Essential Data Structures & Algorithms: A Comprehensive Guide
Data Structure Fundamentals
Deque vs. Circular Queue
A deque (double-ended queue) offers efficient insertion and deletion from both ends, making it suitable for scenarios like sliding window problems. Circular queues, on the other hand, restrict operations to specific ends (rear for enqueue, front for dequeue).
Big O Notation
Big O notation describes an algorithm’s upper bound running time, classifying algorithms by their worst-case behavior. This allows for time complexity comparisons irrespective of hardware or implementation.
Shortest Path vs. Minimum Spanning Tree
Shortest path algorithms find the minimum distance between two nodes, while minimum spanning trees (MSTs) connect all nodes with minimal total edge weight, avoiding cycles.
QuickSort vs. MergeSort
QuickSort is generally faster due to in-place sorting and cache performance. However, poor pivot selection can degrade performance to O(n²). MergeSort, while stable and predictable at O(n log n), requires extra space.
Big Theta Notation
Big Theta (Θ) provides an exact bound, representing both upper and lower limits of an algorithm’s time complexity, offering a more precise analysis than Big O.
Advantages of Postfix Notation
Postfix notation simplifies expression evaluation using a stack by eliminating parentheses and following a left-to-right approach, bypassing operator precedence concerns.
Choosing Data Structures
Key factors include time and space complexity, access patterns (random/sequential), mutability, and specific needs like ordered storage or dynamic resizing.
Asymptotic Notation
Asymptotic notations describe algorithm running time relative to input size:
- Big O (O): Upper bound (worst-case).
- Big Omega (Ω): Lower bound (best-case).
- Big Theta (Θ): Exact bound (when upper and lower bounds match).
Example: Linear search has a Big O of O(n).
Recursion
Recursion involves a function calling itself to solve sub-problems. It requires a base case to prevent infinite loops.
Example: Factorial calculation (n! = n × (n-1)!).
Linked Lists
Linked lists store nodes with data and pointers to the next node. They offer dynamic sizing and efficient insertions/deletions but lack random access and have memory overhead.
Hash Tables
Hash tables store key-value pairs using hash functions. Collisions are resolved through chaining or open addressing.
Insertion Sort
Insertion sort iteratively inserts elements from the unsorted part into the correct position within the sorted part.
Graph Representation
Adjacency List
Each vertex has a list of connected nodes. Efficient for sparse graphs.
Adjacency Matrix
A 2D array indicates connections between vertices. Efficient for dense graphs.
Stack Operations
A stack follows LIFO (Last In, First Out). Push adds an element to the top, and pop removes the top element.
Binary Search Trees (BSTs)
BSTs maintain sorted order. Insertion and deletion follow specific rules to preserve this order.
Breadth-First Search (BFS)
BFS explores graphs level by level using a queue. Time complexity: O(V + E), Space complexity: O(V).
Stack and Queue Applications
Stacks are used for function calls, undo operations, and expression evaluation. Queues manage printer queues, task scheduling, and customer service systems.
Infix to Postfix Conversion
Infix expression K+L-MN+(O^P)W/U/VT+Q converts to postfix: KL MN * OP ^ WU / V * T + Q.
Graph Terminology
- Closed Path: A sequence of vertices starting and ending at the same vertex, without repeating edges.
- Strongly Connected Graph: A directed graph where every vertex is reachable from every other vertex.
- Cycle: A path starting and ending at the same vertex with distinct edges and vertices (except start/end).
- Adjacency Matrix: A 2D array representing graph connections.
Quick Sort
Quick Sort involves partitioning an array around a pivot and recursively sorting sub-arrays.
Graph Traversals and Algorithms
- DFS: Explores depth-first.
- BFS: Explores breadth-first.
- Prim’s Algorithm: Finds a minimum spanning tree.
- Dijkstra’s Algorithm: Finds the shortest path between nodes.