08-heaps

Min-Heap Insert (Sift Up)

Insert one value into a min-heap and restore the parent-child order by sifting upward.

Algorithm

Steps

  1. Store the heap in an array.
  2. Compare parent and child indexes instead of building explicit tree nodes.
  3. Swap only when the heap order is violated.
  4. Print the deterministic final heap state for replay comparison.

Complexity

  • Time: O(log n)
  • Space: O(1) extra
sift up A new value starts at the end of the array and swaps with its parent while it is smaller.

Visual walkthrough

Lua DSA Implementation

basic.lua 
local function list_string(values) return "[" .. table.concat(values, ", ") .. "]" end
local function heap_insert(heap, value)
  table.insert(heap, value)
  local child = #heap
  while child > 1 do
    local parent = math.floor(child / 2)
    if heap[parent] <= heap[child] then break end
    heap[parent], heap[child] = heap[child], heap[parent]
    child = parent
  end
end
local function heap_pop(heap)
  local smallest = heap[1]
  heap[1] = table.remove(heap)
  local parent = 1
  while true do
    local left = parent * 2
    local right = left + 1
    if left > #heap then break end
    local child = left
    if right <= #heap and heap[right] < heap[left] then child = right end
    if heap[parent] <= heap[child] then break end
    heap[parent], heap[child] = heap[child], heap[parent]
    parent = child
  end
  return smallest
end
local heap = {2, 4, 7, 9, 6}
heap_insert(heap, 1)
print(list_string(heap))

The heap is still an array, but the tree view makes parent-child swaps visible. The labels use the pinned replay states from the lesson.

Step 1 - Append 1

The new value 1 starts at index 5 and compares with its parent value 7.

Array state [2, 4, 7, 9, 6, 1] as a heap-shaped tree.2i04i17parent9i36i41new

Step 2 - Swap with 7

Because 1 is smaller than 7, those array slots swap.

After the first sift-up swap: [2, 4, 1, 9, 6, 7].2parent4i11child9i36i47i5

Step 3 - Swap with 2

1 is also smaller than 2, so it moves to the root and the heap order is restored.

Final heap after insert: [1, 4, 2, 9, 6, 7].1root4i12i29i36i47i5

Implementation notes

  • local heap = {2, 4, 7, 9, 6} is a Lua table representing a min-heap with 1-based indexes.
  • heap_insert(heap, 1) mutates that same table; Lua passes the table reference into the helper.
  • Insert uses table.insert(heap, value), so the checked value 1 is appended at slot #heap, producing [2, 4, 7, 9, 6, 1].
  • local child = #heap starts at the appended slot.
  • The parent index is math.floor(child / 2), which matches 1-based heap layout.
  • The loop boundary is while child > 1 do, so sift-up stops at the root.
  • This is a min-heap: if heap[parent] <= heap[child] then break end stops when parent order is already valid.
  • Swaps use Lua multiple assignment: heap[parent], heap[child] = heap[child], heap[parent].
  • The trace swaps 1 with parent 7, giving [2, 4, 1, 9, 6, 7], then swaps 1 with parent 2, giving [1, 4, 2, 9, 6, 7].
  • print(list_string(heap)) renders the final heap as [1, 4, 2, 9, 6, 7].

Output

[1, 4, 2, 9, 6, 7]