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Subset sums are quantities that can be efficiently calculated using the Subset Summation Tree, and are used in the calculation of correlations.
For each rotation, this expression can be efficiently calculated using P forward and one inverse Fast Fourier transforms.
The summation in the likelihood function Equation (1) can be efficiently calculated using Felsenstein's pruning algorithm (Felsenstein, 2003).
Following the literature (Andrew and Goldberg, 1998), a flow with the maximum value can be efficiently calculated using the push-relabel or the binary blocking flow algorithm.
This optimal path between a pair of nodes can then be efficiently calculated using Dijkstra's algorithm for shortest paths (Cormen et al., 2000).
Since this method considers only state transition between two adjacent positions, the optimal alignment can still be efficiently calculated using dynamic programming.
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But the goal of our project was not so much to precisely estimate the maximal k-mismatch substring lengths, but rather to define a distance measure on sequences that can be efficiently calculated and that can be used to obtain biologically meaningful results.
The LBI can be efficiently calculated with the same message passing techniques we used to calculate the posterior fitness distribution.
(4.4) The optimal iteration step size (rho_{k}) can be efficiently calculated through solving (4.2).
As previously shown, the probability distribution of PWM scores can be efficiently calculated by convolution [50].
In (2), the OFDM symbols are related to modulation symbols through an inverse discrete Fourier transform (IDFT) efficiently calculated using the inverse fast Fourier transform (IFFT).
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