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

C# DSA Implementation

basic.cs 
using System;
using System.Collections.Generic;
using System.Linq;
class Program {
    static string ListString(IEnumerable<int> values) => "[" + string.Join(", ", values) + "]";
    static void HeapInsert(List<int> heap, int value) {
        heap.Add(value);
        int child = heap.Count - 1;
        while (child > 0) {
            int parent = (child - 1) / 2;
            if (heap[parent] <= heap[child]) break;
            (heap[parent], heap[child]) = (heap[child], heap[parent]);
            child = parent;
        }
    }
    static int HeapPop(List<int> heap) {
        int smallest = heap[0];
        heap[0] = heap[^1];
        heap.RemoveAt(heap.Count - 1);
        int parent = 0;
        while (true) {
            int left = parent * 2 + 1, right = left + 1;
            if (left >= heap.Count) break;
            int child = left;
            if (right < heap.Count && heap[right] < heap[left]) child = right;
            if (heap[parent] <= heap[child]) break;
            (heap[parent], heap[child]) = (heap[child], heap[parent]);
            parent = child;
        }
        return smallest;
    }
    static void Main() { var heap = new List<int> {2, 4, 7, 9, 6}; HeapInsert(heap, 1); Console.WriteLine(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.

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

Output

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

Implementation notes

  • List<int> is the array-backed heap store; heap.Add(value) may grow its managed backing array, which the CLR later reclaims through GC. The heap uses integer index arithmetic, with child > 0 guarding the parent lookup and (child - 1) / 2 selecting the parent slot.
  • The tuple assignment swaps int values in place, so the replay can show the appended value moving from the last slot to the root through two sift-up swaps.