Algorithm Analysis Fundamentals

A way to express the growth rate of an algorithm as the input size increases. It focuses on the dominant term and ignores constant factors and lower-order terms. Examples include O(n), O(n^2), O(log n).

Describes the best possible performance of an algorithm for any given input size. It's the minimum time required to complete the task.

Describes the worst possible performance of an algorithm for any given input size. It's the maximum time required to complete the task.

Describes the average performance of an algorithm for a typical input size. It takes into account all possible inputs and their probabilities.

A method used to measure how much memory an algorithm needs during its execution, based on the input size.

Used to compare algorithms performing the same task by analyzing their time and space complexities. Helps in choosing the algorithm best suited for a particular application.

A way to express the time complexity of an algorithm using the Big-O notation. It tells you how the runtime grows as the input size increases.

The process of breaking down an algorithm into smaller steps and analyzing the time complexity of each step. This helps you understand how the algorithm's complexity is affected by the input size.

A binary tree where each node's value is greater than or equal to its children, ensuring the largest element is at the root.

A binary tree where each node's value is less than or equal to its children, ensuring the smallest element is at the root.

An operation that restores the heap property after an insertion or deletion. It ensures the heap structure remains valid.

Data structures that prioritize elements, allowing efficient retrieval of the maximum or minimum value.

A data structure that supports adding elements (insertion) and removing the maximum element (extraction).

A data structure that supports adding elements (insertion) and removing the minimum element (extraction).

A highly efficient sorting algorithm that uses a binary heap to sort an array. It leverages the heap property for efficient retrieval of the smallest/largest elements.

An algorithm that finds the shortest paths from a source node to all other nodes in a graph. It uses a min-heap to efficiently select the next node to explore from the priority queue of unexplored nodes.

A data structure representing a set of vertices (nodes) and edges connecting them. It's used to model relationships between objects.

A point or node in a graph. It represents an entity or object in the network.

A line connecting two vertices in a graph. It represents a relationship or connection between objects.

The number of edges connected to a particular vertex in a graph.

A sequence of vertices connected by edges in a graph, starting and ending at a specified vertex.

A representation of a graph using a 2D array where rows and columns represent vertices and the values indicate the presence or absence of an edge between them.

A representation of a graph using a list where each vertex has a list of its adjacent vertices.

A graph where edges have a direction, indicating a one-way connection between vertices.

A problem-solving technique where a problem is broken down into smaller, similar subproblems, solved independently, and then combined to get the final solution. Key characteristics include dividing the problem into smaller subproblems, recursively solving the subproblems, and combining the solutions to solve the original problem.

An efficient sorting algorithm that works by recursively dividing the unsorted array into two halves, sorting each half, and then merging the sorted halves back together.

A recursive algorithm that partitions an unsorted array around a pivot element and then recursively sorts the subarrays. Its recurrence relation is T(n) = T(k) + T(n-k-1) + O(n), where k is the position of the pivot element.

An algorithm for matrix multiplication that divides the matrices into smaller submatrices, recursively multiplies the submatrices, and then combines the results. It achieves a time complexity of O(n^log2(7)) compared to the standard O(n^3) complexity.

A problem in computational geometry that aims to find the smallest convex polygon that encloses a given set of points. It's significant because it finds applications in pattern recognition, image processing, and computer graphics.

A problem-solving technique where the optimal solution is built step-by-step, making locally optimal choices at each stage without reconsidering past decisions.

Given a set of jobs with deadlines and profits, the objective is to find the subset of jobs that maximizes the total profit while meeting the deadlines.

A spanning tree of a graph with the minimum possible total edge weight.

A graph problem where the goal is to find the shortest paths from a single source vertex to all other vertices in the graph.

The principle that states that making the best choice locally at each step will lead to the overall optimal solution.

A problem where items have both weights and values. The goal is to maximize the total value of items selected without exceeding the weight limit.

A variant of the knapsack problem where we can select fractions of items.

A variant of the knapsack problem where we can only select items entirely.

A problem where you need to find the shortest possible route that visits every city exactly once and returns to the starting city. It's a famous optimization problem with wide applications in logistics, transportation, and circuit board design.

An optimization problem that deals with finding the largest complete subgraph (a clique) within a given graph. It's applicable to identifying groups of interconnected individuals in social networks, finding dense clusters in data sets, and analyzing communication patterns.

A problem in graph theory that asks for the minimum number of colors needed to color the vertices of a graph such that no two adjacent vertices have the same color. It relates to scheduling tasks on resources, allocating frequencies in wireless communication networks, and assigning registers in compilers.

A class of problems where finding the optimal solution is computationally very expensive, and it becomes exponentially harder as the problem size increases. For example, determining if a graph has a clique of a given size is NP-hard because checking all possible subsets of vertices is time-consuming.

A method of solving NP-hard problems by finding approximate solutions that are close to the optimal solution within a reasonable time frame. They allow practical solutions for problems that are too complex to solve exactly.

A technique to solve NP-hard problems by simplifying the problem into smaller subproblems that can be solved more efficiently. Then, the solutions to the subproblems are combined to obtain an approximate solution for the original problem.

A class of problems that can be verified in polynomial time, meaning that if you are given a potential solution, you can check if it's correct in a reasonable time. However, finding a solution itself can be computationally very expensive.

A process of converting one NP-complete problem into another NP-complete problem. If a problem can be reduced to another NP-complete problem, then it is itself NP-complete, confirming its computational difficulty.