Question

Let S be a sequence of n distinct integers stored in an array as array elements S[1], S[2], · · · , S[n]. Use the technique of dynamic programming to find the length of a longest ascending subsequence of entries in S. For example, if the entries of S are 11, 17, 5, 8, 6, 4, 7, 12, 3, then one longest ascending subsequence is 5, 6, 7, 12.

Specifically:

(a) define a proper function and find the recurrence for this function;

(b) show that the principle of optimality holds with respect to the function;

Answer 1

Solution

Lets consider Ci is the length of a longest increasing sub sequence in S[1,...i] it will end at

xi=s[i] where 1<=i<=n

the meaning of the S[i] is the last element in the longest   increasing sub sequence in the S[1,...i]

Now the recurrence is

First we need to compute C1,C2,C3,...Cn

then we need to find max_1<=i<=n{Ci}

if you compute Ci that will take O(i) time

It will take O(n²) times to get all of the Ci

For finding the max it need O(n) time

So,

the total time is O(n²)

For finding ascending sub sequence,we need to book keeping like the following

First we need to Create an array K[1,2...n] such that

K[i] contains k in that

S[k]<S[i]

&

Ci=Ck+1

--

all the best

If the Cm is the largest among the values C1,C2,C3,....Cn

then the last value in the longest ascending subsequence is S[m]

&

We find the subsequence recursively from C_k[m]