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Bubble Sort Analysis

Temporal Manifold Navigation

Analysis Entry

Initializing...

Step 1 of 0
Scanning Manifold
Active Displacement
Ordered Manifold

Manifold Specification

Monotonic Bubbling

Bubble Sort is a stable, comparison-based algorithm that iteratively transforms a manifold into a monotonic sequence. It operates by repeatedly stepping through the list, comparing adjacent elements and swapping them if they are in the wrong order.

The name "Bubble Sort" comes from the way smaller elements "bubble" to the top of the list (beginning of the array) while larger elements sink to the bottom.

Asymptotic Bounds

Temporal

O(N²)

Spatial

O(1)

Stability Lemma

Bubble sort is Stable; it preserves the relative order of equal elements. This is a critical property when sorting complex manifolds with multi-key dependencies.

Implementation Guide

Swapping Logic

Iterate through the array multiple times. In each pass, compare adjacent elements and swap if $A[j] > A[j+1]$.

cpp
void bubbleSort(vector<int>& arr) {
    int n = arr.size();
    for (int i = 0; i < n - 1; i++) {
        for (int j = 0; j < n - i - 1; j++) {
            if (arr[j] > arr[j + 1]) {
                swap(arr[j], arr[j + 1]);
            }
        }
    }
}

Cognitive Challenges

Apply this lemma to real-world complexity protocols.