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[1..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.lua
Replay: real traced execution (multi-file project)
local arr = {3, 1, 4, 1, 5, 9, 2, 6}
local total = 0
local i = 1
while i <= #arr do
	total = total + arr[i]
	i = i + 1
end
print(total)

  1. arr ← [3, 1, 4, 1, 5, 9, 2, 6]

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

  2. total ← 0

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

  3. total ← 3

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step0 → 3total1i3arr[i]

  4. total ← 4

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step3 → 4total2i1arr[i]

  5. total ← 8

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step4 → 8total3i4arr[i]

  6. total ← 9

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step8 → 9total4i1arr[i]

  7. total ← 14

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step9 → 14total5i5arr[i]

  8. total ← 23

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step14 → 23total6i9arr[i]

  9. total ← 25

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step23 → 25total7i2arr[i]

  10. total ← 31

    4while i <= #arr do5	total = total + arr[i]6	i = i + 1
    values this step25 → 31total8i6arr[i]

Trace Output

trace.lua
Replay: real traced execution (multi-file project)
local arr = {3, 1, 4, 1, 5, 9, 2, 6}
local total = 0
local i = 1
while i <= #arr do
	local before = total
	total = total + arr[i]
	print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
	i = i + 1
end
print(string.format("final total = %d", total))

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step3totalstep 1: arr(1)=3 total 0 -> 3stdout0before3arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step4totalstep 2: arr(2)=1 total 3 -> 4stdout3before1arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step8totalstep 3: arr(3)=4 total 4 -> 8stdout4before4arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step9totalstep 4: arr(4)=1 total 8 -> 9stdout8before1arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step14totalstep 5: arr(5)=5 total 9 -> 14stdout9before5arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step23totalstep 6: arr(6)=9 total 14 -> 23stdout14before9arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step25totalstep 7: arr(7)=2 total 23 -> 25stdout23before2arr[i]

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

    5local before = total6total = total + arr[i]7print(string.format("step %d: arr(%d)=%d total %d -> %d", i, i, arr[i], before, total))
    values this step31totalstep 8: arr(8)=6 total 25 -> 31stdout25before6arr[i]

  9. stdout ← final total = 31

    9end10print(string.format("final total = %d", total))
    values this stepfinal total = 31stdout31total

Complexity

  • Time: O(n)
  • Space: O(1)

Implementation notes

  • Lua: use the explicit while i <= #arr do loop with total = 0 and a manual 1-based index. The stdlib does not ship a one-call sum for sequences — even table.unpack(arr) would just spread the data; we keep the loop visible.
  • local 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 ipairs or for k, v in ipairs(arr) do that hide the running update.
  • Lua tables are 1-indexed; the replay shows i, arr[i], and total before and after each addition, with i ranging 1..8 to match the actual loop counter.