Arrays and Iteration

Array Sum (Linear Scan)

Walk an array once, accumulating each element into a running total. This is the canonical single-pass linear scan and the simplest possible loop invariant: after step i, total equals the sum of arr[0..i].

Algorithm

The canonical input from the lesson spec is @arr = (3, 1, 4, 1, 5, 9, 2, 6). After eight passes the running total is 31.

linear scan Visit each element exactly once in index order.
running total `total` accumulates the sum as the loop advances.

Basic Implementation

basic.pl
Replay: real traced execution (multi-file project)
use strict; use warnings;
my @arr = (3, 1, 4, 1, 5, 9, 2, 6);
my $total = 0;
my $i = 0;
while ($i < scalar @arr) {
	$total = $total + $arr[$i];
	$i = $i + 1;
}
print "$total\n";

  1. arr ← [3, 1, 4, 1, 5, 9, 2, 6]

    1use strict; use warnings;2my @arr = (3, 1, 4, 1, 5, 9, 2, 6);3my $total = 0;
    values this step[3, 1, 4, 1, 5, 9, 2, 6]arr

  2. total ← 0

    2my @arr = (3, 1, 4, 1, 5, 9, 2, 6);3my $total = 0;4my $i = 0;
    values this step0total[3, 1, 4, 1, 5, 9, 2, 6]arr

  3. total ← 3

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step0 → 3total0i3arr[i]

  4. total ← 4

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step3 → 4total1i1arr[i]

  5. total ← 8

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step4 → 8total2i4arr[i]

  6. total ← 9

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step8 → 9total3i1arr[i]

  7. total ← 14

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step9 → 14total4i5arr[i]

  8. total ← 23

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step14 → 23total5i9arr[i]

  9. total ← 25

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step23 → 25total6i2arr[i]

  10. total ← 31

    5while ($i < scalar @arr) {6	$total = $total + $arr[$i];7	$i = $i + 1;
    values this step25 → 31total7i6arr[i]

Trace Output

trace.pl
Replay: real traced execution (multi-file project)
use strict; use warnings;
my @arr = (3, 1, 4, 1, 5, 9, 2, 6);
my $total = 0;
my $i = 0;
while ($i < scalar @arr) {
	my $before = $total;
	$total = $total + $arr[$i];
	printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
	$i = $i + 1;
}
printf("final total = %d\n", $total);

  1. total ← 3, stdout ← step 0: arr(0)=3 total 0 -> 3

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step3totalstep 0: arr(0)=3 total 0 -> 3stdout0before3arr[i]

  2. total ← 4, stdout ← step 1: arr(1)=1 total 3 -> 4

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step4totalstep 1: arr(1)=1 total 3 -> 4stdout3before1arr[i]

  3. total ← 8, stdout ← step 2: arr(2)=4 total 4 -> 8

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step8totalstep 2: arr(2)=4 total 4 -> 8stdout4before4arr[i]

  4. total ← 9, stdout ← step 3: arr(3)=1 total 8 -> 9

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step9totalstep 3: arr(3)=1 total 8 -> 9stdout8before1arr[i]

  5. total ← 14, stdout ← step 4: arr(4)=5 total 9 -> 14

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step14totalstep 4: arr(4)=5 total 9 -> 14stdout9before5arr[i]

  6. total ← 23, stdout ← step 5: arr(5)=9 total 14 -> 23

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step23totalstep 5: arr(5)=9 total 14 -> 23stdout14before9arr[i]

  7. total ← 25, stdout ← step 6: arr(6)=2 total 23 -> 25

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step25totalstep 6: arr(6)=2 total 23 -> 25stdout23before2arr[i]

  8. total ← 31, stdout ← step 7: arr(7)=6 total 25 -> 31

    6my $before = $total;7$total = $total + $arr[$i];8printf("step %d: arr(%d)=%d total %d -> %d\n", $i, $i, $arr[$i], $before, $total);
    values this step31totalstep 7: arr(7)=6 total 25 -> 31stdout25before6arr[i]

  9. stdout ← final total = 31

    10}11printf("final total = %d\n", $total);
    values this stepfinal total = 31stdout31total

Complexity

  • Time: O(n)
  • Space: O(1)

Implementation notes

  • Perl: use the explicit while ($i < scalar @arr) loop with $total = 0 and a manual 0-based index. The stdlib does not ship a one-call sum for arrays — List::Util::sum would hide the loop the lesson is teaching and would also require a CPAN-style use.
  • my @arr = (3, 1, 4, 1, 5, 9, 2, 6); documents the fixed-content array; the manual $i = $i + 1 step keeps the iteration without leaning on foreach my $val (@arr) that hides the running index.
  • use strict; use warnings; at the top keeps the lesson on the typo-safe path; every $total / $i / @arr reference is my-declared.
  • The replay shows i, arr[i], and total before and after each addition, matching the lesson spec's state-transition table.