FFmpeg/doc/viterbi.txt

This is a quick description of the Viterbi aka dynamic programming
algorithm.

Its reason for existence is that Wikipedia has become very poor on
describing algorithms in a way that makes it useable for understanding
them or anything else actually. It tends now to describe the very same
algorithm under 50 different names and pages with few understandable
by even people who fully understand the algorithm and the theory behind.

Problem description: (that is what it can solve)
assume we have a 2d table, or you could call it a graph or matrix if you
prefer

    O   O   O   O   O   O   O

    O   O   O   O   O   O   O

    O   O   O   O   O   O   O

    O   O   O   O   O   O   O


That table has edges connecting points from each column to the next column
and each edge has a score like: (only some edge and scores shown to keep it
readable)


    O--5--O-----O-----O-----O-----O
     2   / 7   / \   / \   / \   /
      \ /   \ /   \ /   \ /   \ /
    O7-/--O--/--O--/--O--/--O--/--O
     \/ \/ 1/ \/ \/ \/ \/ \/ \/ \/
     /\ /\ 2\ /\ /\ /\ /\ /\ /\ /\
    O3-/--O--/--O--/--O--/--O--/--O
      / \   / \   / \   / \   / \
     1   \ 9   \ /   \ /   \ /   \
    O--2--O--1--O--5--O--3--O--8--O



Our goal is to find a path from left to right through it which
minimizes the sum of the score of all edges.
(and of course left/right is just a convention here it could be top down too)
Similarly the minimum could be the maximum by just flipping the sign,
Example of a path with scores:

    O   O   O   O   O   O   O

>---O.  O   O  .O-2-O   O   O
      5.     .7      .
    O   O-1-O   O   O 8 O   O
                       .
    O   O   O   O   O   O-1-O---> (sum here is 24)


The Viterbi algorithm now solves this simply column by column
For the previous column each point has a best path and a associated
score:

    O-----5     O
     \
      \
    O  \  1     O
        \/
        /\
    O  /  2     O
      /
     /
    O-----2     O


To move one column forward we just need to find the best path and associated
scores for the next column
here are some edges we could choose from:


    O-----5--3--O
     \      \8
      \       \
    O  \  1--9--O
        \/  \3
        /\     \
    O  /  2--1--O
      /     \2
     /        \
    O-----2--4--O

Finding the new best paths and scores for each point of our new column is
trivial given we know the previous column best paths and scores:

    O-----0-----8
     \
      \
    O  \  0----10
        \/
        /\
    O  /  0-----3
      /     \
     /        \
    O     0     4


the Viterbi algorithm continues exactly like this column for column until the
end and then just picks the path with the best score (above that would be the
one with score 3)


Author: Michael Niedermayer
Copyright LGPL