Algorithm
Let 
be a sequence of states 
belonging to a finite set of possible states. Let us denote 
the sequence space, i.e. the set of all possible sequences of states.
Optimal matching algorithms work by defining simple operator algebras that manipulate sequences, i.e. a set of operators 
. In the most simple approach, a set composed of only three basic operations to transform sequences is used:
- one state
is inserted in the sequence 
- one state is deleted from the sequence
and - a state
is replaced (substituted) by state 
, 
.
Imagine now that a cost
is associated to each operator. Given two sequences 
and 
, the idea is to measure the cost of obtaining from using operators from the algebra. Let 
be a sequence of operators such that the application of all the operators of this sequence 
to the first sequence gives the second sequence : 
where 
denotes the compound operator. To this set we associate the cost 
, that represents the total cost of the transformation. One should consider at this point that there might exist different such sequences that transform into ; a reasonable choice is to select the cheapest of such sequences. We thus call distance
that is, the cost of the least expensive set of transformations that turn into . Notice that 
is by definition nonnegative since it is the sum of positive costs, and trivially 
if and only if 
, that is there is no cost. The distance function is symmetric if insertion and deletion costs are equal 
; the term indel cost usually refers to the common cost of insertion and deletion.
Considering a set composed of only the three basic operations described above, this proximity measure satisfies the triangular inequality. Transitivity however, depends on the definition of the set of elementary operations.
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