Static mapping is the assignment of parallel processes to the processing elements (PEs) of a parallel system, where the assignment does not change during the application's lifetime. In our scenario we model an application's computations and their dependencies by an application graph. This graph is first partitioned into (nearly) equally sized blocks. These blocks need to communicate at block boundaries. To assign the processes to PEs, our goal is to compute a communication-efficient bijective mapping between the blocks and the PEs. This approach of partitioning followed by bijective mapping has many degrees of freedom. Thus, users and developers of parallel applications need to know more about which choices work for which application graphs and which parallel architectures. To this end, we not only develop new mapping algorithms (derived from known greedy methods). We also perform extensive experiments involving different classes of application graphs (meshes and complex networks), architectures of parallel computers (grids and tori), as well as different partitioners and mapping algorithms. Surprisingly, the quality of the partitions, unless very poor, has little influence on the quality of the mapping. More importantly, one of our new mapping algorithms always yields the best results in terms of the quality measure maximum congestion when the application graphs are complex networks. In case of meshes as application graphs, this mapping algorithm always leads in terms of maximum congestion AND maximum dilation, another common quality measure.
One can apply a wide range of optimization techniques to the topology mapping problem. Hoefler and Snir @cite_24 employ (among others) the Reverse Cuthill-McKee (RCM) algorithm, originally devised for minimizing the bandwidth of a sparse matrix @cite_13 . If both @math and @math are sparse, the simultaneous optimization of both graph layouts can lead to good mapping results @cite_0 .
@cite_24 @cite_0 @cite_13
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The steadily increasing number of nodes in high-performance computing systems and the technology and power constraints lead to sparse network topologies. Efficient mapping of application communication patterns to the network topology gains importance as systems grow to petascale and beyond. Such mapping is supported in parallel programming frameworks such as MPI, but is often not well implemented. We show that the topology mapping problem is NP-complete and analyze and compare different practical topology mapping heuristics. We demonstrate an efficient and fast new heuristic which is based on graph similarity and show its utility with application communication patterns on real topologies. Our mapping strategies support heterogeneous networks and show significant reduction of congestion on torus, fat-tree, and the PERCS network topologies, for irregular communication patterns. We also demonstrate that the benefit of topology mapping grows with the network size and show how our algorithms can be used in a practical setting to optimize communication performance. Our efficient topology mapping strategies are shown to reduce network congestion by up to 80 , reduce average dilation by up to 50 , and improve benchmarked communication performance by 18 . This paper considers the vehicle routing problem with time windows, where the service of each customer must start within a specified time interval. We consider the Lagrangian relaxation of the constraint set requiring that each customer must be served by exactly one vehicle yielding a constrained shortest path subproblem. We present a stabilized cutting-plane algorithm within the framework of linear programming for solving the associated Lagrangian dual problem. This algorithm creates easier constrained shortest path subproblems because less negative cycles are introduced and it leads to faster multiplier convergence due to a stabilization of the dual variables. We have embedded the stabilized cutting-plane algorithm in a branch-and-bound search and introduce strong valid inequalities at the master problem level by Lagrangian relaxation. The result is a Lagrangian branch-and-cut-and-price (LBCP) algorithm for the VRPTW. Making use of this acceleration strategy at the master problem level gives a significant speed-up compared to algorithms in the literature based on traditional column generation. We have solved two test problems introduced in 2001 by Gehring and Homberger with 400 and 1000 customers respectively, which to date are the largest problems ever solved to optimality. We have implemented the LBCP algorithm using the ABACUS open-source framework for solving mixed-integer linear-programs by branch, cut, and price. In this paper, we develop methods to ''sample'' a small realistic graph from a large Internet topology. Despite recent activity, modeling and generation of realistic graphs resembling the Internet is still not a resolved issue. All previous work has attempted to grow such graphs from scratch. We address the complementary problem of shrinking an existing topology. In more detail, this work has three parts. First, we propose a number of reduction methods that can be categorized into three classes: (a) deletion methods, (b) contraction methods, and (c) exploration methods. We prove that some of them maintain key properties of the initial graph. We implement our methods and show that we can effectively reduce the nodes of an Internet graph by as much as 70 while maintaining its important properties. Second, we show that our reduced graphs compare favorably against construction-based generators. Finally, we successfully validate the effectiveness of our best methods in an actual performance evaluation study of multicast routing. Apart from its practical applications, the problem of graph sampling is of independent interest. In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT &T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.