# RogerBW's Blog

The Weekly Challenge 257: Smaller than Echelon 25 February 2024

I’ve been doing the Weekly Challenges. The latest involved list processing and matrix testing. (Note that this ends today.)

You are given a array of integers, `@ints`.

Write a script to find out how many integers are smaller than current i.e. foreach `ints[i]`, count `ints[j] < ints[i]` where ```i != j```.

There's an obvious O(n²) way of doing this, but I thought it was cleaner to sort.

Scala:

``````  def smallerthancurrent(a: List[Int]): List[Int] = {
``````

Set up a sorted list of inputs, and a map that'll contain the solution.

``````    val s: List[Int] = a.sortWith(_ < _)
var l = mutable.Map.empty[Int, Int]
``````

For each item in the sorted list, store the index, which is the number of items smaller than that. (Use the first occurrence of the item; I added the test case with a duplicate value.)

``````    for ((n, i) <- s.zipWithIndex) {
if (!l.contains(n)) {
l += (n -> i)
}
}
``````

Then just read off the values in the order of the original list.

``````    a.map { n => l(n) }.toList
}
``````

Given a matrix M, check whether the matrix is in reduced row echelon form.

A matrix must have the following properties to be in reduced row echelon form:

1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this the leading 1.

2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.

3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

4. Each column that contains a leading 1 has zeros everywhere else in that column.

I propsed a bunch of the test cases here because I wanted to test each rule individually. Raku:

``````sub reducedrowechelon(@a) {
``````

We'll be using the position of the first 1 in each row quite a bit. That goes in `@leadingone`.

``````    my @leadingone;
``````

Inspect each row:

``````    for @a -> @row {
``````

Set the leading-one position to -1, indiciating no 1 found.

``````        my \$lp = -1;
``````

For each item in the row:

``````        for 0 .. @row.end -> \$cn {
my \$cell = @row[\$cn];
``````

If it's a 1, store that position and exit.

``````            if (\$cell == 1) {
\$lp = \$cn;
last;
``````

If it's not a zero, bail out (rule 1).

``````            } elsif (\$cell != 0) {
return False;
}
}
``````

``````        @leadingone.push(\$lp);
}
``````

If there are trailing rows of all-zero, ignore them.

``````    while (@leadingone[*-1] == -1) {
}
``````

Now sort the remaining leading-one positions.

``````    my @c = @leadingone.sort({\$^a <=> \$^b});
``````

If there are any all-zero rows, they weren't trailing. Bail out by rule 2.

``````    if (@c[0] == -1) {
return False;
}
``````

If the sorted list doesn't match the original, the leading ones aren't progressively to the right. Bail out by rule 3.

``````    if (!(@c eqv @leadingone)) {
return False;
}
``````

Finally take a columnwise slice of each column that has a leading 1 in it. Test for 1s and 0s in the right places (rule 4).

``````    for @c -> \$i {
my @col = @a.map({\$_[\$i]}).sort({\$^a <=> \$^b});
if (@col[*-1] != 1 ||
@col[*-2] != 0 ||
@col[0] != 0) {
return False;
}
}
return True;
}
``````

I spot two possible problems here:

(1) the rule-3 test will pass two leading-ones of equal value. But the rule-4 test will remove those because they won't be unique in the column. I'd rather rule 3 be fully tested in its own segment of the codex.

(2) the rule-4 test should reject a negative value found in the same column as a leading 1 (`@col[0] != 0` above). The existing test cases don't fail in that case (this is my fault since I proposed most of them).

Full code on github.