Knjiga apstrakata - Mathematical Institute of the Serbian Academy of ...

Functional Decomposition as an Optimization Technique ∗Edin H. MulalićFaculty of Electronic EngineeringUniversity of Niš, Serbiaedinmulalic@yahoo.comMiomir S. StankovićFaculty of Ocupational SafetyUniversity of Niš, Serbiamiomir.stankovic@gmail.com1 IntroductionA basic operation such as calculating the value of a function is in the heart of any problem solving process. In specializedsystems where the speed of calculation is of great importance, various optimization techniques are applied. There is nogeneral recipe for successful optimization, it usually requires problem dependant heuristic. One can express a functionto be calculated in different ways, use various decomposition techniques, use hardware implementation (or a combinationof software and hardware implementation), improve speed of data structures used in the algorithm, use precomputedvalues, use approximative solutions etc. In this paper, we will explore the possibility of taking advantage of functionaldecomposition combined with using underlying probability distribution of input variables for determining which valuesshould be used for pre computation.2 PreliminariesLet’s suppose that we have given a finite commutative ring (R, +, ∗), where R = {r 1 , r 2 , ..., r K }, K ∈ N. We want toevaluate function f : R N → R, N ∈ N (f(x) = f(x 1 , x 2 , ..., x N ) where x i ∈ R for 1iN.) Computing the function valuefor a specific input requires time T c (f). If we have a memory of limited size M, we would be able to pre compute andstore function value for up to M values of input parameter combinations (input vectors). Assuming that reading a valuefrom the memory requires time T M and that T M is significantly less than T c (f), with this approach we can cut downthe average time of evaluating the function f. Note that the term memory in this context can denote a physical memoryor a convenient data structure. Let’s assume that there is an underlying probability distribution of input variables, sothat probability of x i = r j is denoted as p ij , where ∑ k=Kk=1 p ik = 1, 1iN. We will also assume that input parameters haveindependent distributions. The information about the distribution can be used to find M most probable combinations ofinput parameters and use them for precomputed and stored values. Obviously, that will minimize the expected time ofevaluating the function which is given by formula:E f [T ] = T c (f) − P (X M )(T c (f) − T M ) (1)where X M is set of all input vectors used for pre computation and P (X M ) is probability that an input vector is used forpre computation.Of course, this is not the only way of using the memory resource. Depending on the usage of the system, this might besatisfying solution. But there are some issues in this approach. First, is it possible to use memory resource in a differentway to reduce average evaluation time even more? And second, how can we affect more than M input vectors? One wayto approach to these two problems is functional decomposition. A decomposition ∆(f) of a function f is set of functions∆(f) = {F, f 1 , f 2 ..., f D }, such thatf(x) = F (f 1 (x 1 ), f 2 (x 2 ), ..., f D (x D )) (2)where x i (1iD) are vectors formed as sub vectors of the initial vector x.decompositions of formIn this paper, from now on, we are consideringf(x) = f 1 (x 1 ) ⊕ 1 f 2 (x 2 ) ⊕ 2 ... ⊕ D−1 f D (x D ) (3)where ⊕ i ∈ {+, ∗}, 1iD − 1. Basically, that means that we are not examining hierarchical decompositions of form g(h(y)).∗ Supported by Ministry of Sci. & Techn. Rep. Serbia, the project III 04400626