Elementary Sorting Algorithms Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Elementary Sorting Algorithms notes for GATE DA: 25 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Elementary Sorting Algorithms

Chapter Roadmap

Elementary Sorting Algorithms

Topic 1: Bubble, Insertion and Selection Sort Pass Analysis

Master the invariants, intermediate states, and exact operation counts for the three elementary sorts. (Current Topic)

Topic 2: Insertion Sort Swaps and Nearly Sorted Arrays

Deep dive into shifts, swaps, and the mathematics of inversions in partially sorted data.

The Core Idea of Pass Analysis

The Concept of a "Pass"

A pass is one complete iteration of the algorithm's primary loop. In exam questions, you are rarely asked to sort the whole array. You are asked to determine the state of the array after exactly $k$ passes.

The Golden Rule

Do not trace the whole algorithm. Identify the invariant: what is mathematically guaranteed to be in its final sorted position after pass $k$ ?

Bubble Sort: Locks elements at the end.
Selection Sort: Locks elements at the beginning.
Insertion Sort: Sorts a prefix of the array.

Bubble Sort: Mechanics and Invariants

Bubble Sort: Mechanics & Invariants

Repeatedly step through the list, compare adjacent elements, and swap them if they are in the wrong order.

The Pass Invariant: After pass

k

, the

k

largest elements are in their final sorted positions at the end of the array.

Comparison Count

In the standard implementation (without early termination), the number of comparisons in pass $k$ is strictly deterministic:

Pass 1: $n - 1$ comparisons
Pass 2: $n - 2$ comparisons
Pass $k$ : $n - k$ comparisons

Total comparisons for

k

passes =

\sum_{i = 1}^{k} (n - i) = k n - \frac{k ( k + 1 )}{2}

Insertion Sort: Mechanics and Invariants

Insertion Sort: Mechanics & Invariants

Divide the array into a "sorted" and "unsorted" region. Repeatedly take the first element of the unsorted region and insert it into the correct position in the sorted region by shifting larger elements to the right.

The Pass Invariant: After pass

k

, the first

k + 1

elements of the array are sorted relative to each other.
(Note: Pass 1 processes the 2nd element, so after pass 1, the first 2 elements are sorted).

Comparison Count

The number of comparisons is data-dependent.

In pass $k$ , the algorithm compares the key against elements in the sorted prefix until it finds the correct spot.
Comparisons range from $1$ (best case, already in place) to $k$ (worst case, needs to go to the front).

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Elementary Sorting Algorithms Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Elementary Sorting Algorithms notes for GATE DA: 25 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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