Dijkstra’s shortest path algorithm | GeeksforGeeks

Summary

In this video, the Dijkstra's algorithm is explained for finding the shortest path in a graph. The algorithm uses a greedy approach with two sets: one for finalized vertices and another for unprocessed ones. By iteratively selecting the vertex with the smallest distance, the algorithm updates adjacent vertices' distances. The output can either be just the shortest distances or the paths as well. Time complexity is analyzed, showing that with a binary heap, the algorithm runs in O(V + E log V), which can be optimized further using a Fibonacci heap.

Takeaways

• 😀 Dijkstra's algorithm is used to find the shortest path from a source vertex to every other vertex in a graph.
• 😀 The algorithm uses a greedy approach and maintains two sets: one for vertices with finalized shortest paths and one for vertices with paths yet to be finalized.
• 😀 A priority queue (often implemented with a binary heap) is used for efficient extraction of the vertex with the smallest distance at each step.
• 😀 The algorithm starts by initializing distances as infinity, except for the source vertex, which is set to zero.
• 😀 In each iteration, the vertex with the smallest tentative distance is extracted from the priority queue and added to the finalized set.
• 😀 For each adjacent vertex of the extracted vertex, the algorithm compares the current known distance with the distance through the current vertex and updates it if shorter.
• 😀 The algorithm can output just the distances or both the distances and the shortest path using a parent array.
• 😀 The time complexity of Dijkstra's algorithm is O(V + E log V), where V is the number of vertices and E is the number of edges, when using a binary heap.
• 😀 Optimizing Dijkstra's algorithm with a Fibonacci heap reduces the time complexity to O(V log V + E), improving performance for dense graphs.
• 😀 The algorithm's performance depends on how the graph is represented; using an adjacency list is more efficient than using an adjacency matrix.
• 😀 The Dijkstra algorithm is useful for finding the shortest path in both directed and undirected graphs, with the number of adjacent vertices (edges) influencing the complexity.

Q & A

What is the Dijkstra's shortest path algorithm?

Dijkstra's algorithm is a greedy algorithm used to find the shortest paths from a source vertex to all other vertices in a graph. It uses a priority queue to iteratively update distances and finalize the shortest paths.

What are the two main sets in Dijkstra's algorithm?

The two main sets in Dijkstra's algorithm are: 1) The set of vertices for which the shortest path has been finalized. 2) The set of vertices for which the shortest path is yet to be finalized.

How does Dijkstra's algorithm update the distances?

The algorithm iterates through the vertices, extracting the vertex with the smallest distance from the priority queue, and then updates the distances of its adjacent vertices if a shorter path is found.

What role does the priority queue play in Dijkstra's algorithm?

The priority queue is used to efficiently extract the vertex with the minimum distance and to update the distances of adjacent vertices. It supports operations like extract minimum and decrease key, which are essential for the algorithm's performance.

Why do we need to maintain a parent array in Dijkstra's algorithm?

The parent array keeps track of the predecessor of each vertex in the shortest path, allowing us to reconstruct the exact path from the source vertex to any other vertex.

How does the algorithm determine the shortest path from the source to a given vertex?

To find the shortest path, the algorithm traces back from the target vertex to the source using the parent array, following the parent pointers until reaching the source.

What is the time complexity of Dijkstra's algorithm when using a binary heap?

The time complexity of Dijkstra's algorithm using a binary heap is O((V + E) log V), where V is the number of vertices and E is the number of edges.

How can the time complexity of Dijkstra's algorithm be optimized?

The time complexity can be optimized by using a Fibonacci heap, which allows the decrease key operation to be performed in constant time, reducing the overall complexity to O(V log V + E).

What happens if the graph is not represented using an adjacency list?

If the graph is not represented using an adjacency list, the time complexity may increase because the algorithm would not be able to efficiently access all adjacent vertices, leading to a potentially quadratic time complexity.

What is the difference between an undirected and a directed graph in terms of Dijkstra's algorithm?

In Dijkstra's algorithm, an undirected graph will have two edges for every pair of connected vertices (one in each direction), while a directed graph has one edge per connection. The number of adjacent vertices (E) will affect the algorithm's performance, especially in terms of time complexity.