Data Structures: Arrays and Linked Lists

• Data structures are fundamental concepts in computer science
• They are used to organize, manage, and store data in a computer so that it can be accessed and modified efficiently
• Choosing the right data structure for a task can significantly impact the performance of an algorithm

Basic Data Structures

• Arrays are the most basic data structure
• Arrays are a contiguous block of memory locations
• Arrays store elements of the same data type
• Elements in an array are accessed using their index (position)
• Arrays have a fixed size, which must be declared in advance
• Accessing an element by index is O(1)
• Inserting or deleting elements in the middle of the array can be O(n) because you might need to shift subsequent elements
• Linked lists consist of nodes
• Each node contains data and a pointer (or link) to the next node in the sequence
• Linked lists do not require contiguous memory allocation
• Linked lists are dynamic in size, and can grow or shrink during runtime
• Singly linked lists are unidirectional
• Doubly linked lists are bidirectional
• Circular linked lists' last node points back to the first
• Insertion and deletion are O(1) if the position is known
• Searching is O(n) because you may need to traverse the list
• Stacks are a LIFO (Last-In, First-Out) data structure
• The last element added to the stack is the first one to be removed
• Operations include Push (add an element to the top), Pop (remove the top element), Peek (view the top element), and IsEmpty
• Stacks can be implemented using arrays or linked lists
• Common applications include expression evaluation, function call management, and undo mechanisms
• Queues are a FIFO (First-In, First-Out) data structure
• The first element added to the queue is the first one to be removed
• Operations include Enqueue (add an element to the rear), Dequeue (remove the element from the front), Peek (view the front element), and IsEmpty
• Queues can be implemented using arrays or linked lists
• Common applications include task scheduling, print queues, and breadth-first search

Advanced Data Structures

• Trees are hierarchical data structures consisting of nodes connected by edges
• Topmost node is called the root, and nodes with no children are called leaves
• Each node has at most two children (left child and right child) in a Binary Tree
• Binary Search Tree (BST) is a binary tree where the value of each node is greater than all values in its left subtree and less than all values in its right subtree
• Operations include insertion, deletion, searching, with efficiency depending on tree balance
• Traversal methods include in-order, pre-order, and post-order
• Used for representing hierarchical relationships, searching, and sorting
• Heaps are specialized tree-based data structures that satisfy the heap property
• In a min-heap, the value of each node is less than or equal to the value of its children
• In a max-heap, the value of each node is greater than or equal to the value of its children
• Heaps are commonly implemented using arrays
• Operations include insert, delete (remove the root), peek (find min/max element)
• Applications include priority queues and heap sort
• Hash tables are data structures that implement an associative array abstract data type, which maps keys to values
• Hash functions are used to compute an index into an array of buckets or slots, from which the desired value can be found
• Collisions happen when two different keys hash to the same index, and must be resolved using techniques like chaining or open addressing
• Operations include insert, delete, search with an average case O(1) if the hash function distributes keys uniformly
• Applications include caching, symbol tables, and database indexing
• Graphs are data structures consisting of nodes (vertices) connected by edges
• Graphs can be directed (edges have a direction) or undirected (edges are bidirectional)
• Representation can be done via an adjacency matrix, or adjacency list
• Traversal can be done using Breadth-first search (BFS), or Depth-first search (DFS)
• Applications include social networks, route finding, and network analysis.

Abstract Data Types (ADTs)

• An ADT is a high-level description of a data structure
• An ADT specifies the operations that can be performed on the data, and the logical properties of those operations, without specifying how the data structure is implemented
• Examples: List, Set, Map

Data Structure Operations

• Insertion means adding a new element to the data structure
• Deletion means removing an element from the data structure
• Searching means finding a specific element in the data structure
• Traversal means visiting each element in the data structure
• Sorting: Arranging the elements in a specific order

Factors Affecting Data Structure Choice

• Data volume is the amount of data to be stored
• Frequency of operations refers to how often insertion, deletion, and search operations are performed
• Memory constraints refers to the amount of available memory
• Performance requirements refers to the required speed of operations
• Ease of implementation refers to the complexity of implementing and maintaining the data structure

Common Data Structure Applications

• Databases for indexing and storing records
• Operating systems for memory management and process scheduling
• Networking for routing and packet processing
• Graphics for representing images and 3D models
• Artificial intelligence for knowledge representation, and search algorithms

Time Complexity

• Time complexity measures the amount of time taken by an algorithm to run as a function of the input size
• Common notations:
  • O(1): Constant time
  • O(log n): Logarithmic time
  • O(n): Linear time
  • O(n log n): Linearithmic time
  • O(n^2): Quadratic time
  • O(2^n): Exponential time
• Understanding time complexity is crucial for selecting the right data structure and algorithm for optimal performance

Space Complexity

• Space complexity measures the amount of memory space used by an algorithm as a function of the input size
• It includes space used by the input data, auxiliary space, and any temporary variables
• Minimizing space complexity is important, especially when dealing with large datasets or limited memory resources

Sorting Algorithms

• Sorting algorithms arrange elements of a list or array in a specific order (e.g., ascending or descending)
• Bubble sort is a simple comparison-based sorting algorithm
• Bubble sort repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order
• Bubble Sort has a time complexity of O(n^2)
• Insertion sort builds the final sorted array one item at a time
• Insertion sort is much less efficient on large lists than more advanced algorithms
• Insertion Sort has a time complexity of O(n^2)
• Merge sort is a divide and conquer algorithm
• Merge sort divides the unsorted list into n sublists, each containing one element
• Merge sort repeatedly merges sublists to produce new sorted sublists until there is only one sublist remaining
• Merge Sort has a time complexity of O(n log n)
• Quick sort is a divide and conquer algorithm
• Quick sort picks an element as a pivot and partitions the given array around the picked pivot
• Quick Sort has a time complexity of average case O(n log n), worst case O(n^2)
• Heap sort is a comparison-based sorting algorithm
• Heap sort first transforms the list into a heap data structure, then repeatedly removes the largest element from the heap and inserts it into the sorted array
• Heap Sort has a time complexity of O(n log n)