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We discuss multicasting for the n-cube network and its close variants, the Chord and the Binomial Graph (BNG) Network. We present simple transformations and proofs that establish that the sp-multicast (shortest path) and Steiner tree problems for the n-cube, Chord and the BNG network are NP-Complete.

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We propose optimal routing algorithms for Chord [1], a popular topology for routing in peer-to-peer networks. Chord is an undirected graph on 2b nodes arranged in a circle, with edges connecting pairs of nodes that are 2k positions apart for any k ≥ 0. The standard Chord routing algorithm uses edges in only one direction. Our algorithms exploit the bidirectionality of edges for optimality. At the heart of the new protocols lie algorithms for writing a positive integer d as the difference of two non-negative integers d′ and d″ such that the total number of 1-bits in the binary representation of d′ and d″ is minimized. Given that Chord is a variant of the hypercube, the optimal routes possess a surprising combinatorial structure.

A circulant graph with n nodes and jumps j<sub>1</sub>, j<sub>2</sub>,..., j<sub>m</sub> is a graph in which each node i, 0 les i les n-1, is adjacent to all the vertices i plusmn j<sub>k</sub> mod n , where 1 les k les m. A binomial graph network (BMG) is a circulant graph where jk is the power of 2 that is less than or equal to n. This paper presents an optimal (shortest path) two-terminal routing algorithm for BMG networks. This algorithm uses only the destination address to determine the next hop in order to stay on the shortest path. Unlike the original algorithms, it does not require extra space for routing tables or additional information in the packet. The experimental results show that the new optimal algorithm is significantly faster than the original optimal algorithm.

A routing strategy for sending a message from a node (called source) to any number of other nodes (called its destinations) in a multicomputer system is studied. The strategy requires that every destination receive the source message in a minimum number of time steps while generating a minimum amount of traffic. This strategy is formulated as finding a particular subtree (called an optimal communication tree) in a graph G, where G represents the underlying topology of the system. We prove that the problem of finding such a tree is NP-hard even if G is bipartite. The problem is then considered for the hypercube multicomputers. Optimal and suboptimal solutions are finally discussed.

A natural communication problem in a multicomputer system, such as the hypercube, is that a processor (called the source) wants to send a message to a number of other processors (destinations). A message-routing paradigm for such a multidestination communication has been formulated as finding a subgraph called an Optimal Communication Tree (OCT). We prove that the problem of finding an OCT is NP-hard for the n-cube graph as well as for a graph whose maximum degree is at most three. Heuristics for finding suboptimal communication trees for the hypercube multicomputer are discussed.

The problem of determining a phylogeny of maximum parsimony from a given set of protein sequences is defined. It is shown that this problem is what is called, in computer science, NP-complete. The implication of this result is that it is equivalent in difficulty to a host of other problems in combinatorial optimization which are notorious for their intractability. This implies that it is more fruitful to attempt to develop heuristic techniques (which do not guarantee maximum parsimony but which do run in reasonable computer time) than to try to develop exact algorithms for phylogeny construction

A natural communication problem in a multicomputer system, such as the hypercube, is that a processor (called the source) wants to send a message to a number of other processors (destinations). A message-routing paradigm for such a multidestination communication has been formulated as finding a subgraph called an optimal distance-preserving tree (ODPT). We prove that the problem of finding an ODPT is NP-hard both for the n-cube graph as well as for a graph whose maximum degree is at most three.

Complexity issues intrinsic to certain fundamental data dissemination problems in high per- formance network topologies are discussed. In particular, we study the p-pairwise edge disjoint shortest paths problem. An ecien t algorithm for the case when every source point is at a distance at most two from its target is presented and for pairs at a distance at most three we show that the problem is NP-complete.

Optimal Routing in Binomial Graph Networks

- T Angkun
- G Bosilca
- B Vander Zanden
- J Dongarra

Angkun, T., Bosilca, G., Vander Zanden, B., and Dongarra, J., "Optimal Routing in Binomial Graph
Networks," Proceedings of the Eight International Conference on Parallel and Distributed Computing,
Applications and Technologies (PDCAT'07), 363 -369, 2007.

Optimal routing in binomial graph networks

- Angkun