mathclasstutor.com

An algorithm is a well-defined procedure or set of rules designed to solve a specific problem or perform a computation. Here are the basics:
Definition and Characteristics of an Algorithm
Finite Sequence: An algorithm must have a finite number of steps.
Well-Defined Instructions: Each step must be precisely stated and unambiguous.
Input: An algorithm may have zero or more inputs.
Output: An algorithm must produce at least one output.
Effectiveness: Each step of an algorithm must be basic enough to be performed, ideally, in a finite amount of time.
Key Properties of Algorithms
Correctness: An algorithm should correctly solve the problem for all valid inputs.
Efficiency: The algorithm should make optimal use of resources like time and space.
Finiteness: It should terminate after a finite number of steps.
Generality: It should work for a wide range of inputs.
Algorithm Design Techniques
Divide and Conquer: Breaking a problem into smaller sub-problems, solving each sub-problem individually, and combining their solutions.
Dynamic Programming: Solving problems by breaking them down into simpler subproblems and storing the results to avoid redundant calculations.
Greedy Algorithms: Making a series of choices, each of which looks the best at the moment.
Backtracking: Trying out different solutions and discarding those that do not meet the required conditions.
Branch and Bound: Systematically considering and discarding branches of a tree representing all possible solutions.
Algorithm Complexity
Time Complexity: The amount of time an algorithm takes to complete as a function of the length of the input.
Big O Notation (O): Describes the upper bound of the time complexity (worst-case scenario).
Big Omega Notation (Ω): Describes the lower bound (best-case scenario).
Big Theta Notation (Θ): Describes the exact bound (average-case scenario).
Space Complexity: The amount of memory an algorithm uses relative to the input size.
Common Algorithm Examples
Sorting Algorithms: Methods for arranging data in a particular order (e.g., Bubble Sort, Merge Sort, Quick Sort).
Searching Algorithms: Techniques for finding specific data within a dataset (e.g., Linear Search, Binary Search).
Graph Algorithms: Solutions for problems related to graph structures (e.g., Dijkstra’s Algorithm for shortest path, Depth-First Search, Breadth-First Search).
String Algorithms: Methods for processing and manipulating strings (e.g., Knuth-Morris-Pratt algorithm for pattern matching).
Pseudocode
A high-level description of an algorithm that uses the structural conventions of programming languages but is intended for human reading rather than machine reading.