Time and Space Complexity of Dijkstra’s Algorithm

The time complexity of Dijkstra’s Algorithm is typically O(V²) when using a simple array implementation or O((V + E) log V) with a priority queue, where V represents the number of vertices and E represents the number of edges in the graph. The space complexity of the algorithm is O(V) for storing the distances and predecessors for each node, along with additional space for data structures like priority queues or arrays.

Aspect	Complexity
Time Complexity	O((V + E) log V)
Space Complexity	O(V)

Let’s explore the detailed time and space complexity of the Dijkstra’s Algorithm:

Time Complexity of Dijkstra’s Algorithm:

Best Case Time Complexity: O((V + E) log V)

• This best-case scenario occurs when using an optimized data structure like a Fibonacci heap for implementing the priority queue.
• The time complexity is determined by the graph’s number of vertices (V) and edges (E).
• In this scenario, the algorithm efficiently finds the shortest paths, with the priority queue operations optimized, leading to the overall time complexity of O((V + E) log V).
• This scenario is typically encountered when the graph is sparse, meaning it has relatively few edges compared to vertices.

Average Case Time Complexity: O((V + E) log V)

• The average-case time complexity of Dijkstra’s algorithm is typically the same as the best-case scenario, O((V + E) log V).
• This is because Dijkstra’s algorithm performs well on most real-world graphs, which are often neither extremely sparse nor fully connected.
• The algorithm efficiently finds shortest paths in graphs with varying densities, finding a balance between the quantity of edges and vertices.
• In practice, this average complexity is encountered in a wide range of scenarios, making Dijkstra’s algorithm a reliable choice for many shortest path problems.

Worst Case Time Complexity: O((V²) log V)

• In the worst-case scenario, Dijkstra’s algorithm operates less efficiently, typically when using a simple priority queue or an array-based implementation.
• This occurs when the graph is dense, with many edges, and the priority queue operations become less efficient due to the lack of optimization.
• The time complexity in this case is determined by the number of vertices squared (V²) and logarithmic factors related to priority queue operations, resulting in O((V²) log V).
• The worst-case scenario often arises in fully connected graphs or graphs with many edges between each pair of vertices.

Auxiliary Space Complexity of Dijkstra’s Algorithm:

The auxiliary space complexity of Dijkstra’s algorithm is typically O(V) to O(E + V), where V is the number of vertices and E is the number of edges in the graph, depending on the implementation and data structures used.

The auxiliary space complexity of Dijkstra’s algorithm primarily depends on the data structures used for implementation, particularly the priority queue for managing vertices with their associated distances.