The Floyd-Warshall Algorithm is a dynamic programming technique used to find the shortest paths between all pairs of vertices in a weighted graph, accommodating both positive and negative edge weights, but not negative cycles. It operates with a time complexity of O(n^3), making it suitable for dense graphs with a relatively small number of vertices.

Floyd-Warshall Algorithm

dynamic programming technique

shortest paths

all pairs of vertices

A weighted graph is a type of graph in which each edge has an associated numerical value, called a weight, which often represents costs, distances, or capacities. These weights are crucial for algorithms that find optimal paths, such as Dijkstra's or Prim's, which are used in network routing, resource allocation, and various optimization problems.

weighted graph

positive and negative edge weights

Negative cycles in a graph refer to cycles whose edges sum to a negative value, causing algorithms like Bellman-Ford to detect the possibility of infinitely reducing the path cost. These cycles are critical in optimization and network flow problems, as they indicate potential for arbitrage or inconsistencies in the system being modeled.

negative cycles

Time complexity is a computational concept that measures the amount of time an algorithm takes to complete as a function of the length of the input. It provides a high-level understanding of the efficiency of an algorithm, helping in the comparison and selection of algorithms based on performance criteria.

time complexity

O(n^3)

A dense graph is characterized by having a large number of edges in relation to its vertices, often approximating the maximum number of edges possible. This type of graph can prove challenging to handle computationally due to its potential complexity and large data size, making efficient algorithms crucial for analysis.

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