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.dart
Replay: real traced execution (multi-file project)
void main() {
  final arr = <int>[3, 1, 4, 1, 5, 9, 2, 6];
  var total = 0;
  for (var i = 0; i < arr.length; i++) {
    total = total + arr[i];
  }
  print(total);
}

  1. arr ← [3, 1, 4, 1, 5, 9, 2, 6]

    1void main() {2  final arr = <int>[3, 1, 4, 1, 5, 9, 2, 6];3  var total = 0;
    values this step[3, 1, 4, 1, 5, 9, 2, 6]arr

  2. total ← 0

    2final arr = <int>[3, 1, 4, 1, 5, 9, 2, 6];3var total = 0;4for (var i = 0; i < arr.length; i++) {
    values this step0total[3, 1, 4, 1, 5, 9, 2, 6]arr

  3. total ← 3

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step0 → 3total0i3arr[i]

  4. total ← 4

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step3 → 4total1i1arr[i]

  5. total ← 8

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step4 → 8total2i4arr[i]

  6. total ← 9

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step8 → 9total3i1arr[i]

  7. total ← 14

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step9 → 14total4i5arr[i]

  8. total ← 23

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step14 → 23total5i9arr[i]

  9. total ← 25

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step23 → 25total6i2arr[i]

  10. total ← 31

    4for (var i = 0; i < arr.length; i++) {5  total = total + arr[i];6}
    values this step25 → 31total7i6arr[i]

Trace Output

trace.dart
Replay: real traced execution (multi-file project)
void main() {
  final arr = <int>[3, 1, 4, 1, 5, 9, 2, 6];
  var total = 0;
  for (var i = 0; i < arr.length; i++) {
    final before = total;
    total = total + arr[i];
    print('step $i: arr[$i]=${arr[i]} total $before -> $total');
  }
  print('final total = $total');
}

  1. total ← 3, stdout ← step 0: arr[0]=3 total 0 -> 3

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $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

    5final before = total;6total = total + arr[i];7print('step $i: arr[$i]=${arr[i]} total $before -> $total');
    values this step31totalstep 7: arr[7]=6 total 25 -> 31stdout25before6arr[i]

  9. stdout ← final total = 31

    8  }9  print('final total = $total');10}
    values this stepfinal total = 31stdout31total

Complexity

  • Time: O(n)
  • Space: O(1)

Implementation notes

  • Dart: use an explicit for (var i = 0; i < arr.length; i++) loop. Calling arr.reduce((a, b) => a + b) or arr.fold(0, ...) would hide the iteration the lesson is teaching.
  • The replay shows i, arr[i], and total before and after each addition, matching the lesson spec's state-transition table. The trace variant prints step i: arr[i]=V total B -> T for each visit and a closing final total = 31 summary.