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

The natural question here is how to find decomposition which minimizes the average time of calculation for function f.But before that, we have to find a way to efficiently find the average time for one particular decomposition ∆(f). Themain issue is how to divide the available memory resource among functions f i in the optimal way. The next section givesanswer to that question.3 Optimal resource distributionThe average time for evaluating the function f is given by formulaWe are looking to maximise ω(D), whereE f,∆ [T ] = T c (∆(f)) −ω(q) =q∑j=1D∑j=1P (X (j,mj)M)(T c (f j (x j )) − T M ) (4)P (X (j,mj)M)(T c (f j (x j )) − T M ) (5)Each function f j from a decomposition ∆(f) can count on m j memory locations and ∑ Dj=1 m jM, 0m j M. Since P (X (j,mj)M)depends on m j , finding minimal average time E f,∆ [T ] requires optimal configuration [m 1 , m 2 , ..., m D ]. One way to solvethis is by brute force, but the problem with this approach is exponential complexity. Here, we will propose the algorithmbased on dynamic programming which solves the problem in polynomial time.Step 1. For each 1jD calculate P ij = P (X (j,i)M) where m j = i, 0il j and l j = min{M, length of vector x j }. Once again,this can be done by brute force, but exponential complexity can be avoided by reducing this problem to finding M + 1best paths in a trellis. Therefore, the complexity of this procedure is O(DMK 2 N).Step 2. Let’s define Q ij = P ij (T c (f j (x j )) − T M ). Let’s define matrix Ω and its element Ω[i, j, k] as max{ω(j)} such thatm j = i and ∑ ja=1 m a = k. Now we can find max{ω(D)} by the following procedure with complexity O(M 3 D).Initialization. For 0il 1 , 0kM{Q i1 if i = kΩ[i, 1, k] =(6)0 otherwise.Recursion. For 2jD, 0il j , ikMStopping. Let’s define M ′ as M ′ = min{M, ∑ Da=1 l a}. Then, max{ω(D)} = maxM ′Reconstruction. For D − 1j1ReferencesΩ[i, j, k] = Q ij + max0i ′ i Ω[i′ , j − 1, k − i] (7)Ψ[i, j, k] = argmaxΩ[i ′ , j − 1, k − i] (8)i ′ ,0i ′ iΩ[i, D, M ′ ], m D = maxΩ[i, D, M ′ ], k D =0il D i,0il Dm j = Ψ[m j+1 , j + 1, k j+1 ], k j = k j+1 − m j+1 (9)[1] L. Huang, D. Chiang, Better k-best Parsing, Proceedings of the 9th International Workshop on Parsing Technologies,2005.[2] D. Brown, D. Golod, Decoding HMMs Using the k Best Paths: Algorithms and Applications, APBC, 2010.[3] W. Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality, Wiley Series in Probabilityand Statistics, 2007.27