Shortest Paths

Preview

1. What is the shortest path algorithm and how does it work?
2. How is the shortest path problem defined and what are the common applications?
3. What are the differences between Dijkstra’s and Bellman-Ford’s algorithms for finding the shortest path?
4. How are negative edge weights handled in finding the shortest path?
5. How can we find multiple shortest paths between two nodes in a graph?

Explain

Shortest path is a problem in graph theory that deals with determining the path with the lowest cost or length between two points in a graph. In other words, it is finding the path with the minimum sum of edge weights.

For example, suppose you have a map of a city with various streets connecting different locations, and you want to find the shortest path from your house to your workplace. The city map can be represented as a graph, where each location is a node and each street connecting two locations is an edge. The edges may have a weight that represents the distance in miles, time taken, or the cost of taking that street. To find the shortest path, we need to determine the path with the minimum sum of edge weights from the starting node (your house) to the ending node (your workplace).

There are different algorithms to solve the shortest path problem, such as Dijkstra’s algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm. These algorithms use different strategies to find the shortest path from one node to another in an efficient way.

Keypoint

• Shortest paths refer to the shortest distances between vertices in a graph.
• The distance between two vertices is the sum of the weights of the edges on the path connecting them.
• Shortest paths can be found using algorithms such as Dijkstra’s algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm.
• Dijkstra’s algorithm is a greedy algorithm that iteratively selects the vertex with the lowest distance from the source and updates the distances of its neighbors.
• Bellman-Ford algorithm can handle negative edge weights and detects negative cycles but has a time complexity of O(VE).
• Floyd-Warshall algorithm computes the shortest paths between all pairs of vertices in O(V^3) time.
• The use of priority queues can improve the efficiency of shortest path algorithms.

Review

1. What is the shortest path algorithm used to find the shortest path between two nodes in an unweighted, undirected graph?
   Answer: The Breadth-First Search algorithm (BFS) can be used to find the shortest path between two nodes in an unweighted, undirected graph.

2. In Dijkstra’s algorithm, the priority queue is initialized with the source node set to zero, and all other nodes set to what value?
   Answer: In Dijkstra’s algorithm, the priority queue is initialized with the source node set to zero, and all other nodes are set to infinity.

3. What is Bellman-Ford algorithm used for?
   Answer: The Bellman-Ford algorithm is used for finding the shortest paths from a single source vertex to all other vertices in a directed, weighted graph.

4. Which algorithm is used to compute shortest paths in a network where there are negative edge weights?
   Answer: The Bellman-Ford algorithm can be used to compute shortest paths in a network where there are negative edge weights.

5. What is a drawback of Floyd-Warshall algorithm?
   Answer: A drawback of the Floyd-Warshall algorithm is that it requires O(n^3) time complexity, which can be too computationally expensive for very large graphs.