Lecture Notes on Data Structures and Algorithms

Introduction to Data Structures

• Definition: A data structure is a way of organizing data so it can be used efficiently.
• Importance:
  • Essential for creating fast algorithms.
  • Helps manage and organize data naturally.
  • Makes code cleaner and easier to understand.

Abstract Data Types

• Abstract Data Type (ADT): An abstraction that provides an interface for a data structure without specifying implementation details.
• Example: Modes of transportation (walking, biking) are like data structures; the abstract type is the concept of transportation.

Computational Complexity

• Key Questions:
  • How much time does an algorithm need to finish?
  • How much space does it require?
• Big O Notation: Used to express the worst-case scenario for time and space complexity.
  • Common Complexities:
    • Constant Time: O(1)
    • Logarithmic Time: O(log n)
    • Linear Time: O(n)
    • Quadratic Time: O(n^2)
    • Exponential: O(2^n)

Data Structures Overview

Arrays

• Static Arrays: Fixed length data containers that are contiguous in memory.
  • Usage: Store objects, buffers, lookup tables, dynamic programming.
• Dynamic Arrays: Can grow or shrink in size, often implemented with static arrays.
  • Complexity:
    • Access: O(1)
    • Search: O(n) worst-case
    • Insert/Delete: O(n) (due to shifting), O(1) for append.

Linked Lists

• Structure: Nodes with pointers, can be singly or doubly linked.
• Usage: Efficient for insertions/deletions compared to arrays.
  • Complexity:
    • Access: O(n)
    • Insert/Delete: O(1) if node reference is known.

Stacks

• Definition: Last In First Out (LIFO) structure.
• Operations: Push, Pop, Peek.
• Complexity: O(1) for all operations.
• Applications: Undo mechanisms, parsing expressions.

Queues

• Definition: First In First Out (FIFO) structure.
• Operations: Enqueue, Dequeue, Peek.
• Complexity: O(1) for all operations.
• Applications: Task scheduling, breadth-first searching.

Hash Tables

• Definition: Maps keys to values using a hash function to ensure efficient access.
• Collision Resolution Techniques:
  • Separate Chaining: Uses linked lists to store multiple items at the same index.
  • Open Addressing: Probes through the table to find an open slot.
• Complexity:
  • Average case: O(1) for insert, delete, search.
  • Worst case: O(n) if many collisions occur.

Trees

• Binary Trees: Each node has at most two children.
  • Binary Search Trees (BST): Nodes are arranged such that left children are smaller, right children are larger.
• Traversals:
  • Pre-order, In-order, Post-order, Level-order.
  • In-order traversal of BST yields sorted order of elements.

Fenwick Trees

• Purpose: Efficient range query and point update operations.
• Operations: Insert, update, query in O(log n) time.
• Construction: Can be built from an array in O(n) time.

Unique Substrings via Suffix Arrays

• Suffix Array: An array of indices that represent sorted suffixes of a string.
• LCP Array: Tracks the longest common prefix of adjacent suffixes.
• Finding Unique Substrings: Use the LCP values to deduce duplicates and count unique substrings.

Union-Find Data Structure

• Purpose: Tracks a set of elements partitioned into disjoint sets.
• Operations: Union and Find.
• Applications: Useful in network connectivity, Kruskal's algorithm for minimum spanning trees.
• Complexity: Amortized O(1) for union and find operations with path compression.

AVL Trees

• Definition: A self-balancing binary search tree where the difference in heights of left and right subtrees is at most one.
• Operations: Insert, delete, search all in O(log n) time.
• Rotations: Used to maintain balance after insertions/deletions.

Indexed Priority Queue

• Purpose: Allows for priority queues with efficient key-value updates and deletions.
• Operations: Insert, remove, update in O(log n) time.
• Applications: Useful in scheduling algorithms, event simulation.

Conclusion

• Understanding and mastering these data structures and their complexities is crucial in computer science, algorithms, and programming.