8 Questions
Which of the following is the time complexity for finding the minimum spanning tree (MST) in a sparse graph?
O(E Log V)
When using an adjacency matrix representation for graphs, what is the space complexity?
O(V²)
Why does Dijkstra's algorithm require non-negative edge weights?
To guarantee that the shortest paths are correctly identified
Which of the following best describes the approach of Dijkstra's algorithm?
Expands the set of visited vertices and updates their tentative distances
In a dense graph, which of the following is the better representation for graph construction?
Adjacency Matrix
What is the primary reason for directly adding vertices to the MST while using an adjacency matrix?
To avoid cycle formation
Which of the following is not handled by Dijkstra's algorithm?
Negative edge weights
For which type of graphs is an adjacency matrix more suitable compared to other representations?
Dense graphs
Study Notes
Design and Analysis of Algorithms
- Adjacency Matrix vs Sparse Graphs: Two different approaches for designing and analyzing algorithms, each suitable for specific graph types.
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Adjacency Matrix:
- Time complexity: O(V²)
- Space complexity: O(V²)
- Suitable for dense graphs
- Builds MST by continuously adding vertices to the growing MST
- Cycle handling: Does not allow cycles since it adds vertices directly to the MST
-
Sparse Graphs:
- Time complexity: O(Log V)
- Space complexity: O(E + V)
- Suitable for sparse graphs
- Builds MST by continuously adding edges to the MST
- Cycle handling: Checks for cycles before adding an edge to the MST
Dijkstra's Algorithm (Single Source Shortest Path Algorithm)
- Overview: Finds the shortest path from a single source vertex to all other vertices in a weighted, directed or undirected graph with non-negative edge weights.
-
Key Properties:
- Single source shortest path problem
- Finds the shortest path from a given vertex to all remaining vertices
- Each edge has a non-negative cost
- Length of the path is the sum of edge costs
-
Working Process:
- Iteratively expands the set of visited vertices and updates their tentative distances
- Finds the shortest path to all vertices
-
Example:
- A motorist finding the shortest path from their place to another place among multiple routes
- Graph Example: A square with 4 vertices, multiple edges with weights, and the algorithm finds the shortest path between vertices.
Discover the key differences between adjacency matrix and sparse graphs in terms of time and space complexity, use cases, graph construction, and cycle handling
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