Dijkstra's Algorithm

Preview

1. What is Dijkstra’s Algorithm and how does it work?
2. What are the key properties of Dijkstra’s Algorithm and how are they relevant to its usefulness in solving shortest-path problems?
3. Can Dijkstra’s Algorithm handle negative edge weights in a graph? If not, what modifications might be needed to address this situation?
4. What are some common applications of Dijkstra’s Algorithm in real-world scenarios?
5. How does Dijkstra’s Algorithm compare to other shortest-path algorithms, such as the Bellman-Ford algorithm or the Floyd-Warshall algorithm, in terms of efficiency and accuracy?

Explain

Dijkstra’s algorithm is an algorithm for finding the shortest path between two vertices in a weighted graph. It solves the single-source shortest path problem by iteratively expanding the frontier set of vertices until the destination is reached. At each iteration, the algorithm chooses the vertex with the smallest tentative distance from the source and adds it to the frontier set.

Here is an example of how Dijkstra’s algorithm works:

Suppose we have a weighted graph with five vertices and the following edges:

• Vertex A has edges to B (weight 10) and D (weight 5)
• Vertex B has edges to C (weight 1) and D (weight 2)
• Vertex C has an edge to E (weight 4)
• Vertex D has edges to B (weight 3), C (weight 9), and E (weight 2)
• Vertex E has no outgoing edges

We want to find the shortest path from vertex A to vertex E using Dijkstra’s algorithm.

The algorithm starts by initializing the distance from the source vertex to all other vertices as infinite except for the source vertex, which has a distance of zero. We also create a set of visited vertices that is initially empty.

1. Set the tentative distance to A to be 0 and add A to the visited set.
2. Choose the vertex with the smallest tentative distance (in this case A), which is already in the visited set.
3. For each neighbor of A (B and D), calculate the distance from A to the neighbor (the weight of the edge plus the tentative distance to A). If this distance is smaller than the current tentative distance to the neighbor, update the tentative distance to the smaller value.
4. Add B to the visited set, as it has the smallest tentative distance among the neighbors of A.
5. Repeat steps 2-4 for the next vertex in the visited set, which is B.
6. For each neighbor of B (C and D), calculate the distance from A to the neighbor (the weight of the edge plus the tentative distance to B). If this distance is smaller than the current tentative distance to the neighbor, update the tentative distance to the smaller value.
7. Add D to the visited set, as it has the smallest tentative distance among the neighbors of B.
8. Repeat steps 2-4 for the next vertex in the visited set, which is D.
9. For each neighbor of D (B, C, and E), calculate the distance from A to the neighbor (the weight of the edge plus the tentative distance to D). If this distance is smaller than the current tentative distance to the neighbor, update the tentative distance to the smaller value.
10. Add E to the visited set, as it has the smallest tentative distance among the neighbors of D.
11. The algorithm terminates when the destination vertex (E) is added to the visited set. The shortest path from A to E is the sequence of vertices that follow the tentative distance values backwards from E to A.

In this case, the shortest path from A to E is A -> D -> E with a total distance of 7.

Keypoint

1. Dijkstra’s algorithm is a graph search algorithm that solves the single-source shortest path problem.

2. It is used to find the shortest path between a starting node and all other nodes in the graph.

3. The algorithm maintains a set of visited nodes and a set of unvisited nodes.

4. Initially, the distance of the starting node is set to zero, and the distances of all other nodes are set to infinity.

5. The algorithm then explores the neighboring nodes of the starting node, updating their distances if a shorter path is found.

6. The algorithm then selects the unvisited node with the shortest distance and repeats the process.

7. The algorithm terminates when all nodes have been visited or when the destination node is reached.

8. The algorithm also maintains a predecessor array that stores the previous node in the shortest path from the starting node.

9. Once the algorithm is complete, the shortest path can be traced back from the destination node by following the predecessor array.

10. Dijkstra’s algorithm works on non-negative weighted graphs and has a time complexity of O(|E|+|V|log|V|) using a min-heap to keep track of the minimum distance.

Review

1. What is Dijkstra’s algorithm?
   Answer: Dijkstra’s algorithm is a shortest path algorithm that finds the shortest path between a source node and all other nodes in a weighted graph.

2. What is the time complexity of Dijkstra’s algorithm?
   Answer: The time complexity of Dijkstra’s algorithm is O(E log V), where E is the number of edges and V is the number of vertices in the graph.

3. How does Dijkstra’s algorithm handle negative weight edges in a graph?
   Answer: Dijkstra’s algorithm does not work with negative weight edges. It assumes that all edges have non-negative weights. However, there are modified versions of Dijkstra’s algorithm that can handle negative weight edges, such as Bellman-Ford algorithm.

4. What is the main difference between Dijkstra’s algorithm and A* algorithm?
   Answer: Dijkstra’s algorithm finds the shortest path between a source node and all other nodes in the graph, while A* algorithm uses heuristics to find the shortest path between a source node and a target node in the graph.

5. Can Dijkstra’s algorithm find the shortest path in a directed graph?
   Answer: Yes, Dijkstra’s algorithm can find the shortest path in a directed graph. However, the shortest path may not be unique if there are multiple paths with the same length.