Distance Based Scoring

• D[s,t]   =  ∑(d[s^'[i],t^'[i]),   i  =  1..l
  • d(x,x)  =  0
  • d(x,y)   >   0
  • d("_","_")  =  0 or undefined
  • d(x,z) < d(x,y) + d(y,z)

• If d(x,y)  =  1 and d(x,"_")  =  1, then D(s,t) is the minimum number of operations required to transform s into t, where the operations are substitution, insertion and deletion. This is called the "edit distance".
• If d(x,y)  >  1 and d(x,"_")  >  1, then it is called the "weighted edit distance".
• D[s,t] is a metric
• D[s,t] is an additive scoring function. We assume positional independence.

Dynamic Programming Algorithm for Global Alignment

• Initialization
  • D[0,s[i]]  =  D[0,s[i-1]] + d(s[i],"_")
  • D[t[j],0]  =  D[t[j-1],0] + d("_",t[j])
  
  
  
• Recurrence
  

  D[i,j]  =   min  {	D[i-1,j] + d(s[i], "_")
  	D[i-1,j-1] + d(s[i], t[j])
  	D[i,j-1] + d("_", t[j])

  
  
• Termination condidtion
• Compute score of all pairs of prefixes in O(m • n) time. D[m,n] gives the score of the optimal alignment.
• Trace back through the alignment matrix in O(m+n) time to obtain the optimal alignment.
• There may be more than one optimal alignment

Similarity Scoring

• p(x,y): similarity of x and y
• p(x,"_"): gap cost
• Score of alignment   =   ∑(p(s'[i], t'[i])), i  =  1..l
  
  
• A simple similarity score that treats all characters equally:
  • p(i, i)   =   M
  • p(i, j)   =   m
  • p(i, "_")   =   g
  
  
  In this case, the recurrence relation is:

  a [i, j]  =  max {	a[i,j-1] + g
  	a[i-1, j-1] + p(i,j)
  	a[i-1,j] + g

  
  
• Require 2g < m < M. If we allow 2g ≥ m then there will be no substitutions.