This presentation offers an in-depth exploration of advanced hashing techniques, designed specifically for students with a foundational understanding of data structures and algorithms

1. Hashing
(Concept & Techniques)

2. What is Hashing?
• Hashing is a technique that maps data of arbitrary size →
fixed-size value (index) using a hash function.
• Enables fast insertion, search, deletion.

3. Hashing Techniques
A. Open Addressing (In-Table Collisions)
• Linear Probing: h'(key, i) = (h(key) + i) % table_size
• Quadratic Probing: h'(key, i) = (h(key) + i^2) % table_size
• Double Hashing: h'(key, i) = (h1(key) + i*h2(key)) % table_size
B. Chaining
• Each index holds a linked list of colliding keys.
C. Perfect Hashing
• Maps n keys n slots
→ without collisions.
• Ideal for static datasets.

4. Hash Table
• Array-like structure using hash function to store keys.
Features:
Fast operations: O(1) average time
Requires collision handling
Pseudo Code (Insertion using Chaining):

5. Hash Function
Converts key index
→
Properties:
• Fast
• Uniform
• Deterministic
Common Types:
• Division: h(key) = key % table_size
• Multiplication: h(key) =
floor(table_size * (key * A % 1))
• String Hashing: ASCII sum,
polynomial rolling hash

6. Advantages of Hashing Techniques
• Fast Access: O(1) average for search, insert, delete
• Efficient Memory: Optimized storage if table size chosen
well
• Flexible: Can store different types of keys
• Widely Used: Databases, caching, symbol tables
Open Addressing Advantages:
• Simple structure, no extra memory
for linked lists
• Cache-friendly
• Easy to implement
Perfect Hashing Advantages:
• No collisions O(1) search
→
guaranteed
• Ideal for static datasets

7. Disadvantages of Hashing
Techniques
• Collisions require handling (chaining or probing)
• Poor hash function clustering & uneven distribution
→
• Fixed-size table overflow or need for resizing
→
• Cannot maintain sorted order of elements
• Open addressing: performance degrades with high load
factor

8. Applications of Hashing
• Database Indexing – fast retrieval of records
• Symbol Tables – in compilers and interpreters
• Caching – storing frequently accessed data
• Password Storage – hashing + salting
• Cryptography – MD5, SHA algorithms
• Sets & Maps – key-value pairs in programming languages

9. AVL Tree
• Self-balancing Binary Search
Tree (BST)
• Balance Factor (BF) =
height(left) - height(right)
• Rotations: LL, RR, LR, RL
Pseudo Code (Insertion):

11. Red- Black Tree
• Self-balancing BST using colors (Red/Black)
Properties:
Root is black
Red nodes cannot have red children
Every path from root leaf has same black height
→
Node Structure:

12. Rotations
Left Right

13. BST Insertion Red-Black Tree Insertion

14. Fixing Violations After Insertion

15. Binomial Heap
• A Binomial Heap is a collection of Binomial Trees that
satisfies the Min-Heap (or Max-Heap) property.
• Each tree in the heap follows heap-order property (parent
children for min-heap).
≤
• Efficient for mergeable priority queues.
Properties:
oA Binomial Tree of order k has 2^k nodes.
oHeight of tree = k
oNumber of nodes at depth i = C(k, i) (binomial coefficient)

16. Basic Operations
Merge Two Heaps Insert Key
Find Minimum Delete Minimum

17. Fibonacci Heap
• A Fibonacci Heap is a collection of heap-ordered trees with
a relaxed structure, allowing very fast amortized
operations.
• Optimized for algorithms that require lots of decrease-key
operations, like Dijkstra’s algorithm.
Properties:
oEach tree satisfies min-heap property (parent children).
≤
oNodes are linked circularly in a root list
oMaintains pointer to the minimum node for O(1) access.
oLazy consolidation — trees are merged only when
necessary.

18. Basic Operations
Insert Node Find Minimum
Extract Minimum
Decrease Key

19. Array, Structure and Binary List
Array: Contiguous elements, random access O(1)
Structure: Groups multiple data types
Binary List: Efficient 0/1 storage used in sets, bitmap
indexing