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.rb
Replay: real traced execution (multi-file project)
arr = [3, 1, 4, 1, 5, 9, 2, 6]
total = 0
i = 0
while i < arr.length
	total = total + arr[i]
	i = i + 1
end
puts total

  1. arr ← [3, 1, 4, 1, 5, 9, 2, 6]

    1arr = [3, 1, 4, 1, 5, 9, 2, 6]2total = 0
    values this step[3, 1, 4, 1, 5, 9, 2, 6]arr

  2. total ← 0

    1arr = [3, 1, 4, 1, 5, 9, 2, 6]2total = 03i = 0
    values this step0total[3, 1, 4, 1, 5, 9, 2, 6]arr

  3. total ← 3

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step0 → 3total0i3arr[i]

  4. total ← 4

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step3 → 4total1i1arr[i]

  5. total ← 8

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step4 → 8total2i4arr[i]

  6. total ← 9

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step8 → 9total3i1arr[i]

  7. total ← 14

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step9 → 14total4i5arr[i]

  8. total ← 23

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step14 → 23total5i9arr[i]

  9. total ← 25

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step23 → 25total6i2arr[i]

  10. total ← 31

    4while i < arr.length5	total = total + arr[i]6	i = i + 1
    values this step25 → 31total7i6arr[i]

Trace Output

trace.rb
Replay: real traced execution (multi-file project)
arr = [3, 1, 4, 1, 5, 9, 2, 6]
total = 0
i = 0
while i < arr.length
	before = total
	total = total + arr[i]
	puts "step #{i}: arr(#{i})=#{arr[i]} total #{before} -> #{total}"
	i = i + 1
end
puts "final total = #{total}"

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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    5before = total6total = total + arr[i]7puts "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

    9end10puts "final total = #{total}"
    values this stepfinal total = 31stdout31total

Complexity

  • Time: O(n)
  • Space: O(1)

Implementation notes

  • Ruby: use the explicit while i < arr.length loop with total = 0 and a manual index. The stdlib arr.sum is fine for production but hides the loop the lesson spec is teaching.
  • 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 each_with_index or inject(:+) that hide the running update.
  • The replay shows i, arr[i], and total before and after each addition, matching the lesson spec's state-transition table.