Then, the two path metrics entering each state at the current sampling time are compared and the path with the minimum metric is selected as the survivor path.Īfter all n samples have been received, the path with the overall minimum path metric is selected as the most likely path and the input sequence associated with that path is the maximum-likelihood sequence estimate x ― ˆ.Ĭonsider three coins labeled 1, 2, and 3. When the current sample is received, path metrics for the two paths leaving each state at the previous sampling time are calculated by adding branch metrics to state metrics. The metric of the minimum metric path is commonly referred to as the state metric. Since all other paths have been eliminated, a ML detector only needs to keep track of the minimum metric path leading to each state at the current sampling time. The Viterbi algorithm systematically eliminates those paths that cannot be part of the most likely path because they diverge and remerge with another path that has a smaller path metric. Consequently, minimizing the squared Euclidean distance is equivalent to minimizing: ∑ i = 1 n r i − y i 2 − r i 2 = ∑ i = 1 n 2 r i y i + y i 2 The branch metrics can be simplified by noting that r i 2 is common to all the branch metrics. The sum of the branch metrics along a path, referred to as the path metric, represents the squared Euclidean distance between the received samples and the ideal samples associated with that path. The four-state trellis for the PR4 channel.Įach branch in the trellis has associated with it an input symbol x i, an ideal value y i, and a branch metric equal to ( r i − y i) 2. His algorithm computes the same parsing tableĪs the CYK algorithm yet he showed that algorithms for efficient multiplication of matrices with 0-1-entries can be utilized for performing this computation.FIGURE 14. Valiant (1975) gave an extension of the CYK algorithm.
This makes it one of the most efficient algorithms for recognizing general context-free languages in practice. O ( n 3 ⋅ | G | ), where n is the length of the parsed string and | G| is the size of the CNF grammar G.
Cocke–Younger–Kasami algorithm (CYK) Class
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