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.cpp 
#include <algorithm>
#include <iostream>
#include <sstream>
#include <vector>
using namespace std;
string listString(const vector<int>& values) {
    stringstream out; out << "[";
    for (size_t i = 0; i < values.size(); i++) { if (i) out << ", "; out << values[i]; }
    out << "]"; return out.str();
}
void heapInsert(vector<int>& heap, int value) {
    heap.push_back(value);
    size_t child = heap.size() - 1;
    while (child > 0) {
        size_t parent = (child - 1) / 2;
        if (heap[parent] <= heap[child]) break;
        swap(heap[parent], heap[child]);
        child = parent;
    }
}
int heapPop(vector<int>& heap) {
    int smallest = heap[0];
    heap[0] = heap.back(); heap.pop_back();
    size_t parent = 0;
    while (true) {
        size_t left = parent * 2 + 1, right = left + 1;
        if (left >= heap.size()) break;
        size_t child = left;
        if (right < heap.size() && heap[right] < heap[left]) child = right;
        if (heap[parent] <= heap[child]) break;
        swap(heap[parent], heap[child]);
        parent = child;
    }
    return smallest;
}
int main() { vector<int> heap = {2, 4, 7, 9, 6}; heapInsert(heap, 1); cout << listString(heap) << "\n"; }

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

  • In C++, the heap is a std::vector<int> initialized as {2, 4, 7, 9, 6} and passed by non-const reference to heapInsert(vector<int>& heap, int value).
  • heap.push_back(value) appends 1 in place; size_t child = heap.size() - 1 tracks the inserted slot.
  • Parent indexes use (child - 1) / 2 with size_t arithmetic. The loop stops at the root or when heap[parent] <= heap[child].
  • Swaps use std::swap on two vector elements, then move child to the parent index. No second heap vector is allocated.
  • The trace records [2, 4, 7, 9, 6], append to [2, 4, 7, 9, 6, 1], swap with parent 7 to [2, 4, 1, 9, 6, 7], then swap with parent 2 to [1, 4, 2, 9, 6, 7].
  • listString(const std::vector<int>&) uses std::stringstream and a size_t loop, and std::cout << ... << "\n" prints [1, 4, 2, 9, 6, 7]. Visible allocation is the heap vector storage and formatting buffer; mutation is vector growth and element swaps.