computing the quartet distance between general trees

7.2. IMPLEMENTATION 51oretic approach.Mailund et al. [14] suggest the use of advanced matrix multiplication methods andsubstantiate their algorithmic results with the best theoretic result within the field ofmatrix multiplication, namely the Coppersmith-Winograd algorithm [7]. As mentionedin Chap. 1, this kind of theoretic result will be met with immediate scepticism by mostprogrammers, since one might suspect that the demands of the theory can never be metby an implementation. In this case because of the size of the input being too small toobserve the improvement.Here the ideal situation would be to find an algorithm that is sub-cubic in theoryand still efficiently implemented in practice. Searching the literature and the web for areasonable method for matrix multiplication I came across the Strassen algorithm [18],which is approximately O(n 2.8 ) and is described as being efficient in practice for largematrices. However, I was not successful in finding an optimized implementation of thealgorithm or a linear algebra library containing one. Since matrix multiplication is notthe focus of this thesis, but rather a smaller piece in the algorithmic puzzle, the time didnot allow me to attempt implementing it on my own. In addition, it seems unreasonablethat I would be competitive with highly optimized libraries.Instead I decided to go with a highly optimized, reliable and robust library. This isa meaningful decision, since the goal of the thesis is to test the practical usefulness ofthe algorithm. What is the use of a theoretic result if it is not practically applicable? Thefinal choice was to use the Basic Linear Algebra Subprograms (BLAS) API 4 , which is a defacto standard for various linear algebra packages. Since my first implementation was inPython, BLAS caught my attention after discovering that it integrates with SciPy/NumPy,which is automatically compiled against the BLAS installation if present on the machine.Furthermore, the Boost.NumericBindings library 5 , provides a generic layer between theBoost.uBlas data types and the BLAS linear algebra routines in C++.The exact routines utilized are some level 3 BLAS calls for matrix-matrix multiplication.In NumPy and Boost.NumericBindings these have general interfaces used throughthe calls C = dot(A,B) and void gemm(A,B,C) respectively. Both solutions require abit of setup using the right type of matrix container structure, but with the libraries thisis not difficult. I will not provide further details, but merely refer to the code which isavailable (see App. B).To get an idea of how the performance improved, a comparison has been made of thefour implementations used, namely the two used in the prototypes and the two BLAS in-4 The BLAS specification: http://www.netlib.org/blas/5 Boost.NumericBindings: http://svn.boost.org/svn/boost/sandbox/numeric_bindings-v1/libs/numeric/bindings/doc/index.html