A Multi-Objective Optimization Scheme for Multicast Routing: A Multitree Approach

Telecommunication Systems 27:2–4, 229–251, 2004 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

A Multi-Objective Optimization Scheme for MulticastRouting: A Multitree Approach

YEZID DONOSO [email protected] Science Department, Universidad del Norte, Barranquilla, Colombia

RAMON FABREGAT and JOSE L. MARZO {ramon;marzo}@eia.udg.esInstitut d’Informàtica i Aplicacions, Universitat de Girona, Girona, Spain

Abstract. In this paper, we propose a multi-objective traffic engineering scheme using different distribu-tion trees to multicast several flows. The aim is to combine into a single aggregated metric, the followingweighting objectives: the maximum link utilization, the hop count, the total bandwidth consumption, andthe total end-to-end delay. Moreover, our proposal solves the traffic split ratio for multiple trees. We for-mulate this multi-objective function as one with Non Linear programming with discontinuous derivatives(DNLP). Results obtained using SNOPT solver show that several weighting objectives are decreased andthe maximum link utilization is minimized. The problem is NP-hard, therefore, a novel SPT algorithm isproposed for optimizing the different objectives. The behavior we get using this algorithm is similar towhat we get with SNOPT solver. The proposed approach can be applied in MPLS networks by allowingthe establishment of explicit routes in multicast events. The main contributions of this paper are the opti-mization model and the formulation of the multi-objective function; and that the algorithm proposed showspolynomial complexity.

Keywords: mathematical programming, optimization, traffic engineering, load balancing, multicast

1. Introduction

Traffic engineering is concerned with optimizing the performance of operational net-works. The main objective is to reduce congestion hot spots and improve resource uti-lization. This can be achieved by setting up explicit routes over the physical networkin such a way that the traffic distribution is balanced across several traffic trunks [Kimet al., 29]. This load balancing technique can be achieved by a multicommodity networkflow formulation [Ahuja et al., 7; Bazaraa et al., 11, 12] which leads to the traffic beingshared over multiple routes between the ingress node and the egress nodes in order toavoid link saturation and hence the possibility of congestion.

When we translate this balancing technique into a mathematical formulation, themain objective is to minimize the maximum link utilization. When the network is con-gested, minimizing the maximum link utilization involves

(1) minimizing the congestion of links,

230 DONOSO ET AL.

(2) reducing the total packet delay, and

(3) minimizing the total packet loss.

One solution is the multipath approach, in which the data is transmitted over differentpaths to achieve the aggregated, end-to-end bandwidth requirement. Several advantagesof using multipath routing are discussed in [Chen and Chan., 18]: the links do not getoverused and therefore do not get congested, and so has the potential to aggregate band-width, allowing a network to support higher data transfer than that which is possible withany one path. However, multiple paths may require more total network bandwidth re-sources, i.e. sum of assigned bandwidth at each link of the paths, than the single shortestpath.

We can also have per-flow multipath routing where an originating node uses multi-ple paths for the same flow, i.e. each flow is split along multiple subflows. The split ratiois fed to the routers which divide the traffic of the same ingress-egress nodes pair intomultiple paths. Several papers [Rost and Balakrishnan, 35; Cetinkaya and Knightly, 17;Sridharan et al., 39; Vutukury and Garcia, 41; Cho et al., 19] address the splitting multi-path problem of unicast traffic, motivated by its importance in any complete traffic engi-neering solution. Traffic splitting is executed for every packet in the packet-forwardingpath. A simple method to partition the input traffic is on a per-packet basis, for example,in a round-robin fashion. However, this method suffers from the possibility of excessivepacket reordering and is not recommended in practice. Villamizar [40] tries to balancethe load among multiple LSPs according to the loading for each path. In MPLS net-works [Rosen et al., 34] multiple paths can be used to forward packets belonging to thesame “forwarding equivalent class” (FEC) by explicit routing. The distribution of loadamong a set of alternate path is determined by the amount of number space from a hashcomputation allocated to each path. Effective use of load balancing requires good trafficdistribution schemes. In [Cao et al., 16] the performance of several hashing schemesfor distributing traffic over multiple links while preserving the order of packets within aflow is studied. Although hashing-based load balancing schemes have been proposed inthe past, [Cao et al., 16] is the first comprehensive study of their performance using realtraffic traces. In [Rost and Balakrishnan, 35] propose a multi-path transmission betweensources and destinations. The current configurations in computer networks provide anopportunity for dispersing traffic over multiple paths to decrease congestion. In thiswork dispersion involves the (1) splitting; and (2) forwarding of the resulting portionsof aggregate traffic along alternate paths. The authors concentrate on (1): methods thatallow a network node to subdivide aggregate traffic, and they offer a number of trafficsplitting policies which divide traffic aggregates according to the desired fractions of theaggregate rate. Their methods are based on semi-consistent hashing of packets to hashregions as well as prefix-based classification.

Multicast connections are connections between one or more senders and a num-ber of members of a group. The aim of multicasting is to be able to send data from asender to the members of a group in an efficient manner [Ammar et al., 8]. Many mul-ticast applications, such as audio and videoconferencing or collaborative environments

MULTI-OBJECTIVE OPTIMIZATION SCHEME 231

and distributed interactive simulation, have multiple quality-of-service requirements onbandwidth, packet delay, packet loss, cost, etc. In multicast transmission, the load bal-ancing consists of traffic being split (using the multipath approach) across multiple trees,between the ingress node and the set of egress nodes.

The MPLS community has focused its effort mainly on the label switching of uni-cast IP traffic, leaving the sections on multicast in the main MPLS documents [Rosenet al., 34] and [Andersson et al., 9] virtually empty, and is to be addressed in futurestudies. Based on this, there are some proposals about supporting multicast in MPLSnetworks. A framework for MPLS multicast traffic engineering proposed by [Oomset al., 32] gives an overview about the application of MPLS techniques to IP multicast.Another proposal is aggregated multicast [Fei et al., 25] where instead of constructinga tree for each individual multicast session, one can have multiple individual multicastsessions sharing a single aggregated tree to reduce multicast state and, correspondingly,tree maintenance overhead. Using MPLS with multicast has many benefits not only forreducing multicast forwarding states but also for traffic engineering and QoS issues.

In [Boudani and Cousin, 13] approach, multicast packets will be sent froma branching node router to another, but authors are not obliged to follow the samepath constructed by conventional multicast protocols. By using different LSPs, loadbalancing is ensured. Another proposals explain how to distribute labels for unidirec-tional multicast trees [Ooms et al., 32] and for bidirectional trees’ label distribution [Cuiet al., 21]. Boudani et al. [14] proposes a simulator for multicast routing over an MPLSnetwork by extending MPLS Network Simulator (MNS) [Ahn and Chun, 5].

To provide MPLS traffic engineering [Awduche et al., 10] for a point-to-multipoint(P2MP) application in a efficient manner in a large scale environment, P2MP TE mech-anisms are required [Aggarwal et al., 4]. Existing MPLS point-to-point (P2P) mecha-nisms have to be enhanced to support P2MP TE LSP setup. Yasukawa et al. [43] presentsa set of requirements for P2MP TE extensions to MPLS. The computation of P2MP TEpaths is implementation dependent and is beyond the scope of the solutions that are builtwith [Yasukawa et al., 43] as a guideline. The path information which could have beencomputed by some off-line or on-line algorithms.

This paper is based on two theories: the multi-objective optimization and the max-imum flow theories. The multi-objective optimization proposed is based on weightingmethods [Cohon, 20], the aim of which is to obtain an efficient solution for severalweighting objectives, where any improvement in one objective can only be achieved atthe expense of another. In our proposal, the main objective is to minimize the maximumlink utilization (MLU, denoted generally as α), the hop count (HC), the total bandwidthconsumption (BC), and the total end-to-end delay (DL). Furthermore, we propose aload-balancing scheme to create multiple trees. The solution obtained in this paper willbuild multiple trees for transporting several multicast flows. With this load balancingtechnique, each flow is split across multiple LSPs [Wang et al., 42] depending on thesolution obtained.

232 DONOSO ET AL.

Figure 1. Flow from N0 to {N5,N8,N11} is split into two sub flows, and each one is sent alongone “multiple LSP”: {(0, 1), (1, 6), (6, 5), (6, 9), (9, 8), (9, 11)} and {(0, 2), (2, 7), (7, 8), (2, 4), (4, 5),

(4, 10), (10, 11)}. The flow fraction along each “multiple LSP” is 0.72 and 0.28, respectively. Note that thetotal flow coming from each egress node is 1.

The maximum flow problem is given by: in a capacitated network, we wish to sendas much flow as possible between an ingress node and egress node without exceedingthe capacity of any link. These algorithms are of two types:

(1) Augmenting path algorithms that maintain mass balance constraints at every node ofthe network other than the ingress and egress nodes. These algorithms incrementallyaugment flow along paths from the ingress node to the egress node.

(2) Preflow-push algorithms that flood the network so that some nodes have excesses(or buildup of flow).

These algorithms incrementally relieve flow from nodes with excesses by sendingflow from the node forward toward the egress node or backward toward the ingressnode [Ahuja et al., 6, 7]. In our algorithm, we use the flow maximum theory whenwe send the demand of the flow through the different trees with its bandwidth bound.Table 1 summarizes several maximum flow algorithms.

The rest of this paper is organized as follows: in section 2, we describe some re-lated studies. In section 3, we propose the multi-objective load-balancing scheme. Insection 4, we propose a heuristic algorithm to solve the multi-objective problem pre-sented previously. In section 5, we present a performance analysis. Finally, in section 6,we give our conclusions and suggestions for further study.

2. Related work

Various traffic engineering solutions using techniques to balance loads by multiple routeshave been designed and analyzed in different studies which attempt to optimize a costfunction subject to constraints induced from QoS requirements of the application.


Table 1Summary of maximum flow algorithms

Algorithm Author(s) Running time Features

Labeling [Ford andFulkerson(1956)]

O(nmU) 1. Maintains a feasible flow and augmentsflows along directed paths in the residualnetwork from node s to node t

2. Easy to implement and very flexible3. Running time is pseudopolynomial: the

algorithm is not very efficient in practice

Aug

men

ting

path

algo

rith

m Dinic [Dinic(1970)]

O(n2m) 1. This algorithm augments flow alongthose paths P in the layered network

2. This algorithm does not allow traversingthe links of the layered network in the op-posite direction

Capacity scaling [Gabow(1985);Ahuja andOrlin(1991)]

O(nm log U) 1. A special implementation of the labelingalgorithm

2. Augments flows along paths from node s

to node t with sufficiently large residualcapacity

3. Unlikely to be efficient in practice

Successive shortestpath algorithm

[Ahuja andOrlin(1991)]

O(n2m) 1. Another special implementation of the la-beling algorithm

2. Augments flow along the shortest di-rected path from node s to node t in theresidual network

3. Uses distance labels to identify shortestpaths from node s to node t .

4. Relatively easy to implement and very ef-ficient in practice

Karzanov [Karzanov(1974)]

O(n3) 1. A special implementation of the preflow-push algorithm, but pushes flow from theingress node to the egress node using lay-ered networks instead of distance labels

Generic preflow-push [Goldbergand Tarjan(1986)]

O(n2m) 1. Maintains a pseudoflow; performs push/relabel operations at active nodes

2. Very flexible; can examine active nodesin any order

3. Relatively difficult to implement becausean efficient implementation requires theuse of several heuristics

FIFO preflow-push [Goldbergand Tarjan(1986)]

O(n3) 1. A special implementation of the genericpreflow-push algorithm

2. Examines active nodes in the FIFO order3. Very efficient in practice

234 DONOSO ET AL.

Table 1(Continued).


Highest-labelpreflow-push

[Goldberg andTarjan (1986)]

O(n2√m) 1. Another special implementation of thegeneric preflow-push algorithm

2. Examines active nodes with the highestdistance label

3. Possibly the most efficient maximumflow algorithm in practice

Pre

flow

-pus

hal

gori

thm

s

Lowest-labelpreflow-push

[Goldberg andTarjan (1986)]

O(n2m) 1. Another special implementation of thegeneric preflow-push algorithm

2. Examines active nodes with the highestdistance label

3. Possibly the most efficient maximumflow algorithm in practice

Excess scaling [Ahuja andOrlin (1989)]

O(nm + n2 log U) 1. A special implementation of the genericpreflow-push algorithm

2. Performs push/relabel operations atnodes with sufficiently large excessesand among these nodes, selects a nodewith the smallest distance label

3. Achieves an excellent running time with-out using sophisticated data structures

Stack scaling [Ahuja (1989)] O(nm+((n2 log U)/

(log log U)))

1. The Stack-scaling algorithm scales ex-cesses by a factor of k � 2 (i.e., re-duces the scale factor by a factor of k

from one scaling phase to another), andalways pushes flow from a large excessnode with the highest distance label

Wave scaling [Ahuja (1989)] O(nm + n2√log U) 1. The wave-scaling algorithm scales ex-

cesses by a factor of 2 and uses a parame-ter L whose value is chosen appropriately

N – number of nodes, m – number of augmentation; how many times a path is found and the residualcapacity of this path increased. U = max{Uij, (i, j) ∈ A}; maximum link capacity.

In [Rao and Batsell, 33] consider two generic routing algorithms that plan mul-tipaths, consisting of possibly overlapping paths, wherein bandwidth can be reserved,and guaranteed, once reserved, on the links. The first problem deals with transmit-ting a message of finite length from ingress node to egress node within r units of time.A polynomial-time algorithm is proposed and the results of a simulation are used toillustrate its applicability. The second problem deals with transmitting a sequence ofsome units at such a rate that maximum time difference between the two units receivedout of order is limited. The authors show that this second problem is computationally in-tractable, and propose a polynomial-time approximation algorithm. Therefore, a Quality


of Service (QoS) routing via multiple paths under a time constraint is proposed when thebandwidth can be reserved.

In [18], Chen and Chan propose an algorithm to carry out the unicast transmissionof applications requiring minimum bandwidth through multiple routes. The algorithmconsists of five steps:

(a) the multipath P set is initialized as empty;

(b) the maximum flow graph is obtained;

(c) the shortest route from the ingress node to the egress node is obtained;

(d) the bandwidth consumption obtained in the maximum flow of step (b) is decreasedand

(e) step (d) is repeated until the required bandwidth for transmission is reached.

The results presented show very similar end-to-end delay values to those obtained inde-pendently whether or not the load balancing is being applied. However, link utilizationis improved when the load balancing is applied.

In [42] Wang et al. present a multi-objective optimization scheme to transportunicast flows. In this scheme, the maximum link utilization (α) and the selection ofbest routes based on the flow assignation through each link is considered. In this paperis considered a new approach that accomplishes traffic engineering objectives withoutfull mesh overlaying. Instead of overlaying IP routing over the logical virtual network.Traffic engineering objectives such as balanced traffic distribution are achieved throughmanipulating link metrics for IP routing protocols such as OSPF. In this paper, theypresent a formal analysis of the integrated approach, and propose a systematic methodfor deriving the link metrics that convert a set of optimal routes for traffic demands toshortest path with respect to the link weights through. And the link weights can becalculated by solving the dual of a linear programming formulation.

Lee et al. in [31] propose a method for transporting unicast flows. The constraintof a maximum number of hops is added to the minimization of the maximum link uti-lization (α). Moreover, a division of the traffic over multiple routes in a discrete way isestablished. This division simplifies the implementation of the solution. The behaviorsof five approaches are analyzed: shortest path based on non-bifurcation, ECMP (EqualCost Multiple Paths), traffic bifurcation, H hop-constrained traffic bifurcation and Hhop-constrained traffic bifurcation with node affinity. Through the approaches of hop-constrained traffic bifurcation, a minimum value of the maximum link utilization (α) isobtained.

Seok et al. in [37] propose non-bifurcation and bifurcation methods to transportmulticast flows with hop-count constrained. In the analysis of results and simulations,they consider the non-bifurcation methods only. The constraint of consumption band-width is added to the constraints considered in [Rao and Batsell, 33]. In [Lee et al., 31]a heuristic is proposed. The proposed algorithm consists of two parts:

236 DONOSO ET AL.

(1) modifying the original graph to the hop-count constrained version,

(2) finding a multicast tree to minimize the maximum link utilization (α).

Aboelela and Douligeris in [1] propose a fuzzy optimization model for routing inB-ISDN networks. The proposed model challenge is to find routes for flows throughpaths that are not hideously expensive, according to the required QoS and do not penal-ize the other flows already existing or expected to arrive in the network. The model isanalyzed in terms of performance under different routing scenarios. They obtained goodimprovements in performance compared with the traditional single metric routing tech-niques (number of hops or delay based routing). This improvement was achieved whilemaintaining a sufficient low processing overhead. Throughput has been increased andprobability of congestion has been decreased by balancing the load all over the networklinks.

Roy et al. in [36] propose a new multicast tree selection algorithm based on non-dominated sorting technique of genetic algorithm to simultaneously optimize multipleQoS parameters. Simulation results demonstrate that the proposed algorithm is capableof discovering a set of QoS-based near optimal, non-dominated multicast routes withina few iterations. In this paper, they use a non-dominated sorting based genetic algo-rithm (NS-GA) technique to develop an efficient algorithm which determines multicastroutes on-demand by simultaneously optimizing end-to-end delay guarantee, bandwidthrequirements and bandwidth utilization without combining them into a single scalar ob-jective function.

Cui et al. in [22] propose an algorithm MEFPA (Multi-constrained Energy Func-tion based Precomputation Algorithm) for a multi-constrained QoSR problem based onthe analysis of linear energy functions (LEF). They assume that each node s in the net-work maintains a consistent copy of the global network state information. This algorithmtakes care of each QoS metric to b degrees. It then computes B (B = Ck−1

b+k−2) coeffi-cient vector that are uniformly distributed in the k-dimensional QoS metric space, andconstructs one LEF for each coefficient vector. Then based on each LEF, node s usesDijkstra’s algorithm to calculate a least energy tree rooted by s and a part of the QoSrouting table. Finally, s combines the B parts of the routing table to form the completeQoS routing table it maintains. For distributed routing, for a path from s to t , in ad-dition to the destination t and the k weights, the QoS routing table only needs to savethe next hop of each path. For source routing, the end-to-end path from s to t along theleast energy tree should be saved in the routing table. Therefore, when an QoS connec-tion request arrives, it can be routed by looking up a feasible path satisfying the QoSconstraints in the routing table.

Cho et al. in [19] propose two multi-path constraint based routing algorithms forInternet traffic engineering using MPLS. In a normal constraint-based shortest path first(CSPF) routing algorithm, there is a high probability that it can not find a feasible paththrough networks for a large bandwidth constraint which is one of the most significantconstraints for traffic engineering. The proposed algorithms can divide the bandwidthconstraint into two or more subconstraints and find a constrained path for each subcon-


straint, providing there is no single path satisfying the whole constraint. Extensive sim-ulations show that they enhance the success probability of path setup and the utilizationof network resources.

Abrahamsson et al. in [3] propose an intra-domain routing algorithm based onmulti-commodity flow optimization which enable load sensitive forwarding over mul-tiple paths. It is neither constrained by weight-tuning of legacy routing protocols, suchas OSPF, nor requires a totally new forwarding mechanism, such as MPLS. These char-acteristics are accomplished by aggregating the traffic flows destined for the same egressinto one commodity in the optimization and using a hash based forwarding mechanism.The aggregation also results in a reduction of computational complexity which makes thealgorithm feasible for on-line load balancing. Another contribution is the optimizationobjective function which allows precise tuning of the tradeoff between load balancingand total network efficiency.

Sridharan et al. in [39] propose an approach that remedies two main difficulties inoptimal routing. The first is that these protocols use shortest path routing with destinationbased forwarding. The second is that when the protocols generate multiple equal costpaths for a given destination routing prefix, the underlying forwarding mechanism per-forms load balancing across those paths by equally splitting traffic on the correspondingset of next hops. These added constraints make it difficult or impossible to achieve opti-mal traffic engineering link loads. It builds by taking advantage of the fact that shortestpaths can be used to achieve optimal link loads, but it is compatible with both destinationbased forwarding and even splitting of traffic over equal cost paths. Compatibility withdestination based forwarding can be achieved through a very minor extension to the re-sult of [Wang et al., 42], simply by taking advantage of a property of shortest paths andreadjusting traffic splitting ratios accordingly. Accommodating the constraint of evensplitting of traffic across multiple shortest paths is a more challenging task. The solutionwe propose leverages the fact that current day routers have thousands of route entries(destination routing prefixes) in their routing table. Instead of changing the forwardingmechanism responsible for distributing traffic across equal cost paths, we plan to controlthe actual (sub)set of shortest paths (next hops) assigned to routing prefix entries in theforwarding table(s) of a router.

Fortz and Thorup in [26] propose to optimize the weight setting based on the pro-jected demands. They showed that optimizing the weight settings for a given set ofdemands is NP-hard, so they resorted to a local search heuristic. They found weightsettings that performed to within a few percent from that of the optimal general routingwhere the flow for each demand is optimally distributed over all paths between sourceand destination. This contrasts the common belief that OSPF routing leads to congestionand it shows that for the network and demand matrix studied we cannot get a substan-tially better load balancing by switching to the proposed more flexible Multi-protocolLabel Switching (MPLS) technologies.

Cetinkaya and Knightly in [17], introduce Opportunistic Multipath Scheduling(OMS), a technique for exploiting short term variations in path quality to minimize de-lay, while simultaneously ensuring that the splitting rules dictated by the routing protocol

238 DONOSO ET AL.

are satisfied. In particular, OMS uses measured path conditions on time scales of up toseveral seconds to opportunistically favor low-latency high-throughput paths. However,a naive policy that always selects the highest quality path would violate the routing pro-tocol’s path weights and potentially lead to oscillation. Consequently, OMS ensures thatover longer time scales relevant for traffic management policies, traffic is split accordingto the ratios determined by the routing protocol. A model of OMS is developed and anasymptotic lower bound on the performance of OMS as a function of path conditions(mean, variance, and Hurst parameter) for self-similar traffic is derived.

Vutukury and Garcia in [41] propose a traffic engineering solution that, for a givenlong-term traffic matrix, adapts the minimum-delay routing to the backbone networkswhich is practical and is suitable to implement in a Differential Services framework.A simple scalable packet forwarding technique that distinguishes between datagram andtraffic that requires in-order delivery and forwards them accordingly and efficiently isintroduced.

Hachimi et al. in [28] propose a hybrid label aggregation algorithm in order tosolve multicast scalability problem and provide a solution for multicast in MPLS supportDiffServ. In the proposed scheme, one label is assigned per multicast group (logicalaggregation) and different multicast groups sharing the same interface in a router areaggregated locally (physical aggregation). In order to support the proposed algorithm,we proposed a separate treatment and labels space (prime numbers only) for multicasttraffic. The proposed solution allows consuming fewer labels, reducing the forwardingtable and consequently the total packet processing delay.

Kim et al. in [30] suggests a method to improve network performance by appro-priately distributing traffic in accordance with the state of paths, against dynamic trafficpattern occurring in a short time, in a multipath environment. TEAM (TE using anAdaptive Multipath-forwarding) is one of traffic engineering (TE) algorithms that aimsto improve network performance by properly distributing traffic against dynamic trafficpattern occurring in a short time scale. This method monitors states of paths by us-ing probe packet in the ingress node of the network and computes costs of path withmonitored values. Path costs consist of weights given in paths, like packet’s delay andloss rate, the number of hops and the number of LSPs. This enables to adapt to stateof network without a sudden change by tracing neighboring solutions from an existingsolution. Therefore, it is observed that network performance is improved when the totalcosts of paths of the whole network are minimized. Also, by distributing traffic into eachinterface using table-based hashing method, the ordering of packets problem is solved.

Song et al. in [38] proposes an adaptive multipath traffic engineering mechanismnamed LDM (Load Distribution over Multipath). The main goal of LDM is to enhancethe network utilization as well as the network performance by adaptively splitting trafficload among multiple paths. LDM takes a pure dynamic approach not requiring any apriori traffic load statistics. Routing decisions are made at the flow level and trafficproportioning reflects both the length and the load of a path. LDM also dynamicallyselects a few good Label Switched Paths (LSPs) according to the state of the entirenetwork.


Donoso et al. in [23] we presented several static models which are analyzed bycomparing some particular one-objective optimization functions with the multi-objectiveoptimization function.

Table 2 summarizes the main characteristics of these proposals and our own op-timization model which is presented in the following sections. As you can see in thetable 2, none of the above proposals consider how to find the appropriate multiple treesto minimize all four features (maximum link utilization, number of hops, end-to-enddelay and total bandwidth consumption) which we address in the optimization modelproposed in this paper. In this paper, we have decided to work with hop count (HC) in-stead of flow assignation (FA) because HC is more practical in real network topologiesand in this case it is possible to see exactly the amount of total hops in the tree.

While in [Rao and Batsell, 33; Aboelela and Douligeris, 1; Sridharan et al., 39;Fortz and Thorup, 26; Song et al., 38] consider unicast flow; in [Chen and Chan., 18;Wang et al., 42; Lee et al., 31; Cho et al., 19; Abrahamsson et al., 3; Cetinkayaand Knightly, 17; Vutukury and Garcia, 41] this unicast flow is splitting; and in[Seok et al., 37; Roy et al., 36; Cui et al., 22] the flow is multicast but not split-ting, thus our proposal solves the traffic split ratio for multicast flows. The majordifferences between our work and the other multicast works are: first, from the view-point of objectives and function, we are proposing a multi-objective scheme to solvethe optimal multicast routing problem with some constraints, while [Roy et al., 36;Cui et al., 22] propose a scheme without constraints; second, from the viewpoint ofhow many trees are used, we are proposing a multi-tree scheme to optimize the resourceutilization of the network, while [Seok et al., 37; Roy et al., 36] propose only one treeto transmit the flow information; and third, from the viewpoint of traffic splitting, weare proposing the traffic split to transmit the multicast flow information through the loadbalancing using several trees to the same flow, while [Seok et al., 37; Roy et al., 36;Cui et al., 22] do not propose this characteristic.

It should be pointed out that [Song et al., 38; Wang et al., 42; Lee et al., 31; Choet al., 19; Vutukury and Garcia, 41; Kim et al., 30; Seok et al., 37] proposals can beapplied to MPLS networks, in table 2 are in boldface.

3. Optimal multicast routing

The network is modeled as a directed graph G = (N,E), where N is the set of nodesand E is the set of links. We denote by n the number of network nodes, i.e., n = |N |.Among the nodes, we have a source s ∈ N (ingress node) and some destinations T (theset of egress nodes). Let t ∈ T be any egress node. Let (i, j) ∈ E be the link fromnode i to node j . Let f ∈ F be any multicast flow, where F is the flow set and Tf isthe egress nodes subset to the multicast flow f . We denote by |F | the number of flows.Note that T = ⋃

f ∈F Tf .

Let Xtfij be the fraction of flow f to egress node t assigned to link (i, j). Note that

these variables include the egress node t , which is not considered in previous works.

240 DONOSO ET AL.

Table 2Type of transported flow and constraints.

Objectives Constraints Flow Path/ Split HeuristicTrees

[Rao andBatsell, 33]

DL BC Unicast Multi-path

– Ford–Fulkersonmethod

[Aboelela andDouligeris, 1]

MLU HC DL BC Unicast Multi-path

– Fuzzy logic

[Sridharanet al., 39]

MLU BC BC Unicast Multi-path

– Linear pro-grammingand shortestpath

[Fortz andThorup, 26]

MLU BC Unicast Multi-path

– Linear pro-grammingand shortestpath

[Songet al., 38]

HC BC HC Unicast Multi-path

– (Linear)multi-commoditynetwork flowproblem

[Chen andChan., 18]


Split Max-flowand shortestpath

[Wanget al., 42]

MLU BC FA BC Unicast Oneandmulti-plepaths

Split Linear pro-gramming

[Lee et al., 31] MLU HC BC Unicast Multi-path

Split Mixed-integerprogram-ming

[Cho et al., 19] BC BC MSF Unicast Multi-path

Split Max-flowand shortestpath

[Abrahamssonet al., 3]

MLU Unicast Multi-path

Split (Linear)multi-commoditynetwork flowproblem

[Cetinkaya andKnightly, 17]

DL AQS BC Unicast Multi-path

Split Schedulingalgorithm

[Vutukury andGarcia, 41]


Split Nonlinearprogram-ming

[Kim et al., 30] HC DL PL NL Unicast Multi-path

Split Shortest path


Table 2(Continued).

Objectives Constraints Flow Path/ Split HeuristicTrees

[Seok et al.,37]

MLU BC HC BC MSF Multicast Onlyonetree

– Mixed-integerprogram-ming

[Roy et al., 36] DL BC FA Multicast Onlyonetree

– Geneticalgorithms

[Cui et al., 22] Linear energy functions (LEF) Multicast Multi-tree

– Shortest pathtree

[Donosoet al., 23;Donoso etal., 24] andthis paper

MLU HC DL BC BC MSF Multicast Multi-tree

Split Nonlinearprogram-ming,max-flowand shortestpath tree

MLU: maximum link utilization NL: number of LSPsHC: hop count AQS: average queue sizeDL: total delay FA: flows assignationBC: total bandwidth consumption MSF: maximum number of subflows. Handle of flow fraction

by each egress node across a link in the optimizationmodel.PL: packet loss

To include the egress nodes permits to control the bandwidth consumption in each linkwith destination to the set of egress nodes. Therefore, it is possible to maintain exactlythe constraint of flow equilibrium to the intermediate nodes. The problem solution, X

tfij

variables, provides optimum flow values.Let cij be the capacity of each link (i, j). Let bwf be the traffic demand of a flow

f from the ingress node s to Tf . The binary variables Ytfij represent whether link (i, j) is

used (1) or not (0) for the multicast tree rooted at the ingress node s and reaching egressnode subset Tf . Let vij be the propagation delay of link (i, j). Let m be the number ofvariables in the multi-objective function. Let connectionij be the indicator if there is alink between nodes i and j .

The problem of minimizing |F | multicast flows from ingress node s to the egressnodes of each subset Tf is formulated as follows:

Minimize

(r1α + r2

∑f ∈F

∑t∈Tf

∑(i,j)∈E

Ytfij + r3

∑f ∈F

∑(i,j)∈E

bwf maxt∈Tf

(X

tfij

)

+ r4

∑f∈F

∑t∈Tf

∑(i,j)∈E

vijYtfij

)(MHDB model) (1)

242 DONOSO ET AL.

subject to ∑(i,j)∈E

Xtfij −

∑(j,i)∈E

Xtfji = 1, t ∈ Tf , f ∈ F, i = s, (2)

∑(i,j)∈E

Xtfij −

∑(j,i)∈E

Xtfji = −1, i, t ∈ Tf , f ∈ F, (3)

∑(i,j)∈E

Xtfij −

∑(j,i)∈E

Xtfji = 0, t ∈ Tf , f ∈ F, i �= s, i /∈ Tf , (4)

∑f∈F

bwf maxt∈Tf

(X

tfij

)� cijα, α � 0, (i, j) ∈ E, (5)

∑j∈N t∈Tf

Ytfij �

⌈bwf⌊∑

j∈N cij/∑

j∈N connectionij

⌋⌉, i ∈ N, f ∈ F (6)

where

Xtfij ∈ �, 0 � X

tfij � 1, (7)

Ytfij = ⌈

Xtfij

⌉ ={

0, Xtfij = 0,

1, 0 < Xtfij � 1,

(8)

m∑i=1

ri = 1, ri ∈ �, ri � 0, m > 0. (9)

The multi-objective function (MHDB model) (1) defines a function and generatesa single aggregated metric for a combination of weighting objectives.

The main objective consists of minimizing the maximum link utilization (MLU),which is represented as α in (1). In this case, the solution obtained may report longroutes. In order to eliminate these routes and to minimize hop count (HC), the term∑

f ∈F

∑t∈Tf

∑(i,j)∈E Y

tfij is added. This is needed because the objective function may

report only the most congested link and the optimal solution may include unnecessarilylong paths in order to avoid the bottleneck link [Kim et al., 29].

In order to minimize the total bandwidth consumption (BC) over all links, the term∑f ∈F

∑t (i,j)∈E bwf maxt∈Tf

(Xtfij ) is also added.

Furthermore, in order to minimize the total end-to-end propagation delay (DL)over all links, the term

∑f∈F

∑t∈Tf

∑(i,j)∈E vijY

tfij is also added. In [Aboelela and

Douligeris, 1] is showed that delay has three basic components: switching delay, queu-ing delay and propagation delay. The switching delay is a constant value and it can beadded into the propagation value and the queuing delay is already reflected in the band-width consumption. They said that the queuing delay is used as an indirect measure ofbuffer overflow probability (to be minimized). Other computational studies [Aboelelaand Douligeris, 1] has been shown that typically makes little difference whether the costfunction used in routing includes the queuing delay or the much simpler form of the linkutilization.


Constraints (2)–(4) are flow conservation constraints. Constraint (2) ensures thatthe total flow emerging from ingress node to any egress node t at flow f should be 1.Constraint (3) ensures that the total flow coming from an egress node t at flow f shouldbe 1. Constraint (4) ensures that for any intermediate node different from the ingressnode (i �= s) and egress nodes (i /∈ T ), the sum of their output flows to the egress node t

minus the input flows with destination egress node t at flow f should be 0.Constraint (5) is the maximum link utilization constraint. In an unicast connec-

tion, the total amount of bandwidth consumed by all the flows with destination to theegress node t must not exceed the maximum utilization (α) per link capacity cij, thatis,

∑f∈F bwf

∑t∈T X

tfij � cijα, (i, j) ∈ E. Nevertheless, in constraint (5) only the

maximum value of Xtfij for t ∈ Tf must be considered. Though several subflows of the

flow f in the link (i, j) with destination to different egress nodes are sent, in multicastIP specification just one subflow will be sent. The function max in constraint (5) gener-ates discontinuous derivatives. For this reason, the problem should be solved through aGAMS tool for solving DNLP (Nonlinear programming with discontinuous derivatives)such as MINOS, MINOS5, COMOPT, COMOPT2, and SNOPT (GAMS (2004)). TheDNLP problem is the same as the NLP (Nonlinear Programming) problem, except thatnon-smooth functions (abs, min, max) can appear.

Constraint (6) limits the maximum number of subflows in each node by means ofthe capacity of each link and the traffic demand. This formulation represents the amountof necessary links for a demand of traffic Without this constraint, the model could sufferfrom scalability problems, i.e. the label space usage by LSPs would be too high. In[Donoso et al., 23] the right expression of the constraint (6) is a constant value.

Expression (7) shows that the Xtfij variables must be real numbers between 0 and 1.

These variables form multiple trees to transport a multicast flow. The demand betweenthe ingress node and the egress node t may be split over multiple routes. When theproblem is solved without load balancing, this variable will only be able to take values 0and 1, which will show, respectively, whether or not the link (i, j) is used to carryinformation to the egress node t .

Expression (8) calculates Ytfij as a function of X

tfij .

Finally, expression (9) shows that the weighting coefficients ri assigned to the ob-jectives are normalized. These values are calculated through the solution of the opti-mization problem.

The problem presented is NP-hard because the problem of computing the min-imum cost tree for a given multicast group is known as a Steiner tree problem(NP-complete) and this model includes constrained integer and real variables.

4. Proposed heuristic

4.1. Algorithm

In this section we propose a heuristic algorithm to solve the multiobjective model pro-posed in (1). This algorithm, shown in figure 3, receives as parameters, G(N,E), s, Tf ,

244 DONOSO ET AL.

Table 3The weighting objective vector r .

r1 r2 r3 r4

0.997925 0.001 0.001 0.000075

Table 4Distance labels by paths.

Dist. nodes hops mseg Mbps

d1 – 0 0 0d2 P1: {1, 2} 1 10 1.5

P3: {1, 2} 1 10 1.5

d4 P1: {1, 2, 4} 2 15 = 10 + 5 1.5P2: {1, 3, 4} 2 15 = 8 + 7 1.5P3: {1, 2, 3, 4} 3 25 = 10 + 8 + 7 1.5

d3 P2: {1, 3} 1 8 1.5P3: {1, 2, 3} 2 18 = 10 + 8 1.5

bwf and r, where G is the network topology, N is the nodes set, E is the links set, s isthe ingress node for the flow f , Tf is the egress nodes subset, bwf is the transmissionrate for the flow f , and r are the weighting objectives vector. Let d

tfi denote the length of

a SPT from the ingress node to the egress nodes Tf . Let λf be a tree from s to t , t ∈ Tf ,associated to the flow f , which consists of several paths P

tfk with 1 � k � h, where

h is the number of paths. This algorithm consists of two steps: (1) obtaining graph G′with distance based on hops count, bandwidth consumption and delay, and (2) findingthe multicast tree.

Step 1. Obtaining modified graph G′. In this step, all possible paths between theingress node s and every egress node t , t ∈ Tf are looked for. This step consists ofthree nested loops which calculate (for each egress node, t ∈ Tf , for each path P

tfk with

destination to the node t and for each node i, i ∈ Ptfk ) the distance value d

tfi based on

hops count, bandwidth consumption and delay.

Step 2. Finding the multicast tree. The second step consists of finding out, for eachegress node t ∈ Tf , the paths required to transmit the flow of information, accordingto graph G′ with its distance variables, d

tfi , and the available capacity of each path. To

find out the cost of the path, the r ′is obtained with SNOPT (see table 3) in the solution of

the multi-objective function of the MHDB model are taken into account. From amongthe paths that have a capacity greater than zero, the path with the minimum cost isselected and the maximum possible flow is applied, in accordance with the maximumlink utilization constraint (5).

For a simple case with just one egress node (as in the topology shown in figure 2),table 4 shows the values of the distances d

tfi obtained on the first run of the algorithm.


Figure 2. Graph G.

Algorithm MMR (G(N,E), s, Tf , bwf , r):begin

Xtfij ← 0

// Step1: (Obtain modified graph G′) //for each (t ∈ Tf ) do

while (exists a path Ptfk from node sf to node t) do

for each (i ∈ Ptfk ) do

dtfi ← {αp,HCp,DLp,BCP };

endforendwhile

endforα ← min αp

// Step2: (Finding multicast tree) //for each (t ∈ Tf ) do

while (bwf > 0 &&X

tfij ← identify a path P

tfk in d

tfi from node sf to node t

min(r1 ∗ α + r2 ∗ HCP + r3 ∗ DLP + r4 ∗ BCP ) �= 0)) doδ ← min{max_flowij, (i, j) ∈ P

tfk };

increase δ units of flow along Ptfk and update G′;

bwf ← bwf − δ;endwhile

endforend algorithm;

Figure 3. Algorithm MMR (Multicast Multi-objective Routing).

The Xtfij values returned by the algorithm allow us to calculate the values of MLU,

HC, DL and BC according to the multi-objective (1) of the MHDB model.Table 5 summarizes the main features of the algorithms proposed.

246 DONOSO ET AL.

Table 5Summary of the algorithms proposed.


Augmentingpath algorithm

MMR [Donoso et al., 23] O(n3 log n) 1. Find several trees to transmit multicastflow through load balancing

2. The several trees are found through themulti-objective scheme

n – number of nodes.

4.2. Complexity analysis

Step 1. The complexity of outer loop of the algorithm is bound by O(|Tf |). The maxi-mum value is O(n), where n is the number of nodes, because all nodes can be an egressnode. To look for the paths, a second loop through a breadth-first search path and thebest known bound is O(n log n) in a directed graph, where n is the number of nodes inthe graph. In order to assign the distance labels d

tfi to each node belonging to a path

Ptfk , a third loop needs to be carried out, which in the worst case could have n nodes. In

summary, the complexity of this step is O(n3 log n).

Step 2. At most, there may be O(n) nodes as egress nodes and, at most, there may beO(n) paths to each egress node in summary, the complexity of this step is O(n2).

Finally, the complexity of the algorithm proposed would be given by O(n3 log n

+ n2). In which case, what we find is a polynomial solution to the problem of multi-objective optimization that we are describing in this paper. Algorithm proposed in[Seok et al., 37] has this same complexity, but in this paper optimizing several objec-tives.

5. Performance evaluation

5.1. Network topology

The optimization variables (MLU, HC, DL and BC values) are calculated using a GAMSsolver called SNOPT and the results are compared with the heuristic algorithm pre-sented. The topology used is the 14-node NSF (National Science Foundation) network(figure 4). The costs on the links represent the delay and all links have 1.5 Mbps ofbandwidth capacity.

Two flows with the same source, s = N0, are transmitted by each analysis. Theegress nodes subsets are T1 = {N5, N8, N11} and T2 = {N8, N11, N13}. The transmis-sion rates are 256 Kbps, 512 Kbps, 1 Mbps, 1.5 Mbps, 2 Mbps and 2.5 Mbps for eachflow.


Figure 4. NSF network.

5.2. Comparison between MHDB model and heuristic algorithm MMR

To compare the incidence of increments or decrements of MMR with respect to theMHDB-model, different normalized variables, i.e. for MLU, MLUMMR vs MHDB =(MLUMMR − MLUMHDB)/MLUMHDB are calculated.

As the SNOPT solver calculates results to a set of flows and MMR is applied toonly one flow, the MMR algorithm will be run first for a flow (f1) and then for theother flow (f2), i.e. MMR(f1, f2). Figures 5(a) and (b) show normalized values ofMMR(f1, f2) versus MHDB and MMR(f2, f1) versus MHDB respectively. In bothcases, normalized MLU is very near to 0, i.e. the MMR shows the same behavior asthe MHDB model. With respect to the normalized HC variable, the MMR behaves bet-ter than the MHDB model showing a decrement (normalized HC < 0). With respectto the normalized DL variable, the MMR also behaved better. Finally, for the normal-ized BC variable, the MMR behaved worse than the MHDB model, with increases ofbetween 20% and 60%; though it can be observed that as the flow transmission rate isincreased, the aforementioned variables show a tendency to behave in the same way inboth the proposed algorithm and the mathematical model.

This behavior results from the fact that both the mathematical model and the pro-posed algorithm optimize the use of network resources for traffics exceeding the capacityof a single path. These results show that when multiple objectives are optimized, whilesome objectives are improved, others are worsened. Furthermore, we can see that theproposed algorithm (MMR) behaves in a satisfactory way, minimizing some variablesand sacrificing others but, in all cases, with a performance close to that indicated by theMHDB model.

5.3. Computational time

We measured the running time of the SNOPT solver and the MMR algorithm on a Pen-tium 1.2 GHz PC running MS Windows XP with 256 MB of RAM memory. For the

248 DONOSO ET AL.

(a)

(b)

Figure 5. Normalized values MMR vs. MHDB to the flow f2 and f1 (variables MLU, HC, DL, BC).

SNOPT solver, the mean value was 400 msec, the maximum value was 450 msec andthe minimum value was 320 msec. In the tests carried out, we did not observe significantincreases in execution times when the MHDB model was considered. For MMR algo-rithm the mean value was 65 msec, the maximum value was 90 msec and the minimumvalue was 40 msec.

6. Conclusions

In this paper, we have presented multi-tree routing to develop a multicast transmissionwith load balancing, using multiple trees. We have employed a multi-objective, load-balancing scheme to minimize: the maximum link utilization (α), the hop count (HC),the total bandwidth consumption (BC), and the total end-to-end delay (DL). By introduc-ing HC, lengthy paths are eliminated. By introducing BC, the bandwidth consumptionby links is minimized. Using DNLP mathematical programming, we obtained the opti-mal set of X

tfij (the fraction of flow f to destination node t assigned to link (i, j)) for the

problem (MHDB).


Furthermore, because finding the solution of this optimal DNLP problem is NP-hard, in this paper, we propose an algorithm that calculates a multi-objective multicasttree in polynomial time O(n3 log n + n2), which is the same computational complexityshown in [Lee et al., 31], but with several objectives in our case. The optimization vari-ables (MLU, HC, DL and BC values) are calculated using a GAMS solver called SNOPTand the results are compared with the heuristic algorithm presented. The simulation re-sults show that the proposed MMR algorithm solves the multi-objective multicast routingwith nearly the same variables as that of the optimal solution obtained with the SNOPTsolver in the MHDB model. Finally, the proposed approach can be applied in MPLS net-works by allowing the establishment of explicit routes in multicast events [Boyle et al.,15; Ooms et al., 32].

For further study, we will consider that the members of the multicast group cannotbe assumed to be static and we will design an algorithm for dynamic traffic. In thiscase, an auxiliary optimization model will be formulated. We will also investigate theapplicability of evolutionary algorithms in the multi-objective load-balancing schemeproposed in this paper.

Acknowledgments

This work was partially supported by the MCyT under the project TIC2003-05567. Thework of Yezid Donoso was supported by the Universidad del Norte (Colombia) undercontract G01 02-MP03-CCBFPD-0001-2001.

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A Multi-Objective Optimization Scheme for Multicast Routing: A Multitree Approach

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