Explain the algorithm to solve 'longest increasing subsequence' problem

Question

I have been trying to understand this algorithm for past two hours, but can't seem to get it. Can someone please explain it in easy to understand manner?

function lis_length(a)
    n := a.length
    q := new Array(n)
    for k from 0 to n:
        max := 0;
        for j from 0 to k, if a[k] > a[j]:
            if q[j] > max, then set max = q[j].
        q[k] := max + 1;
    max := 0
    for i from 0 to n:
        if q[i] > max, then set max = q[i].
    return max;

Answer 1

After the first (double) loop terminates, q[i] is the length of the longest increasing subsequence ending at position i.

To see how the double loop works, suppose q[j] already contained the length of the largest increasing subsequence ending at position j, but only for j between 0 and k-1. Given this, how would you compute q[k]?

Well, you would find all of the j with j < k and a[j] < a[k], see which of the corresponding q[j] values was biggest, add one, and stash that value in q[k]. This is exactly what the inner loop does.

So at entry to the inner loop, q[j] already has the correct values for j between 0 and k-1. And at exit, it also has the correct value for k. So by the time the double loop exits, q[i] has the correct value for all i between 0 and n.

The final loop just selects the largest of those, which is the answer.