Steiner Problem in Multistage Computer Networks

Multistage computer networks are popular in parallel architectures and com munication applications We consider the message communication problem for the two types of multistage networks one popular for parallel architectures and the other popular for communication networks A subset of the problem can be equated to the Steiner tree problem for multistage graphs Inherent complexities of the problem is shown and polynomial time heuristics are devel oped Performance of these heuristics is evaluated using analytical as well as simulation results Introduction Multistage interconnection networks MINs are popular among parallel architecture and or communication network topologies An N log N element MIN consists of log N stages of N elements each A common pictorial view of an N log N MIN is to collect N elements in a stage vertically and arrange log N such stages horizontally one after the other MINs o er a good balance between network cost and performance They are often characterized as intermediate fO N log N g cost networks falling within the two extreme cases fully connected fO N costg and bus connected fO N costg Architectural and other topological properties of MIN may be found in Supported in part by NSF grant CCR S Bhattacharya and B Dasgupta Two Versions of MINs Let Si j denote the i th stage j th row element in an N log N MIN i log N j N We consider source to source wrap around MINs only i e when j S j Slog N j These networks can allow multiple passes using the wrap around connections Depending on the role of intermediate stage elements two types of MINs are possible as outlined below Intermediate stages as switches only This type is popular in parallel archi tecture applications Here the source end leftmost and the destination end rightmost stage constitute of processors while the intermediate elements are bare switches which interconnect various sources and destinations Such MINs are of commercial usage in parallel processors e g the BBN Butter y machine We refer to MINs of this type as type MIN Intermediate stages as processors This type is common in communication net work applications Here the intermediate stage elements are identical to the source or destination stage processors i e they can have their own message tra c Example of such MINs can be found in We refer to MINs of this type as type MIN Communication in MINs Depending on the number of destinations involved in a communication in MIN three types can be classi ed one to one one to many and one to all These are commonly known as routing multicast and broadcast In this article we focus ourselves to the multicast problem for MINs Note that routing broadcast are two special instances of multicast and do not o er any opportunity for tra c reduction The multicast problem speci es a source node and a set of k destination nodes Without loss of generality we assume the source node to be S Destination nodes are spread over the MIN k N k routing k N broadcast Objective of the multicast problem is to transmit the message from the source node to the destination nodes Flow control Mechanism For multihop networks various form of switching and ow control mechanisms have evolved Store and forward is a traditional approach to message communication Vir tual cut through wormhole de ection routing etc have been subsequently proposed A survey can be found in We assume packetized message communication where packets are independently own through the network Our focus is to estimate and possibly reduce the overall tra c overhead in message communications Steiner Problem in Multistage Computer Networks Optimality Criteria in MIN Multicast Two possible criteria to measure the optimality of MIN multicast communication are to minimize one of the following two objective functions the total tra c generated in the network each occupied link of the network counts as one unit of tra c the hops distance between the source node and any destination node The tra c metric makes the problem equivalent to the Steiner problem for MIN while the time metric is a di erent dimension altogether These two metrics work in the dual sense Reducing one increases the other and vice versa Thus we focus on the tra c metric only Considerations along the time metric is an open problem Multistage Interconnection Networks We consider type MINs with the cube network topology These class of networks e g baseline delta generalized cube indirect binary cube omega banyan have been proposed as xed degree alternative to hypercube architecture They are pop ular in switching and communication applications They can also emulate the per formance of hypercube in most applications e g the CCC architecture Let MINd denote a d dimensional generalized MIN Formulation of the Tra c Reduction Problem We consider multicasting on MINd which are unique path networks Given a set of k multicast destinations Di i k and a source node S in MINd the path from S to any particular Di is xed However it is clear that for a given set of multicast destinations the total tra c generated in MINd depends on the relative order in which d di erent dimensions are arranged This leads to our problem formulation as see Section for practical applicability Given a set of destination nodes tra c optimum multicasting in MINd is to nd a permutation of the d dimensions each stage of MINd is allocated to one particular dimension value so that the total tra c is minimized Unfortunately this problem is NP complete as shown by the next theorem Hence we need to investigate the possibility of designing e cient heuristics for this problem Theorem The tra c optimum multicasting problem is NP complete Proof sketch The problem is obviously in NP To show NP hardness one can reduce the space minimized full trie problem which is shown to be NP complete in to this problem Details are available in S Bhattacharya and B Dasgupta Design Issues Any hardware implementation of a MINd would assume an ordering among the d dimensions In such cases online dimension ordering as required by the tra c reduc tion criterion in this paper in a MINd may be argued from the practical viewpoint We identify the following situations as practical applications Communication networks often use MINs Traditional hardware implementa tion of switches at every intermediate stages have been replaced using Wave Time Division Multiplexors WTDM over passive stars The actual interconnection is formed by wavelength frequency or time slot assignment of di erent nodes i e by rmware control A rmware controlled design can be changed without changing the underlying hardware Thus it is possible to re order the dimensions in a MINd dynamically Every stage may have to con gure to at most d possible dimensions for which the wave time assignments can be pre computed and stored If the tra c pattern is known and repetitive as may happen in periodically oc curring similar message communications then from the above optimum dimensional ordering for each multicasting instance one can derive the most common pattern and design the MINd using the corresponding optimum dimensional ordering The idea here is to achieve tra c optimality for most multicasting instances which leads to an overall tra c reduction Hierarchical hypercubes are designed for several practical reasons Such hierarchical designs limit the availability of di erent dimensions at any node Only a certain set of dimensions can be availed at each node This imposes a hierarchy among dimensions in a routing multicasting operation In some other cases even with complete hypercubes routing multicasting is done in hierarchical fashion imposing a arbitrary desired ordering among dimensions With these applications our results and optimality ordering among dimensions can be used as a measure whether or not a particular multicast operation is generating optimal tra c Note that a hypercube with hierarchically ordered dimensions can be treated as aMINd for analysis purpose and results from the latter can be used for the former Greedy Heuristic Let Reachp denote the number of nodes which received a copy of the message at stage p Let kd be the dimension between stage p and stage p We de ne an expansion ratio Fkd Reachp Reachp Intuitively this fraction Fkd indicate how much the size of the multicast destination is increasing at every stage This expansion ratio depends of the dimension stage position and on the set of all prior dimensions served already For the sake of brevity we treat this all previous information as part of the stage information and denote compactly using the stage number position Now the total tra c equals Steiner Problem in Multistage Computer Networks