You are given a directed graph with (possibly negative)...

You are given a directed graph with (possibly negative) weighted edges, in which the shortest path between any two vertices is guaranteed to have at most $k$ edges. Give an algorithm that finds the shortest path between two vertices $u$ and $v$ in $O(k|E|)$ time.

Key Concepts

Edge Relaxation

Edge relaxation is a key operation in many shortest path algorithms, in which the algorithm attempts to improve the current best-known distance to a vertex by considering the distances from its neighboring vertices plus the weight of the edge connecting them. This iterative process helps in gradually converging to the true shortest paths.

Time Complexity Analysis

Time complexity analysis assesses how the running time of an algorithm scales with the size of its input. In the context of the problem, understanding that the shortest path is guaranteed to have at most k edges allows algorithms like a limited version of Bellman-Ford to run in O(k|E|) time, where |E| is the number of edges, ensuring efficiency under the given constraints.

Bellman-Ford Algorithm

The Bellman-Ford algorithm is designed to find shortest paths from a source vertex to all other vertices in a graph that may include negative edge weights. It operates by iteratively relaxing edges, and its structure allows it to handle cases where paths are constrained by a number of edges, making it a natural fit for problems in which the maximum path length is known.

Shortest Path Problem

The shortest path problem involves finding the minimum-cost route between two vertices in a weighted graph. Different algorithms address this problem under various conditions, such as the presence of negative weights or specific restrictions on the number of edges that can be used.

Weighted Edges

Weighted edges in a graph associate a numerical value to each edge, representing factors such as cost, distance, or time. When weights can be negative, it introduces additional complexity as typical greedy approaches may fail, and algorithms must carefully handle potential issues like negative cycles.

Directed Graph

A directed graph is a type of graph where each edge has an orientation, meaning it goes from one vertex (or node) to another in a specified direction. This structure is essential in many applications because it models relationships that are not necessarily reciprocal, and it dictates that the order in which vertices are visited matters when finding paths.

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