Algorithms and Data Structures: A Comprehensive Cheat Sheet
In the realm of programming, understanding algorithms and data structures is pivotal for solving complex problems efficiently. This cheat sheet delves into crucial topics such as Big O notation, space complexity, and various data structures like trees, graphs, and heaps. Let’s dive into these concepts with examples.
Big O Notation
Big O notation describes the time complexity of an algorithm as the input size grows. It provides a high-level understanding of the algorithm's efficiency.
?? O(1) – Constant Time
Example: Accessing an element in an array by index. Regardless of the array size, accessing an element directly by index takes the same amount of time.
?? O(log n) – Logarithmic Time
Example: Binary search in a sorted array. The time to find an element grows logarithmically as the array size increases.
?? O(n) – Linear Time
Example: Linear search in an array. The time taken grows directly proportional to the number of elements in the array.
?? O(n log n) – Log-Linear Time
Example: Merge sort. This sorting algorithm divides the array into halves, sorts each half, and merges them, with time complexity growing proportionally to n log n.
?? O(n2) – Quadratic Time
Example: Bubble sort. Time complexity increases with the square of the input size, as each element is compared with every other element.
Common Data Structures and Their Big O Complexities
Arrays
?? Access: O(1) – Direct access by index.
?? Search: O(n) – Requires iterating through elements.
?? Insertion (at the end): O(1) – Adding an element at the end is constant time.
?? Deletion (at the end): O(1) – Removing the last element is constant time.
Objects
?? Insertion: O(1) – Adding a new key-value pair.
?? Deletion: O(1) – Removing a key-value pair.
?? Access: O(1) – Retrieving the value associated with a key.
?? Search: O(n) – Finding a key requires checking each key-value pair.
Space Complexity
Space complexity measures the amount of memory an algorithm needs relative to the input size.
?? O(1) – Constant Space
Example: An algorithm that uses a fixed amount of memory regardless of the input size.
?? O(n) – Linear Space
Example: An algorithm that stores intermediate results in an array proportional to the input size.
Common Problem-Solving Patterns
Frequency Counter
?? Description: Uses an object or hash table to count occurrences of elements.
Example: Checking if two strings are anagrams by counting the frequency of each character.
Multiple Pointers
?? Description: Employs multiple pointers to solve problems involving subsets.
Example: Two-sum problem, where you use two pointers to find pairs that sum up to a target value.
Sliding Window
?? Description: Maintains a window that moves across a data structure to track subsets.
Example: Finding the maximum sum of a contiguous subarray.
Divide-and-Conquer
?? Description: Divides a problem into smaller subproblems, solves each, and combines the results.
Example: Merge sort, which splits the array into halves, sorts each, and merges them.
Recursion
?? Description: Solves a problem by calling the same function within itself until a base case is met.
Example: Calculating the Fibonacci sequence, where each term is the sum of the two preceding ones.
Understanding the Call Stack
The call stack manages function calls in programming. Each function call is pushed onto the stack, and once the function completes, it is popped off. Deep recursion can lead to a large call stack.
Example: In a recursive function to calculate factorial, each function call adds a new frame to the call stack.
Searching Algorithms
Linear Search
?? Time Complexity: O(n)
Description: Iterates over each element to find a target value.
Example: Searching for a value in an unsorted array by checking each element one by one.
Sorting Algorithms
Bubble Sort
?? Time Complexity: O(n2)
Description: Repeatedly swaps adjacent elements if they are in the wrong order.
Example: Sorting a list of numbers by repeatedly bubbling the largest unsorted element to its correct position.
Selection Sort
?? Time Complexity: O(n2)
Description: Repeatedly finds the minimum element from the unsorted portion and swaps it with the first unsorted element.
Example: Sorting by selecting the smallest remaining element and moving it to the beginning.
Insertion Sort
?? Time Complexity: O(n2)
Description: Builds the sorted array one element at a time by inserting elements in their correct position.
Example: Sorting a hand of playing cards by inserting each card into its correct position.
Merge Sort
?? Time Complexity: O(n log n)
Description: Divides the array into halves, sorts each half, and merges them.
Example: Sorting an array by recursively dividing it into halves, sorting each half, and merging them.
Quick Sort
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?? Time Complexity: O(n log n)
Description: Picks a pivot element, partitions the array around the pivot, and sorts the partitions.
Example: Sorting an array by choosing a pivot and recursively sorting the elements around it.
Radix Sort
?? Time Complexity: O(nk)
Description: Sorts numbers by their digits, starting from the least significant digit.
Example: Sorting a list of numbers by processing each digit, starting from the least significant to the most significant.
Data Structures
Singly Linked List
?? Operations: Insertion, deletion, traversal.
Complexity: O(n) for search, O(1) for insertion/deletion at the beginning.
Doubly Linked List
?? Operations: Supports both forward and backward traversal.
Complexity: Similar to singly linked list, but with O(1) for backward traversal.
Stacks
?? Description: LIFO (Last In, First Out) structure.
Operations: Push (O(1)), Pop (O(1)).
Queues
?? Description: FIFO (First In, First Out) structure.
Operations: Enqueue (O(1)), Dequeue (O(1)).
Tree Data Structures
Binary Search Tree (BST)
?? Description: Each node has at most two children; left child is smaller, right child is larger.
Complexity: O(log n) for insert, delete, search (average case).
Tree Traversal
?? In-Order: Left → Node → Right
?? Pre-Order: Node → Left → Right
?? Post-Order: Left → Right → Node
?? Breadth-First Search (BFS): Level by level traversal.
?? Depth-First Search (DFS): In-depth traversal (pre-order, in-order, post-order).
Binary Heaps
Binary Heaps are complete binary trees used to implement priority queues.
?? Min-Heap: Parent nodes are smaller than their children.
?? Max-Heap: Parent nodes are larger than their children.
Graphs
Graph Terminology
?? Vertices (nodes) and Edges (connections).
?? Adjacency Matrix: 2D array representation of a graph.
?? Adjacency List: A list where each vertex points to a list of its neighbors.
Graph Traversal
?? Depth-First Search (DFS): Explore as far as possible along a branch.
?? Breadth-First Search (BFS): Explore all neighbors level by level.
Advanced Concepts
Dijkstra's Shortest Path Algorithm
?? Description: Finds the shortest path from a source vertex to all other vertices in a graph.
Complexity: O(E log V), where E is the number of edges and V is the number of vertices.
Dynamic Programming (DP)
Memoization
?? Description: A top-down approach where results of expensive function calls are stored and reused.
Tabulation
?? Description: A bottom-up approach where solutions to subproblems are filled in a table.
Example: Calculating the Fibonacci sequence using dynamic programming to avoid redundant calculations.
String and Array Manipulations
String Pattern Matching
?? Description: Searching for a substring within a string using algorithms like KMP (Knuth-Morris-Pratt).
Array Manipulation
?? Description: Operations like reversing, rotating, and finding the maximum or minimum values in an array.
Hash Tables
?? Description: Store key-value pairs with average time complexity O(1) for insert, delete, and access.
?? Collision Handling: Techniques like chaining (linked lists) or open addressing (probing) to resolve collisions.
By mastering these algorithms and data structures, you'll be well-equipped to tackle a wide range of coding challenges with efficiency and effectiveness!
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