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Butterflies are an important motif found in bipartite graphs that provide a structural way for finding dense regions within the graph. Beyond counting butterflies and enumerating them, other metrics and peeling for bipartite graphs are designed around counting butterfly motifs. The importance of counting butterflies has led to many works on efficient implementations for butterfly counting, given certain situational or hardware constraints. However, most algorithms are based on first counting the building block of the butterfly motif, and from that calculating the total possible number of butterflies in the graph. In this paper, using a linear algebra approach, we show that many provably correct algorithms for counting butterflies can be systematically derived. Moreover, we show how this formulation facilitates butterfly peeling algorithms that find the k-tip and k-wing subgraphs within a bipartite graph. 

We approach the problem of implementing mixed-datatype support within the general matrix multiplication ( gemm ) operation of the BLAS-like Library Instantiation Software framework, whereby each matrix operand A , B , and C may be stored as single- or double-precision real or complex values. Another factor of complexity, whereby the matrix product and accumulation are allowed to take place in a precision different from the storage precisions of either A or B , is also discussed. We first break the problem into orthogonal dimensions, considering the mixing of domains separately from mixing precisions. Support for all combinations of matrix operands stored in either the real or complex domain is mapped out by enumerating the cases and describing an implementation approach for each. Supporting all combinations of storage and computation precisions is handled by typecasting the matrices at key stages of the computation—during packing and/or accumulation, as needed. Several optional optimizations are also documented. Performance results gathered on a 56-core Marvell ThunderX2 and a 52-core Intel Xeon Platinum demonstrate that high performance is mostly preserved, with modest slowdowns incurred from unavoidable typecast instructions. The mixed-datatype implementation confirms that combinatorial intractability is avoided, with the framework relying on only two assembly microkernels to implement 128 datatype combinations. 

We describe a learning process that uses one of the simplest examples, matrix-matrix multiplication, to illustrate issues that underlie parallel high-performance computing. It is accessible at multiple levels: simple enough to use early in a curriculum yet rich enough to benefit a more advanced software developer. A carefully designed and scaffolded set of exercises leads the learner from a naive implementation towards one that extracts parallelism at multiple levels, ranging from instruction level parallelism to multithreaded parallelism via OpenMP to distributed memory parallelism using MPI. The importance of effectively leveraging the memory hierarchy within and across nodes is exposed, as do the GotoBLAS and SUMMA algorithms. These materials will become part of a Massive Open Online Course (MOOC) to be offered in the future. 

In this paper, we describe a trilogy of Massive Open Online Courses (MOOCs) that together expose knowledge and skills of fundamental importance to HPC. Linear Algebra: Foundations to Frontiers (LAFF) covers topics found in an introductory undergraduate course on linear algebra. It links abstraction in mathematics to abstraction in programming, with many enrichments that connect to HPC. LAFF-On Programming for Correctness introduces how to systematically derive programs to be correct. Of importance to HPC is that this methodology yields families of algorithms so that the best one for a given situation can be chosen. Programming for HPC (working title) is in the design stage. We envision using a very simple example, matrix-matrix multiplication, to illustrate how to achieve performance on a single core, on multicore and many-core architectures, and on distributed memory computers. These materials lower barriers into HPC by presenting insights, supports, and challenges to novices and HPC experts while scaling access to the world.