Min-Heap Insert (Sift Up)

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

Algorithm

Steps

Store the heap in an array.
Compare parent and child indexes instead of building explicit tree nodes.
Swap only when the heap order is violated.
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

Swift DSA Implementation

basic.swift

func listString(_ values: [Int]) -> String { "[" + values.map(String.init).joined(separator: ", ") + "]" }
func heapInsert(_ heap: inout [Int], _ value: Int) {
    heap.append(value)
    var child = heap.count - 1
    while child > 0 {
        let parent = (child - 1) / 2
        if heap[parent] <= heap[child] { break }
        heap.swapAt(parent, child)
        child = parent
    }
}
func heapPop(_ heap: inout [Int]) -> Int {
    let smallest = heap[0]
    heap[0] = heap.removeLast()
    var parent = 0
    while true {
        let left = parent * 2 + 1
        let right = left + 1
        if left >= heap.count { break }
        var child = left
        if right < heap.count && heap[right] < heap[left] { child = right }
        if heap[parent] <= heap[child] { break }
        heap.swapAt(parent, child)
        parent = child
    }
    return smallest
}
var heap = [2, 4, 7, 9, 6]
heapInsert(&heap, 1)
print(listString(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.

Step 2 - Swap with 7

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

Step 3 - Swap with 2

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

Implementation notes

var heap = [2, 4, 7, 9, 6] is a mutable Swift [Int] holding a min-heap in array layout.
heapInsert(_ heap: inout [Int], _ value: Int) mutates the caller's array; heap.append(value) places the new value at the last index.
After appending 1, var child = heap.count - 1 starts at index 5.
The parent index is let parent = (child - 1) / 2, using integer division on Swift Int values.
The while loop continues while child > 0; it stops early when heap[parent] <= heap[child], preserving min-heap order.
Violations use heap.swapAt(parent, child) and then move child = parent. The trace swaps 1 with 7, then with 2.
The replayed heap states are [2, 4, 7, 9, 6], [2, 4, 7, 9, 6, 1], [2, 4, 1, 9, 6, 7], and finally [1, 4, 2, 9, 6, 7].
listString(_:) formats the array with comma separators, so print(listString(heap)) outputs [1, 4, 2, 9, 6, 7].

Output

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