Formulation as a graph problem

The optimization problem stated above may be viewed as a special variant of the single-source shortest path problem in a graph. The underlying graph in this case has a source node and a sink node, corresponding to the blank characters that delimit the given sequence of n characters. The other nodes of the graph are arranged in columns. There is one column for each of the n characters in the word to be recognized. Each column contains m (typically 26) nodes, one for each letter of the alphabet. Each column is ``fully connected'' to the column immediately to the right, i.e. there is a directed arc from each node of a given column to each node of the next column. This reflects the desire to initially allow any possible sequence of n letters.

Word hypotheses correspond to paths in this graph from the source node to the sink node. Each path has an associated cost. In order to make the cost function additive, we simply take the logarithm of the likelihood described in the previous section and split up the result into parts corresponding to various arcs. Specifically, the weight of the arc from the node for letter L_i in column c-1 to the node for letter L_j in column c is

$$\text{cost}((c-1, i),(c, j)) = -\log P(L_j \vert L_i) - \log P(I_c \vert L_j)$$

Notice that we have changed the notation slightly here relative to the notation that was being used previously; namely, L_j now represents the j-th letter of the alphabet, regardless of what position(s) within a word hypothesis it is being considered for. The total likelihood over a given word hypothesis thus corresponds directly to the sum of the weights over the path in the graph that represents the given word hypothesis (you have to change the signs of the weights, then take an exponential at the end to recover the likelihood, but this is straightforward). The second term in the expression for the weights is perhaps more naturally associated with the terminal node of the arc in question rather than with the arc itself; nonetheless, the above description also leads to a correct formulation. In either case, the most likely word hypothesis will correspond to the path with the lowest total cost. In summary, one arrives at the following problem.