Sequence Comparison.pdf

2.4 Dynamic Programming 31
Fig. 2.12 Tabular computation of the length of an LCS of 〈A,L,G,O,R,I,T,H,M〉 and
〈A,L,I,G,N,M,E,N,T〉.
work. These arrows will guide the backtracking process upon reaching the terminal
entry (m,n). Since the time spent for each entry is O(1), the total running time of
algorithm LCS LENGTH is O(mn).
Figure 2.12 illustrates the tabular computation. The length of an LCS of
and
〈A,L,G,O,R,I,T,H,M〉
〈A,L,I,G,N,M,E,N,T 〉
is 4.
Besides computing the length of an LCS of the whole sequences, Figure 2.12
in fact computes the length of an LCS between each pair of prefixes of the two
sequences. For example, by this table, we can also tell the length of an LCS between
〈A,L,G,O,R〉 and 〈A,L,I,G〉 is 3.
Algorithm LCS OUTPUT(A = 〈a 1 ,a 2 ,...,a m 〉, prev, i, j)
begin
if i = 0or j = 0 then return
if prev[i, j]=“↖” then
LCS OUTPUT(A, prev, i − 1, j − 1)
print a i
else if prev[i, j]=“↑” then LCS OUTPUT(A, prev, i − 1, j)
else LCS OUTPUT(A, prev, i, j − 1)
end
Fig. 2.13 Delivering an LCS.