Question

given a string consists only of 0s and 1s say 10101
how to find the length of the longest non decreasing sub-sequence??
for example,

for the string,
10101

the longest non decreasing sub sequences are

• 111
• 001
  so you should output 3

for the string
101001

the longest non decreasing sub sequence is

• 0001

so you should output 4
how to find this??

how can this be done when we are provided with limits.sequence between the limit
for example
101001
limits [3,6]
the longest non decreasing sub sequence is

• 001
  so you should output 3

can this be achieved in o(strlen)

Answer 1

Can this be achieved in O(strlen)?

Yes. Observe that the non-decreasing subsequences would have one of these three forms:

0........0 // Only zeros
1........1 // Only ones
0...01...1 // Some zeros followed by some ones

The first two forms can be easily checked in O(1) by counting all zeros and by counting all ones.

The last one is a bit harder: you need to go through the string keeping the counter of zeros that you’ve seen so far, along with the length of the longest string of 0...01...1 form that you have discovered so far. At each step where you see 1 in the string, the length of the longest subsequence of the third form is the larger of the number of zeros plus one or the longest 0...01...1 sequence that you’ve seen so far plus one.

Here is the implementation of the above approach in C:

char *str = "10101001";
int longest0=0, longest1=0;
for (char *p = str ; *p ; p++) {
    if (*p == '0') {
        longest0++;
    } else { // *p must be 1
        longest1 = max(longest0, longest1)+1;
    }
}
printf("%d\n", max(longest0, longest1));

max is defined as follows:

#define max( a, b ) ( ((a) > (b)) ? (a) : (b) )

Here is a link to a demo on ideone.