Lecture Notes on Data Structures and Algorithms

Introduction

• Importance of understanding core concepts in data structures
• Definition of Data Structures:
  • Ways of organizing data for efficient use.

Importance of Data Structures

• Essential for creating fast and powerful algorithms.
• Helps manage and organize data naturally.
• Clean code and understanding appropriate usage of data structures differentiate excellent programmers.

Abstract Data Types (ADT)

• Distinction between data structures and abstract data types.
• Abstract Data Type (ADT): Provides an interface without details of implementation.
• Example: Modes of transportation as an analogy for ADTs.

Computational Complexity

• Common questions:
  • How much time does the algorithm need to finish?
  • How much space does the algorithm require?
• Introduction to Big O notation:
  • Focuses on worst-case scenarios.
  • Constant time O(1), Logarithmic O(log n), Linear O(n), Quadratic O(n²), Exponential O(2ⁿ).
  • Notation Validity: Only concerned with large inputs.

Big O Properties

• Ignore constant factors for large inputs.
• Example function analysis:
  • f(n) = 7 log(n³) + 15n² + 2n + 8, results in O(n³).

Examples of Complexity

• O(1): Direct computation.
• O(n): Looping through all n elements.
• O(n²): Nested loops.
• O(log n): Efficient search algorithms like binary search.

Arrays

• The most commonly used data structure.
• Static Array: Fixed length container, contiguous memory chunks.
• Use cases:
  • Storing temporary objects.
  • Buffers for input/output streams.
  • Lookup tables.

Dynamic Arrays

• Can grow and shrink in size, common operations performed.
• Complexity analysis:
  • Access: O(1), Search: O(n), Insert/Delete: O(n).

Singly and Doubly Linked Lists

• Singly Linked List: Each node points to the next.
• Doubly Linked List: Each node points to the next and previous.
• Use cases and complexity analysis for linked lists.

Stacks

• LIFO (Last In First Out) structure.
• Key operations: Push, Pop, Peek.
• Applications in recursion and graph algorithms.

Queues

• FIFO (First In First Out) structure.
• Key operations: Enqueue, Dequeue, Peek.
• Applications in scheduling and breadth-first search.

Priority Queues

• Handle elements with specific priorities.
• Implemented with heaps (binary heaps, etc.).

Hash Tables

• Key-value pairs with a hash function.
• Collision resolution techniques:
  • Separate Chaining: Store entries in linked lists.
  • Open Addressing: Probing for free slots in the table.
  • Linear Probing: Probing next index.
  • Quadratic Probing: Probing with quadratic function.
  • Double Hashing: Using a second hash function for probing.

Construction of Fenwick Tree

• Data structure for range queries and point updates.
• Supports fast prefix sums.
• Constructing efficiently in linear time.

Suffix Arrays and LCP Arrays

• Suffix Array: Stores sorted suffixes of a string.
• LCP Array: Length of longest common prefixes between sorted suffixes.
• Applications in finding unique substrings.

Conclusion

• Understanding various data structures and algorithms is essential for efficient programming.
• Each structure has its own advantages, complexities, and use cases.