History of the Viterbi Algorithm
02 Dec 2015Required reading: The Viterbi Algorithm: A Personal History by G. David Forney Jr.
The Viterbi algorithm originated as a decoding algorithm for convolution codes. Nowdays the Viterbi algorithm has widespread use, particularly in speech recognition and natural language processing.
The interesting points from the above article (to tempt you into reading it):

Viterbi devised the algorithm to help him teach:
the Viterbi algorithm for convolution codes … came out of my teaching … I found information theory difficult to teach, so I started developing some tools.

The Viterbi algorithm, when first published, was not known to be related to dynamic programming methods and also not known to provide the optimal or maximum likelihood solution. The original paper states that:
this decoding algorithm is clearly suboptimal

It was G. David Forney, Jr who later proved that the Viterbi algorithm was an exact recursive algorithm for the shortest path through a trellis diagram. The relationship to dynamic programming then became clear.
The article also provides various places where the Viterbi algorithm has been used in practice, including the Galileo mission to Jupiter in 1992 (it was used to boost the transmission bandwidth when the primary antenna failed to deploy).
The article also mentions various relationships between algorithms for “codes on graphs” and Pearl’s belief propogation algorithm for Bayesian networks. The following paper is a good reference on this topic:
S. M. Aji and R. J. McEliece, “The generalized distributive law,” IEEE Trans. Inform. Theory, vol. 46, pp. 325343, Mar. 2000.