## Multicast Communication in Multicomputer Networks (1990)

Venue: | IEEE Transactions on Parallel and Distributed Systems |

Citations: | 38 - 5 self |

### BibTeX

@ARTICLE{Lin90multicastcommunication,

author = {Xiaola Lin and Xiaola Lin and Lionel M. Ni and Lionel M. Ni},

title = {Multicast Communication in Multicomputer Networks},

journal = {IEEE Transactions on Parallel and Distributed Systems},

year = {1990},

volume = {4},

pages = {1105--1117}

}

### OpenURL

### Abstract

Efficient routing of messages is the key to the performance of multicomputers. Multicast communication refers to the delivery of the same message from a source node to an arbitrary number of destination nodes. While multicast communication is highly demanded in many applications, it is not directly supported by all existing multicomputers; rather it is indirectly supported by multiple one-to-one or broadcast communications, which result in more network traffic and a waste of system resources. In this paper, we study routing evaluation criteria for multicast communication under different communication paradigms. Multicast communication in multicomputers is formulated as a graph theoretical problem. Depending on the evaluation criteria and communication mechanisms, we study three optimal multicast communication problems, which are equivalent to the finding of the following three subgraphs: optimal multicast path, optimal multicast cycle, and minimal Steiner tree, where the interconnectio...

### Citations

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Citation Context ...n. However, message replication can be implemented by using some hardware approach [12]. If our major concern is to minimize traffic, the multicast problem becomes the well-known Steiner tree problem =-=[13]-=-. Formally, we restate the ST problem as follows. Definition 3 A Steiner tree, S(V; E), for a multicast set K is a subtree of G, such that K ` V (S). A minimal Steiner tree (MST) is a ST with a minima... |

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Citation Context ...the OMP problem for 2D-mesh graphs is similar to the one in the proof of Theorem 1. Thus, we have the following theorem. Theorem 2 The OMP problem is NP-complete for 2D-mesh graphs. Garey and Johnson =-=[18]-=- have proved that the rectilinear Steiner tree (RST) problem is NPcomplete. The RST problem for a set A is a tree structure, composed solely of horizontal and vertical line segments, which interconnec... |

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Citation Context ...supported in part by the NSF grants ECS-88-14027 and MIPS-88-11815. ii 1 Introduction Multicomputers are composed of hundreds or thousands of processors interconnected together in some fixed topology =-=[1]-=-. Each processor has its own memory, and there is no globally shared memory. Message passing is the only means for information exchange between processors. In general, each processor is directly conne... |

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Citation Context ... in which the host graph is a 2D-mesh (non-wraparound) as adopted in Ametek 2010 [7]. Before discussing the 2D-mesh graph, we have to briefly review a more general class of graphs, called grid graphs =-=[17]-=-. Let G 1 be the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are connected if and only if the Euclidean distance between them... |

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Citation Context ...10 0110 5 5 1110 12 12 0111 6 6 1111 11 11 Upon receiving the message, node [2; 5] will execute the greedy ST algorithm for message routing. It will generate D 1 = ([0; 5]) and D 2 = ([2; 3]; [4; 1]; =-=[6; 3]-=-; [7; 4]). One copy of the message with D 1 will be sent to node [1; 5], a bypass node, and another copy of the message with D 2 will be sent to the bypass node [2; 4]. By repeating this procedure at ... |

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Citation Context ...nt, a popular message-based parallel programming language, developed at Caltech also supports multicast communication [3]. A better heuristic multicast algorithm for the hypercube was proposed in [4] =-=[5]-=-. In [4], it was conjectured that finding an optimal multicast communication tree in terms of time and traffic is NP-hard in a hypercube graph. Furthermore, a VLSI router to handle multicast communica... |

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Citation Context ... to 6(k \Gamma 1), where k = jV (G)j. So we have the following theorem. Theorem 5 The OMP problem for n-cube graphs is NP-complete. For the problem of finding a MST in n-cube graphs, it was proved in =-=[19]-=- that the MST problem is NP-complete for n-cube graphs. Theorem 6 The MST problem for n-cube graphs is NP-complete [19]. 6 Heuristic Algorithms for Multicast Communication As shown in previous section... |

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Citation Context ...ation. In second generation multicomputers, more advanced communication mechanisms have been adopted. For example, iPSC-2 [6] adopts circuit switching and Ametek 2010 (Symult) adopts wormhole routing =-=[7]-=- [8]. In these new communication paradigms, the message transmisson time is almost independent of the number of hops between two nodes. A good survey and comparison study of various communication para... |

16 |
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Citation Context ...all first generation hypercube multicomputers and by iPSC-2, whereas the 2D-mesh is used in some second generation multicomputers, such as Ametek 2010. Multicast in bus-based networks can be found in =-=[14]-=-. In the following sections, we will examine the computational complexities of the MP, MC, and ST optimization problems. The OMT problem was originally studied in [4] and has been proven to be NP-comp... |

12 |
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Citation Context ...munication paradigms, the message transmisson time is almost independent of the number of hops between two nodes. A good survey and comparison study of various communication paradigms can be found in =-=[9]-=-. Apparently, the underlying communication paradigm will affect the criterion in evaluating multicast communication schemes. 3 Models for Multicast Communication Graphs will be used to model the under... |

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Citation Context ...ommunication in hypercube multicomputers was proposed in [2]. The Cosmic Environment, a popular message-based parallel programming language, developed at Caltech also supports multicast communication =-=[3]-=-. A better heuristic multicast algorithm for the hypercube was proposed in [4] [5]. In [4], it was conjectured that finding an optimal multicast communication tree in terms of time and traffic is NP-h... |

8 |
Multicast Communication in Multiprocessor Systems
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Citation Context ... to multiple neighboring nodes involves too much overhead and is usually undesirable. Thus, the routing method does not allow each processor to replicate the message passing by. Also, as indicated in =-=[11]-=-, a multicast path model provides better performance than the tree model when there is a contention in the network. From communication technology point of view, the multicast path model is more suitab... |

5 |
Graph Theory. Readings
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Citation Context ...nication schemes. 3 Models for Multicast Communication Graphs will be used to model the underlying topology of multicomputers. We will closely follow the graph theoretical terminology and notation of =-=[10]-=-; terms not defined here can be found in that book. Let graph G(V; E) denote a graph with node set V and edge set E. When G is known from context, the sets V (G) and E(G) will be referred to as V and ... |

5 |
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Citation Context ...iner Tree (ST) Problem Both OMC and OMP assume that the message will not be replicated by any node during transmission. However, message replication can be implemented by using some hardware approach =-=[12]-=-. If our major concern is to minimize traffic, the multicast problem becomes the well-known Steiner tree problem [13]. Formally, we restate the ST problem as follows. Definition 3 A Steiner tree, S(V;... |

5 |
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(Show Context)
Citation Context ...timization problems. The OMT problem was originally studied in [4] and has been proven to be NP-complete for n-cube graphs [15]. Several heuristic algorithms have been proposed for the MT problem [4] =-=[16]-=-. The OMT problem will not be discussed further here. 4 Optimal Multicast in Mesh Topology We first consider the case in which the host graph is a 2D-mesh (non-wraparound) as adopted in Ametek 2010 [7... |

2 |
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(Show Context)
Citation Context ...1101 10 10 0110 5 5 1110 12 12 0111 6 6 1111 11 11 Upon receiving the message, node [2; 5] will execute the greedy ST algorithm for message routing. It will generate D 1 = ([0; 5]) and D 2 = ([2; 3]; =-=[4; 1]-=-; [6; 3]; [7; 4]). One copy of the message with D 1 will be sent to node [1; 5], a bypass node, and another copy of the message with D 2 will be sent to the bypass node [2; 4]. By repeating this proce... |

2 |
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Citation Context ...he source node and u 1 through u k are destination nodes. 6.1 Heuristic Routing Algorithms for MP and MC The problem of finding an optimal MC (or MP) seems to be similar to traveling salesman problem =-=[20]-=-. In finding an MC (or MP) for multicast routing, however, the destination nodes cannot be visited in an arbitrary order since the host graph is mesh graph or n-cube graph. As a result, some well know... |

1 |
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(Show Context)
Citation Context ...ly, we have output D 1 = (000101; 010101; 000001; 001101; 101001; 110001). By repeating the greedy ST algorithm for bypass node replicate node or destination source node replicate node or [6,4] [4,1] =-=[4,2]-=- [7,4] [7,3] [6,3] [5,3] [4,3] [3,3] [1,5] [0,5] [2,3] [2,4] [2,5] [2,6] [2,7] Figure 10. A complete ST routing pattern in an 8 \Theta 8 mesh. message routing (Fig. 7) to each message receiving node, ... |

1 |
The complexity of optimal distance-preserving trees
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(Show Context)
Citation Context ...ions, we will examine the computational complexities of the MP, MC, and ST optimization problems. The OMT problem was originally studied in [4] and has been proven to be NP-complete for n-cube graphs =-=[15]-=-. Several heuristic algorithms have been proposed for the MT problem [4] [16]. The OMT problem will not be discussed further here. 4 Optimal Multicast in Mesh Topology We first consider the case in wh... |