Computer Science Theory

Bubble Sort has a worst-case time complexity of O(n^2) when the input is already sorted in reverse order, but it has a best-case time complexity of O(n) when the input is already sorted.

Finding the minimum element in an unsorted array requires examining all elements, resulting in a time complexity of O(n).

A heap is a data structure that maintains a partially ordered tree, allowing for efficient retrieval of the minimum or maximum element in logarithmic time.

In a graph with n vertices and no self-loops, the maximum number of edges is n(n-1) because each vertex can be connected to every other vertex except itself.

Dijkstra's Algorithm is used to find the shortest path between two nodes in a weighted graph by iteratively selecting the node with the smallest distance from the starting node.

Goldbach's Conjecture states that any sufficiently large integer can be expressed as the sum of three primes, but it remains unproven.

The complexity class P represents problems that can be solved by a deterministic Turing machine in polynomial time.

The 3-SAT Problem asks whether a given Boolean formula in conjunctive normal form with three literals per clause is satisfiable.

The Ford-Fulkerson Algorithm is used to find the maximum flow in a network by iteratively finding augmenting paths and increasing the flow along those paths.

Computability Theory studies the limits of computation and the relationship between computation and information.

Depth-First Search can be used to find the connected components of a graph by traversing the graph in a depth-first manner and identifying the connected components as the subgraphs that are reachable from each starting vertex.

Lagrange's Four-Square Theorem states that any sufficiently large integer can be expressed as the sum of four squares.

The complexity class EXP represents problems that can be solved by a deterministic Turing machine in exponential time.

The Hamiltonian Cycle Problem asks whether a given graph contains a Hamiltonian cycle, which is a cycle that visits every vertex exactly once.

Kruskal's Algorithm is used to find the minimum spanning tree of a weighted graph by iteratively adding edges to the tree in ascending order of weight.