Graph Theory Questions Long
In Graph Theory, the term 'shortest path' refers to the path between two vertices in a graph that has the minimum total weight or cost. This weight or cost can be defined based on various factors, such as the distance between vertices, the time taken to travel between them, or any other relevant metric.
Dijkstra's algorithm, named after its inventor Edsger W. Dijkstra, is a popular algorithm used to find the shortest path between two vertices in a weighted graph. It is particularly efficient for finding the shortest path in graphs with non-negative edge weights.
The algorithm starts by assigning a tentative distance value to every vertex in the graph, with the source vertex having a distance of 0 and all other vertices having a distance of infinity. It then iteratively selects the vertex with the smallest tentative distance and updates the distances of its neighboring vertices. This process continues until the algorithm has visited all vertices or until the destination vertex is reached.
During each iteration, Dijkstra's algorithm selects the vertex with the smallest tentative distance and explores its neighboring vertices. If the distance to a neighboring vertex can be improved by going through the current vertex, the algorithm updates the tentative distance of that neighboring vertex. This is done by comparing the sum of the current vertex's distance and the weight of the edge connecting it to the neighboring vertex with the neighboring vertex's current tentative distance. If the sum is smaller, the tentative distance is updated.
The algorithm continues this process until all vertices have been visited or until the destination vertex is reached. Once the algorithm terminates, the shortest path from the source vertex to any other vertex can be determined by backtracking from the destination vertex using the recorded distances.
Dijkstra's algorithm guarantees to find the shortest path in a graph with non-negative edge weights. However, it may not produce correct results if the graph contains negative edge weights or cycles. In such cases, alternative algorithms like Bellman-Ford or Floyd-Warshall should be used.