Suppose we are given an array, A of n positive numbers $($or a string of n characters$).$ A $=[$a$1,$a$2,....,$an$].$ An increasing subsequence of A is a subsequence, $\{$aia$,$ ai$2,....,$ aik$\},$ of elements from A such that ij $<$ ij$+1$ and aij $<$ aij$+1$ for each $1<=$ j $<$ k$.$ Notice that the elements in the increasing subsequence need not be contigous in A$.$ Given an increasing subsequence of A$,$ we can compute the length of this subsequnce. The longest increasing sub