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.cs
Replay: real traced execution (multi-file project)
using System;

class Program {
	static void Main() {
		int[] arr = new int[] { 3, 1, 4, 1, 5, 9, 2, 6 };
		int total = 0;
		for (int i = 0; i < arr.Length; i++) {
			total = total + arr[i];
		}
		Console.WriteLine(total);
	}
}

  1. arr ← [3, 1, 4, 1, 5, 9, 2, 6]

    4static void Main() {5	int[] arr = new int[] { 3, 1, 4, 1, 5, 9, 2, 6 };6	int total = 0;
    values this step[3, 1, 4, 1, 5, 9, 2, 6]arr

  2. total ← 0

    5int[] arr = new int[] { 3, 1, 4, 1, 5, 9, 2, 6 };6int total = 0;7for (int i = 0; i < arr.Length; i++) {
    values this step0total[3, 1, 4, 1, 5, 9, 2, 6]arr

  3. total ← 3

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step0 → 3total0i3arr[i]

  4. total ← 4

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step3 → 4total1i1arr[i]

  5. total ← 8

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step4 → 8total2i4arr[i]

  6. total ← 9

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step8 → 9total3i1arr[i]

  7. total ← 14

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step9 → 14total4i5arr[i]

  8. total ← 23

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step14 → 23total5i9arr[i]

  9. total ← 25

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step23 → 25total6i2arr[i]

  10. total ← 31

    7for (int i = 0; i < arr.Length; i++) {8	total = total + arr[i];9}
    values this step25 → 31total7i6arr[i]

Trace Output

trace.cs
Replay: real traced execution (multi-file project)
using System;

class Program {
	static void Main() {
		int[] arr = new int[] { 3, 1, 4, 1, 5, 9, 2, 6 };
		int total = 0;
		for (int i = 0; i < arr.Length; i++) {
			int before = total;
			total = total + arr[i];
			Console.WriteLine(
				$"step {i}: arr[{i}]={arr[i]} total {before} -> {total}");
		}
		Console.WriteLine($"final total = {total}");
	}
}

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

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step3totalstep 0: arr[0]=3 total 0 -> 3stdout0before3arr[i]

  2. total ← 4, stdout ← step 1: arr[1]=1 total 3 -> 4

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step4totalstep 1: arr[1]=1 total 3 -> 4stdout3before1arr[i]

  3. total ← 8, stdout ← step 2: arr[2]=4 total 4 -> 8

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step8totalstep 2: arr[2]=4 total 4 -> 8stdout4before4arr[i]

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

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step9totalstep 3: arr[3]=1 total 8 -> 9stdout8before1arr[i]

  5. total ← 14, stdout ← step 4: arr[4]=5 total 9 -> 14

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step14totalstep 4: arr[4]=5 total 9 -> 14stdout9before5arr[i]

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

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step23totalstep 5: arr[5]=9 total 14 -> 23stdout14before9arr[i]

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

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step25totalstep 6: arr[6]=2 total 23 -> 25stdout23before2arr[i]

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

    8int before = total;9total = total + arr[i];10Console.WriteLine(
    values this step31totalstep 7: arr[7]=6 total 25 -> 31stdout25before6arr[i]

  9. stdout ← final total = 31

    12	}13	Console.WriteLine($"final total = {total}");14}
    values this stepfinal total = 31stdout31total

Complexity

  • Time: O(n)
  • Space: O(1)

Implementation notes

  • C#: use the explicit for (int i = 0; i < arr.Length; i++) loop with int total = 0;. LINQ's arr.Sum() is fine for production but hides the loop the lesson spec is teaching.
  • int[] arr = new int[] { 3, 1, 4, 1, 5, 9, 2, 6 }; documents the fixed-size managed reference array contract; the CLR owns allocation/GC and bounds-checks each arr[i] read. arr.Length gives the explicit count without leaning on a helper that hides the iteration.
  • The replay shows i, arr[i], and total before and after each addition, matching the lesson spec's state-transition table.