Abstract 1 The communication cost plays a key role in the perfor- mance of many parallel algorithms. In the particular case of the one-sided Jacobi method for symmetric eigenvalue and eigenvector computation the communication cost of previously proposed algorithms is mainly determined by the particular ordering being used. In this paper we pro- posed two novel Jacobi orderings: the permuted-BR order- ing and the degree-4 ordering, aimed at efficiently exploit- ing the multi-port capability of a hypercube. It is shown that the former is nearly optimal for some scenarios and the latter outperforms previously known orderings by a factor of two. 1. Introduction The one-sided Jacobi method for symmetric eigen- value computation is very suited for its application on a multicomputer since it exhibits a high parallelism and potentially low communication requirements [5]. The one-sided method uses a series of similarity transforma- tions to make the original symmetric matrix converge to a diagonal form. Every similarity transformation zeroes one off-diagonal element and its symmetric. The elements of the matrix can be zeroed in any order, giving place to dif- ferent Jacobi orderings and, as a result, to different one-sided Jacobi algorithms. In addition to the conver- gence rate, different Jacobi orderings differ in the commu- nication requirements of the resulting parallel algorithm. In multicomputers, the communication overhead plays an important role on the performance of any particular algorithm [1]. In this paper, we focus on multicomputers with a hypercube interconnection topology and with multi- ple ports per node [14]. In such scenario one may design algorithms that communicate multiple messages simulta- neously through different links of the same node (commu- nication parallelism), which may result in a significant reduction in the communication overhead. One-sided Jacobi algorithms previously proposed for hypercubes 1. This work was supported by the Ministry of Education and Science of Spain (CICYT TIC-429/95) make a poor utilization of the multi-port capability because, at any given time, the information to be commu- nicated by every node is sent through one (or at most two) link, constraining in this way the exploitation of communi- cation parallelism. In [9] a method was developed to design parallel algo- rithms that efficiently exploit the multi-port capability in hypercubes. The method requires the specification of the original problem in the form of a CC-cube algorithm, whose properties will be described later. The method reor- ganizes the computation in a systematic way to introduce the appropriate level of communication parallelism in order to efficiently exploit the multi-port capability. It will be shown in this paper that one-sided Jacobi computation can take the form of a CC-cube algorithm. Thus, the method in [9] can be used to reduce the commu- nication cost in multi-port hypercubes. We will show first that the impact of the method when applied to known Jacobi algorithms is limited by the structure of the Jacobi ordering used in these algorithms. The key contribution of this paper is the proposal of two novel Jacobi orderings, the permuted-BR and the degree-4 orderings, which enable an efficient exploitation of the multi-port capability, through the use of the method described in [9]. The permuted-BR ordering has a performance that tends asymptotically (for large matrices) to 80% of a lower bound. The degree-4 ordering has a worse asymptotic performance but it behaves better for small matrices. In this case, it reduces the communication overhead of the algorithm to the half when compared with previous Jacobi orderings. The rest of this paper is organized as follows. Section 2 provides a brief review of the one-sided Jacobi method and the orderings proposed for its implementation on a hypercube multicomputer. Section 3 presents the novel orderings. Section 4 evaluates the performance of the pro- posed orderings. Finally, the main conclusions are summa- rized in section 5. 2. Preliminaries In this paper we focus on efficient algorithms for eigenvalues and eigenvectors computation of symmetric matrices through the one-sided Jacobi method on mulit- computers with a hypercube interconnection topology and Jacobi Orderings for Multi-Port Hypercubes Dolors Royo, Antonio González and Miguel Valero-García Universitat Politècnica de Catalunya Department of Computer Architecture Jordi Girona 1-3, Mòdul D6 08034 Barcelona (Spain) Email:{dolors,antonio,miguel}@ac.upc.es Authorized licensed use limited to: IEEE Xplore Customer. Downloaded on October 13, 2008 at 10:59 from IEEE Xplore. Restrictions apply.