An Efficient Approach for Constructing Dynamic Multicast Trees

In multicast transmissions network resources are more efficiently used than multiple point-to-point transmissions. However, the problem of creating optimal multicast trees, named the Steiner Tree problem in Networks (SPN), is NP-complete, and to
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  An Efficient Approach for Constructing Dynamic Multicast Trees G. Aiello, S. Gaglio, P. Storniolo and A. Urso  Abstract   —In multicast transmissions network resources aremore efficiently used than multiple point-to-point transmissions.However, the problem of creating optimal multicast trees,named the Steiner Tree problem in Networks (SPN), is NP-complete, and to solve it heuristic methods are generallyused. In this paper we propose the adoption of the RandomNeural Network (RNN) model for the solution of the dynamicversion of the Steiner Tree Problem in Networks. The RandomNeural Network is adopted as a heuristic capable of improvingsolutions achieved by previously proposed dynamic algorithms.The RNN model is adapted in order to map the network charac-teristics during a dynamic multicast transmission. The proposedapproach is validated by means of extensive experiments. I. INTRODUCTIONIn the last years multimedia networking applicationssuch as distance education, remote collaboration, video-on-demand services and teleconferencing will becomewidespread, relying on the ability of the network to providemulticast services effectively and efficiently. The multicasttransmission refers to the capability of sending the same datatowards multiple destinations. A distribution tree connectingsource and destinations must be selected on the graph, whichrepresents the network topology. Two different criteria can beadopted for the optimization of the multicast tree. The firstexploits the concept of the shortest path and it is known as ”shortest path tree” . The shortest path [1] can be defined as the  ”minimum cost path”  that connects two hosts inthe network. Another criterion of optimization consists of minimizing the cost of the overall tree. This is equivalent tofind the optimal Steiner Tree and it is universally referred asthe Steiner Tree Problem in Networks  (SPN) . This problemhas been proved to be NP-Complete in its decisional version.However, there exist several heuristics [12] that allow todetermine sub-optimal solutions in polynomial time. Theliterature presents, also, some post-processing techniques [6],which are capable of improving previously determined so-lutions. A further classification of the multicast problemconsists of the following two categories:  static multicast  problem , in which the destination subset is fixed, and the best Steiner tree has to be determined;  dynamic multicast  problem , in which the destination subset can dynamicallychange because of join or delete requests. In this paper, wedeal with the dynamic multicast and we try to minimize thespanning tree cost during the overall session. We present the ”Real Time Random Neural Network” (RTRNN)  heuristic, a S.Gaglio and G. Aiello are with DINFO, Dipartimento di IngegneriaInformatica, University of Palermo, Italy A.Urso and P. Storniolo are with ICAR-CNR, Istituto di Calcolo e Retiad Alte Prestazioni, Consiglio Nazionale delle Ricerche, Palermo, Italy { storniolo,urso }  post-processing technique, which improves multicast trees inthe dynamic case. It is based on the Random Neural Network (RNN) [2], which its authors successfully applied to the staticcase of the SPN [3]. Random Neural Networks find their ap- plications in many NP-complete problems, such as the Trav-eling Salesman Problem [5], function approximations [4],cognitive packet networks [7], etc., and their good capabilityof reinforcement learning has been demonstrated [7].In this paper, we show that the Random Neural Network model is applicable to the dynamic multicast problem,and we compare  RTRNN   with three well known dynamicheuristics:  Greedy Heuristic (GH)  [11],  Weighted Greedy Heuristic (WGH)  [11], and   Virtual Trunk Dynamic Multicast (VTDM)  [8]. The remainder of the paper is structured asfollows: in the next section the Random Neural Network model is introduced; in section III the application of the RNNto the dynamic multicast problem is described; section IV isdevoted to the experimental results. Finally, section V tracessome conclusions.II. T HE  R  ANDOM  N EURAL  N ETWORK   M ODEL In the  Random Neural Network   model neurons are char-acterized by a state or potential  k  such that  k i  ≥  0 ,  ∀ i  ∈  S  ,where  S   represents the set of neurons that constitute thenetwork. The neuron potential depends on the potentials of other neurons through the exchange of   spikes .  Spikes  areunitary amplitude signals that have the effect of increasingor decreasing the potentials of neurons in which they arrive.Spikes can be classified as:  excitatory spikes  represented  by  +1 , and   inhibitory spikes  represented by  − 1 . Excitatoryspikes increase the potential of the target neurons, activatingthem, if disabled. Inhibitory spikes decrease the potentialof their target neurons; null potential means that neuron isdisabled, and an inhibitory spike does not produce any effect.A neuron with positive potential can randomly fire either excitatory or inhibitory spikes. The interfering rate  r i , for sending signals to other neurons or outside the network,is exponentially distributed. When the neuron  i  fires, thespike will arrive at neuron  j  with a probability  p + ij  if thespike is excitatory, and with a probability  p − ij  if the spike isinhibitory. Furthermore, a neuron  i  can fire spikes outside thenetwork with a probability  d i . The probability assignment isconstrained by the following equation: d i  +  j [  p + ij  +  p − ij ] = 1  (1)We define the rate, at which neuron  i  sends excitatory spikesto the neuron  j , as  w + ij  =  r i  p + ij  , and the rate, at whichthe neuron  i  sends inhibitory spikes to the neuron j, as  Fig. 1. A neuron set in RNN and thier change of spikes. w − ij  =  r i  p − ij  . The following two parameters describe theRNN model:  p ( k,t ) =  Pr [ K  ( t ) =  K  ] , q  i ( t ) =  Pr [ k i ( t )  >  0]  (2)where  K  ( t ) = ( k 1 ( t ) ,k 2 ( t ) ,...,k n ( t ))  represents the vector of the network state, and   k  is a fixed value of the network  potential. Equations 2 represent, respectively, the distribu-tion of probability of the network state and the marginal probability that neuron  i  is excited at  t  instant. In our heuristic we adopt the last parameter to select the neuronwhich is a potential Steiner vertex candidate. The time-dependent behavior of the model is represented by an infinitesystem of Chapman-Kolmogorov equations for discrete statespace and continuous Markovian systems. Let  t  fixed, thestationary probability distribution associated with the modelis the quantity produced in the following equations:  p ( K  ) = lim t →∞ Prob [ K  ( t ) =  K  ] , q  i  = lim t →∞ Pr [ k i ( t )  >  0] (3)If the neuron  i  is stable[4], the  q  i  quantity is given by: q  i  =  λ + i r i  + λ − i (4)where  λ + i  and   λ − i  represent respectively the excitatory spikesand the inhibitory spikes that neuron  i  receives from othersneurons and from outside the network. Their formalizationis shown in the following equations: λ + i  = n  j =1 q  j r j  p + ji  + Λ i  , λ − i  = n  j =1 q  j r j  p − ji  + λ i  (5)where  Λ i  is the rate of exogenous excitatory signals thatarrive at neuron  i ,  λ i  represents the rate of inhibitory spikesthat neuron  i  receives from outside the network. Both  Λ i  and  λ i  arrive at neuron  i  with a rate that satisfies the Poissondistribution with average value  1 /r i .A neuronal configuration in RNN may be represented as inFigure 1.III. T HE  RNN  MODEL FOR DYNAMIC  SPNThe Steiner Tree Problem can be defined as the oneof finding the minimum cost tree, which spans the nodes belonging to a given subset of all the graph nodes. It can beconsidered as a problem of optimum interconnection, withinthe scope of an assigned topology.  A. The static multicast problem Formally, the Steiner Tree problem in Networks may bedefined as follows: Given:1) a weighted indirect, connected graph  G  = ( V,E,c ) ,where  c  :  E   →  R + is a weight function that mapssubset  E   in a set of positive real numbers;2) a set of destination vertices  Z   ⊆  V  .Find: a  T   tree of minimum cost such that there is a path between every pair of vertices in  Z  .Multicast vertices may be connected via other nodes not belonging to the multicast set. Those nodes are called Steiner vertices. If  | Z  |  =  | V  | , the SPN is equivalent to find the Mini-mum Spanning Tree which is polinomially solvable by either Prim or Kruskal algorithms [12]. If   | Z  |  = 2 , the problemsimply consists of connecting the nodes by the shortest path and it can be solved using the Dijkstra algorithm [1].Although trivial cases can be solved in polynomial time, ithas been proved that instances of the problem where  2  < z < | V  |  are NP-complete, also in the case where the network is planar (with unitary edge costs) [13]. Some algorithmsthat solve small size instances in exponential time have been proposed in the past. However, for real applications, thereis the need of heuristics techniques that solve the problemin polynomial time, providing approximate solutions of theoptimal Steiner tree.  B. The dynamic multicast problem The dynamic version of the multicast problem consistsof finding a sequence of optimal trees, after the executionof a sequence of operations representing node insertions or deletions. Operations are coded in a two component-vector  ( node,action )  that represents the history of multicast ses-sion. We assume that each session starts with the source nodeonly. Starting from a sequence representing the multicastsession in the form  ( n,a )  where  n  ∈  V   is a node belongingto the network and   a  ∈ { 0,1 }  is the action such as insertionor deletion, the dynamic multicast problem can be defined as follows:Given:1) An indirect, connected graph  G  = ( V,E,c )  where  E  is a set of edges and   c  :  E   →  R + is a weight functionmapping edges in  R + , the ”weight” can be defined asthe cost of the link,2) A set of multicast nodes  Z   ⊆  V  ,3) A vector of requests  R  =  { r 1 ,r 2 ,...,r n }  where  r i  ∈ R  is a pair   ( n i ,a i ) .4) A multicast tree  T  ( V  ′ ,E  ′ )  where  V  ′ ⊆  V   and   E  ′ ⊆ E  .Find: The sequence of trees  T  1 ,T  2 ,...,T  m  such thatnodes-member of each tree  T  i  are obtained from  T   modified    by requests  r 1 ,r 2 ,...,r m , and their cost (sum of weights of edges) is minimum between all the possible combinations of  T  i . C. Algorithm description The goal of our algorithm is the efficient updating of amulticast tree after each real time insertion or deletion of other multicast nodes. In order to apply the RNN to dynamicmulticast, we associate a physical meaning to the involved variables. A mapping of a computer network on a RNN can be carried out as follows:1) a neuron represents a node of the real network;2) the weight of the edge connecting neurons i and jdepends on the rate used by neuron  i  to send excitatoryspikes to neuron  j  and it is calculated, as in [5],according to the following equations: w + ij  =  A ′ A ij if A ij   =  ∞ , w − ij  = 0  if A ij  =  ∞  (6)where  A ′ is the network link average cost,  A ij  is thecost of link connecting node  i  to node  j . If   A ij  =  ∞ ,the link   < i,j >  does not exist. Therefore, since  w + ij  isinversely proportional to the cost of the link, it may beviewed as the link capability of transporting multicast packets;3) parameter   w − ij  may be simply viewed as a flag indicat-ing the existence in the network of a link connectingnode  i  to node  j . Consequently,  w − ij  is computed asfollows: w − ij  = 0  if A ij   =  ∞ , w − ij  = 1  if A ij  =  ∞  (7)4) the  k i  potential of the  i  neuron is constant: this meansthat the random neural network is analyzed in steadystate;5) the marginal probability distribution  q  i  may be asso-ciated to the goodness of neuron  i  to be a potentialSteiner vertex candidate, and it is calculated accordingto: q  i  =  j q  j  · w + ij  + Λ i r i  +  j q  j  · w − ij  + λ i (8)In the equation 8 numerator represents the number of packetsentering node  i , and denominator represents the number of  packets that cannot reach node  i ; from the analysis of the previous equation it emerges that with a fixed numerator, the probability that neuron  i  will be chosen as potential Steiner vertex candidate is inversely proportional to the the number of multicast packets that do not enter the  i  neuron. The  q  i value depends on the current tree topology. If a neuron isa multicast node, it must receive all multicast packets and therefore  q  i  = 1 , which, according to [2], means neuroninstability.Let  X   be the set of Steiner vertices of   T  ,  Z   the set of multicast nodes, and   Y   the set of vertices used for the neuralnetwork configuration, the algorithm works as follows: ini-tially, it builds a tree using one of the existing heuristics(Greedy Heuristic, Weighted Greedy Heuristic or VirtualTrunk Dynamic Multicast); then,  RTRNN   is applied and, if during its execution, an insertion request is received by node i , (meaning that node  i  must receive all multicast packets)  q  i is set to 1. Afterwards node  i  will be inserted into the  Y   set; RTRNN   will take this into account at the next iteration, and it will find a new tree called   T  RNNtemp , which will becomethe updated tree after the request will be satisfied. In order to calculate  T  RNNtemp , a further temporary tree,  T  NEW   iscreated, primarily, to connect the  i  node to the tree, by meansof the shortest path between them; then all  q  j  s.t.  j / ∈  Y   arecalculated, in order to choose the neuron  k  corresponding tothe largest  q  k . Such a node will be appended to  Y   as  best Steiner vertex candidate . If during the  RTRNN   executiona disconnect request is received by the node  j , it will belabeled as  “not for Z” , the tree will be pruned and, if node  j  is not a leaf node, node  j  will be labeled as  Steiner vertex .Afterwards, node  j  will be deleted from the  Y   set, and   q  j will be set to 0 in order to compute all  q  k  s.t.  k / ∈  Y   and to choose the neuron  s  with the largest  q  s . At the end of requests  RTRNN   is guaranteed to find a tree better thanthe one found by the basic heuristic.In simulations described in the next section, the heuristic wasapplied as a continuous rearrangement algorithm. Further-more, it is possible to disable  RTRNN   during a session,and to resume it whenever a certain number of operationsare executed or when some threshold on the tree cost isexceeded.We assume that one insertion or deletion request can besimultaneously received. Due to RTRNN   model operationalcharacteristics, we adopt a  ForcingConnection  function.This function is needed when the sub-graph induced bythe vertices in  U   =  X   ∪  Y   and by their adjacent edges,is disconnected.  RTRNN   finds all the nodes causing thisdisconnection, and includes into the  U   set all the nodes alongthe shortest path between those nodes and the sub-graphinduced by the tree.  D. Complexity analisys In the proposed algorithm, as described in [3], weightsare initialized only once and the consequent complexity is O ( | V  | 2 ) , where | V  | is the number of network neurons. Then,the initialization of   q   probabilities requires  O ( | V  | )  as time,therefore the overall programming complexity is  O ( | V  | 2 ) .The largest complexity arises from determining the neuronwith the largest value of   q   and this requires a computationof all  q   for   ( | V  |−| Y  | )  neurons requiring  O ( | V  | 2 )  as time.The computation of a Minimum Spanning Tree of a sub-graph induced by vertices in  U   takes a  O ( | U  | )  as time.For each insertion or deletion request, the computationaltime is just  O ( | V  | 2) . The additional functions of   RTRNN  heuristic are negligible and the connection of a node to treeusing shortest path between the node and the tree, requiresa time of   O ( | V  | 2 )  using Dijkstra algorithm [1]. The overallcomputational complexity is thus  O ( | V  |−| Y  || V  | 2 )  whichmeans  O ( | V  | 3 ) .  Fig. 2. Cumulative Competitiveness for Steinlib Networks. IV. E XPERIMENTAL RESULTS In order to validate the proposed algorithm, simulationsare carried out in which it is supposed that sessions begin just with the source node; afterwards, sessions continue withthe insertion or the deletion of the nodes that request either to join or to leave the multicast group. For the simulations, a setof requests equal to the 20% of the number network’s nodesis generated. Furthermore, whenever the cost of the treeexceeds a fixed bound, an experimental measure to comparethe three heuristics with their improved version is carried out.In the dynamic case, because we did not know the optimalcosts, we compare the costs achieved by the RTRNN withthe costs obtained by the Stirring algorithm [6].The cumulative cost competitiveness graphs for both Stein-lib [10] and BRITE networks [9], are shown in figures 2and 3. The Steinlib graphs represent a well known testbed in the literature, whilst BRITE networks are internet likegraphs. The competitiveness, shown in the charts, representsthe ratio between the cost produced by the heuristic and theoptimal cost. By optimal cost we mean the real best solutionfor the static case and the best found solution for the dynamiccase. Figure 2 shows the relevant improvement produced  by  RTRNN   when applied to  GH  ,  WGH   and   VTDM  .Both  GH   + RTRNN   and   WGH   + RTRNN   find alwaysa Steiner tree such that  C  Heur /C  Opt  ≤  1 . 105 .Figure 3 shows that  RTRNN   applied to  BRITE   networksoutperforms the behavior obtained on Steinlib graphs; the GH   + RTRNN   curve converges at 100% for competitive-ness values not much greater than 1.08. Likewise,  RTRNN  is able to improve considerably in real time the sequence of multicast trees found by  VTDM  .Figure 3 also shows that  RTRNN   obtains a better improvement when applied to  WGH  . Finally, although WGH   presents a behavior worse than  GH   on Internet-like networks, RTRNN+WGH has a better behavior than RTRNN   +  GH  .  RTRNN  , therefore, improves  WGH  more than  GH  , and finds multicast trees with cost distantfrom the optimum less than 8%. Another   RTRNN   feature isdue to the limited tree topology changes involved after eachoperation. This constitutes an advantage because a limited re-routing of packets is required. Namely, the tree topol-ogy changes, on average, after each operation are limited  Fig. 3. Cumulative Competitiveness for BRITE Networks.  between 1.5% and 3.5%, satisfying the constraint imposed  by dynamic multicasting.V. C ONCLUSIONS In this paper a dynamic heuristic for the  SPN   has been proposed. The heuristic is based on the adoption of theRandom Neural Network Model for the solution of thedynamic version of the Steiner Tree Problem in Networks,and it is capable of improving the partial solutions obtained in real time.The heuristic has been tested on sets of sample graphs artifi-cially generated and it has been demonstrated that  RTRNN  is an efficient dynamic post-processing method able also tosatisfy the constraints imposed by dynamic multicasting, asthe  real time dynamic  and the  limited re-routing of multicast  packets .R  EFERENCES[1] Dijkstra, E.W., A note on two problems in connections with graphs.  Numerische Mathemat. , 1:269–271, 1959.[2] Gelenbe, E. , Random neural networks with negative and positivesignals and product form solution.  Neural computation , 7:502–511,1989.[3] Gelenbe, E., Ghanwani, A., and Srini-Vasonr, V., Improved neuralheuristics for multicast routing.  IEEE Journal on selected areas incommunications , 15:147–155, 1997[4] Gelenbe, E., Mao, Z. 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Steinlib library. .[11] Waxman, B.M., Routing of multipoint connections.  IEEE Journal on selected areas in communications , 6 n.9, 1988[12] Winter, P., Steiner problem in the networks: a survey.  Networks ,17:129–167, 1987[13] Garey M.R., Graham L.R., and Johnson D.S., The cpmlexity of computing Steiner minimal trees  Siam J. Apll. Math. , vol32, pages835–859, 1977
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