stuck with understanding 2 for loops

Question

i am using this site as a resource http://www.perlmonks.org/?node_id=573138

I am trying to understand about O notation and it gives two examples of searching two arrays for the same element. First example has O(n^2) as does the second, however the second has an enhancement to it so it runs quicker but still maintains the same O notation, I will paste the code samples below. What I would like to know is how they work, I have limited programming knowledge and am most comfortable in java, i can understand the first I think, just two for loops and check, something like;

for (int i = 0; i < arrarysize ; i++){
    for (int j = 0; j < arraysize; j++){
        if(getElementFromArray(i).equals(getElementFromArray(j))){
            //do something
        }
    }
}

but how the second works is beyond me, i just don't get the "enhancement"

for my $i (0 .. $#array) {
    for my $j (0 .. $#array) {
        next if $j == $i;
        # Compare $i, $j
    }
}

for my $i (0 .. $#array - 1) {
    for my $j ($i + 1 .. $#array) {
        # Compare $i, $j
    }
}

Answer 1

Think of it in terms of a rectangle of possible (i, j) values. The first loop compares every pair - so it compares (5, 0) and later compares (0, 5) which will obviously just give the opposite result.

The second loop divides that rectangle in half - basically it only checks one "triangle" of it - every value where j > i so it would check (0, 5) but not (5, 0). This avoids redundancy - but it just means that it's checking n*(n-1)/2 values instead of n^2 values - it's still O(n^2).

The second loop in Java would be:

for (int i = 0; i < arraysize - 1; i++) {
    for (int j = i + 1; j < arraysize; j++){
        if(i == j) {
            //do something
        }
    }
}