You are running a campaign to seek funding for educational computers from Sacramento, California to Augusta, Maine. You need to convince each state to adopt a legal measure that would do so. You know you aren't going to win every state, so you want to visit the states that are most likely to consider your measure. You (unlike me) decide to plan out the trip so that it will be most efficient. You start thinking. From Sacramento, there are a number of capital cities (Phoenix, AZ; Carson City, NV; Salem, OR). You number each possibility and assign a likelihood to each.
From each of your first cities, there are a number of second cities that you can drive to. You number each secondary city and assign an additional likelihood. Immediately you realize that both from Salem, OR and from Carson City, NV you can drive to Boise, ID. You don't need to keep both paths (Sacramento-Carson City-Boise and Sacramento-Salem-Boise); you can eliminate one of them and continue on. So, you pick the one with the highest overall likelihood of adoption and move on.
You continue to eliminate possibilities in this manner. By the time you arrive in Augusta, you'll probably have a handful of remaining paths you could take to get from Sacramento to Augusta. You can then decide which of the paths is best.
For more on viterbi algorithm please read the attachment
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