) ) if and only if
(16) - 2 has a flow of value D D . > ? & , which can be coloured by Proof. Suppose first that admits dipaths from to each B integers from )G in6F!"such that no two dipaths sharing the same arc have the same "!" 4a way color. Define the values 2 , for each arc 768"9 & which belongs to D a dipath D in
(17) - 2 of. color + ( ) +) ) ). These values can be extended in a natural way to a flow of value 2 D D in - 2 . By construction of
(18) - 2 , this Conversely, assume that there exists a+ flow 2 of value M ? & ? & 1+ implies that there is for each node an incoming arc with flow one. Hence for each ? !
(19) ! ! there is exactly one + such that the node has an incoming arc with flow one in . In that > ! ? ! there must be a dipath from to consisting of arcs all having flow one, and > we can assign ? & the , corresponding dipath in the color + . Thereby are defined dipaths in from to each which are coloured by ) colours as required. K @ N H K @ , and an Theorem 4 Let be a digraph and @ JaKmulticast instance in . Then I H. optimal solution to the WDM problem @ can be found in polynomial time. Proof. In view of lemma 1 and the above proposition, it will suffice to show that the network K- 2 , N H JK @ , has a flow of value D D . We shall do this by showing that every cut has capacity with ) D D , cf. [3]. at least B G 1 < 5B G in - 2 , and let 1 !8 1 " ;! and 1 < !8 1 < " ;! , for ) +) ) , and Consider a cut 1 + + let 1 1 " and 1 < +- 1 < " + . Clearly, we may assume that each > !4& 1 !" ) + ) ) , otherwise the capacity of the cut will be infinite. Then the capacity is 5 1 B G 1 < !B G 8! 3 1 ! 1 < ! #!= 3 1 ! 1 < + #,D 1 +ED B ? ! D + M? ! & 1 G . If ! denotes the copy of in ! , then ?C& For !=each != 3 1 ! 1 < + 5 3 1 1 < + , and that D 1 ! " F! D , let 51 D 1 D . Note < 1 + . For every 3 1 1 < + is if ?O& 1 + , and is D 1 D ifalso ? & that ? & , we have D 1 D ) ) , and hence 5
(20) 'D 1 D ) ) D 1 +D , i.e., D 1 +ED -
(21) 4D 1 D . Summarizing, we obtain: 5 1 (B G 1 < !B G ! 3 1 ! 1 < ! #
(22) D 1 &D # ) 5
(23) D 1 D ! 3 1 ! 1 < ! # ) 5 D 1 D ! < ! # D ! ! D ! 3 1 1 ) 1 " JK the inequality (1), we have D 1 ! " 9! !D # ) 3 1 ! 1 < ! D D Recall that we set ) N H B @ . < By for each + , , and thus 5 1 G 1 B5 G D D . JK @ (and solve Our proof implies, in an obvious way, a flow-based algorithm to compute I H K. the WDM problem time, for any multicast instance @ in any digraph . (Of @ ) inKpolynomial @ and the congestion problem.) course, the same applies to N H Proposition 3. RR n˚3276.
(24) 6. B. Beauquier, P. Hell, S. Pérennes. . (a). . . (c) .
(25) . (b).
(26) . . (p). (p). (p). . . . . (p). >. . .
(27) . (p). (p). . (p). . . . . . B 9
(28) "? G. Figure 1: (a) A digraph with originator and where . (b) The network (c) The network . All capacities not marked are equal to one. All arc orientations are omitted.. 2.
(29) .. References [1] B. Beauquier, J-C. Bermond, L. Gargano, P. Hell, S. Pérennes, and U. Vaccaro. Graph problems arising from wavelength–routing in all–optical networks. In Proc. of nd Workshop on Optics and Computer Science, April 1997. [2] N. K. Cheung, K. Nosu, and G. Winzer. Special issue on dense WDM networks. Journal on Selected Areas in Communications, 8, 1990. [3] L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 1962. [4] P. E. Green. Fiber–Optic Communication Networks. Prentice–Hall, 1993. [5] D. Minoli. Telecommunications Technology Handbook. Artech House, 1991. [6] B. Mukherjee. WDM-based local lightwave networks, Part 1: Single-hop systems. IEEE Network Magazine, 6(3):12–27, May 1992.. Contents 1 Motivation and Definitions. 3. 2 Multicasting and Network Flows. 4. INRIA.
(30) Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unité de recherche INRIA Sophia Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA ANTIPOLIS Cedex. Éditeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.
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