Data Structures
“Master multi-dimensional address translation, stack evaluation, binary tree degree theorems, AVL rotations, B/B+ trees, binary heaps, and double hashing.”
A foundational, mathematically rigorous study notebook covering row/column-major array address mappings, stack expression evaluation, circular queues, linked list cycle detection, binary tree structural theorems, binary search trees, self-balancing AVL trees, B/B+ tree order calculations, priority heaps, hash collision resolution dynamics, and Disjoint Set Union (DSU) for semester examinations and top-percentile GATE CS/IT & DA performance.
Structured Notes & Exam Lens
A · B.Tech Structured Study Sequence
6 Sections · Full CurriculumArrays, Stacks, Queues & Linked Lists
Address calculation formulas for multi-dimensional arrays, stack infix-to-postfix conversion, circular queue arithmetic, and linked list pointer operations.
Row-major vs column-major indexing, 3D tensor addresses, operator precedence stack mechanics, and recursion frames.
Circular queue boundary arithmetic, Deques, queue via two stacks, Floyd's cycle detection, and list reversals.
Tree Fundamentals, Traversals & Binary Search Trees
Mathematical properties of binary trees, degree formulas (n0 = n2 + 1), Catalan number structural counts, traversal reconstruction, and BST insertion/deletion mechanics.
Handshaking lemma on trees, n0 = n2 + 1 proof, Catalan structural count, and unique tree reconstruction from traversals.
In-order monotonic property, three-case deletion mechanics, successor/predecessor finding, and height degeneracies.
Self-Balancing Trees & Multi-Way Indices
AVL tree balance factor rotations (LL, RR, LR, RL), Fibonacci recurrence for minimal nodes, B-Tree and B+ Tree order, fanout, and disk block size derivations.
Balance factors, single and double rotations (LL, RR, LR, RL), Fibonacci recurrence for minimal nodes, and height bounds.
B-Tree node constraints, split and merge invariants, B+ tree leaf chaining, and disk block order/fanout derivations.
Priority Queues & Binary Heaps
Binary heap array invariants, Max-Heapify mechanics, linear-time Build-Heap algebraic-geometric proof, and Heap Sort.
Array indexing, Max-Heapify mechanics, algebraic-geometric proof of O(n) Build-Heap, and priority queue operations.
Hashing, Collision Dynamics & Disjoint Sets
Open addressing (linear, quadratic, double hashing), clustering phenomena, separate chaining load factors, and Disjoint Set Union with union by rank and path compression.
Division and multiplication hash functions, linear vs quadratic probing, double hashing, and primary/secondary clustering.
Separate chaining, load factor alpha, expected probe lengths, uniform hashing assumption, and universal hash families.
Partition representation, tree rank heuristics, path compression mechanics, and the inverse Ackermann bound O(alpha(n)).
GATE DA: Python Programming
Python language fundamentals and implementations that accompany the DA data-structure and algorithm syllabus. These notes do not assume that C satisfies the Python requirement.
Read and reason about small Python programs, including types, branches, loops, and common expression traps.
Choose a built-in collection by its operations, then trace aliasing and aggregation correctly.
Build small reusable transformations and make data flow explicit.
Implement and trace the named DA search and sorting methods in Python.
Connect abstract data-structure operations to Python representations and their costs.
B · GATE 2027 Syllabus Scope
Official syllabus scope and related lesson coverage; topics outside the syllabus are marked as enrichment.
- •Arrays: 1D, 2D, 3D row-major and column-major address translation, triangular and symmetric matrices
- •Stacks: ADT, infix to postfix/prefix conversion, evaluation, recursion simulation, parenthesis balancing
- •Queues: Linear queue, circular queue wrap-around arithmetic, Deque, queue implementation using stacks
- •Linked Lists: Singly, doubly, circular, pointer reversal, Floyd's cycle detection algorithm
- •Trees: Binary trees, strict/complete/full definitions, degree theorems (n0 = n2 + 1), Catalan structural counts, traversals and unique reconstruction
- •Binary Search Trees: Search, insertion, three-case deletion, inorder successor/predecessor, height bounds
- •Self-Balancing Trees: AVL trees, balance factor, LL/RR/LR/RL rotations, minimum nodes recurrence N(h) = N(h-1) + N(h-2) + 1
- •Multi-Way Trees: B-Trees and B+ Trees, node order definition, fanout, leaf chaining, disk block numericals
- •Heaps & Priority Queues: Binary min/max heap array indexing, Max-Heapify, linear-time Build-Heap, Heap Sort
- •Hashing: Hash functions (division, multiplication), chaining, open addressing (linear, quadratic, double hashing), clustering, probe bounds
- •Disjoint Sets: Disjoint Set Union (DSU), union by rank, path compression, inverse Ackermann complexity
Interactive Labs & Verified PYQs
Revision & Rapid Review Sheets
Authoritative Textbooks & Video Lectures
All notes in this notebook cross-reference standard academic literature. We do not pretend student notes substitute for classical texts — they guide you through them.
Curated lectures from verified educators (Gate Smashers, Neso Academy, Knowledge Gate, and MIT OCW) selected specifically for difficult concept visualizations.
These notes are open and maintained together.
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