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

Java DSA Implementation

Basic.java

import java.util.*;
class Basic {
    static String listString(List<Integer> values) { return values.toString(); }
    static void heapInsert(List<Integer> heap, int value) {
        heap.add(value);
        int child = heap.size() - 1;
        while (child > 0) {
            int parent = (child - 1) / 2;
            if (heap.get(parent) <= heap.get(child)) break;
            Collections.swap(heap, parent, child);
            child = parent;
        }
    }
    static int heapPop(List<Integer> heap) {
        int smallest = heap.get(0);
        heap.set(0, heap.remove(heap.size() - 1));
        int parent = 0;
        while (true) {
            int left = parent * 2 + 1;
            int right = left + 1;
            if (left >= heap.size()) break;
            int child = left;
            if (right < heap.size() && heap.get(right) < heap.get(left)) child = right;
            if (heap.get(parent) <= heap.get(child)) break;
            Collections.swap(heap, parent, child);
            parent = child;
        }
        return smallest;
    }
    public static void main(String[] args) { List<Integer> heap = new ArrayList<>(Arrays.asList(2, 4, 7, 9, 6)); heapInsert(heap, 1); System.out.println(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.

Output

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

Implementation notes

Java stores the heap as a List<Integer> backed by an ArrayList, seeded with new ArrayList<>(Arrays.asList(2, 4, 7, 9, 6)). Heap entries are boxed through Integer.valueOf, so these small fixture values may be cached Integer instances rather than primitive int array slots.
heapInsert mutates the same list with heap.add(value), then starts the new child at heap.size() - 1 and computes each parent as (child - 1) / 2.
This is a min-heap: the sift-up loop stops when child == 0 or heap.get(parent) <= heap.get(child), which unboxes the two Integer values for primitive comparison. Otherwise Collections.swap(heap, parent, child) exchanges the two list positions and continues from the parent index.
The replay-visible states show append [2, 4, 7, 9, 6, 1], then swaps to [2, 4, 1, 9, 6, 7] and [1, 4, 2, 9, 6, 7]. Allocation is the ArrayList storage and any uncached boxed integers, all managed by JVM GC.