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Questions and Answers
What is the runtime of DFS?
What is the runtime of DFS?
- O(log n)
- O(n^2)
- O(n)
- O(n+m) (correct)
What is the runtime of BFS?
What is the runtime of BFS?
- O(n+m) (correct)
- O(n)
- O(log n)
- O(n^2)
What method do you use to check for a removable vertex?
What method do you use to check for a removable vertex?
DFS
What method do you use to check if a graph is bipartite?
What method do you use to check if a graph is bipartite?
What function checks if a path exists?
What function checks if a path exists?
How do you find the shortest cycle in a graph?
How do you find the shortest cycle in a graph?
What method do you use to count connected components?
What method do you use to count connected components?
What method do you use to find the longest connected component?
What method do you use to find the longest connected component?
What method do you use to find the shortest path in an unweighted graph?
What method do you use to find the shortest path in an unweighted graph?
What is the Topological Sort method known as 'Magic'?
What is the Topological Sort method known as 'Magic'?
What is the runtime of Tarjan's Topo Algorithm?
What is the runtime of Tarjan's Topo Algorithm?
What is Kahn's Algorithm used for in Topological Sort?
What is Kahn's Algorithm used for in Topological Sort?
What is the runtime of Kahn's Topo Algorithm?
What is the runtime of Kahn's Topo Algorithm?
What is used to check for time consistency in a graph?
What is used to check for time consistency in a graph?
What is the method to find all Strongly Connected Components?
What is the method to find all Strongly Connected Components?
What is the runtime of Kosaraju's Algorithm?
What is the runtime of Kosaraju's Algorithm?
How do you find a mother vertex using indegree 0?
How do you find a mother vertex using indegree 0?
What algorithm is used for finding the shortest path with weights?
What algorithm is used for finding the shortest path with weights?
What is the runtime of Dijkstra's Algorithm?
What is the runtime of Dijkstra's Algorithm?
What is the runtime of Dijkstra's Algorithm if the graph is connected?
What is the runtime of Dijkstra's Algorithm if the graph is connected?
How can you prove Dijkstra's algorithm?
How can you prove Dijkstra's algorithm?
What is the runtime for extracting min once per vertex in Dijkstra's Runtime?
What is the runtime for extracting min once per vertex in Dijkstra's Runtime?
What is the runtime for building the heap in Dijkstra's algorithm?
What is the runtime for building the heap in Dijkstra's algorithm?
What is the runtime for decreasing the key E times in Dijkstra's algorithm?
What is the runtime for decreasing the key E times in Dijkstra's algorithm?
What method do you use to find the shortest path in a directed acyclic graph (DAG)?
What method do you use to find the shortest path in a directed acyclic graph (DAG)?
What is the method for finding the longest path in a DAG?
What is the method for finding the longest path in a DAG?
What algorithm is similar to Dijkstra's for Minimum Spanning Tree?
What algorithm is similar to Dijkstra's for Minimum Spanning Tree?
What is the runtime of Prim's MST algorithm?
What is the runtime of Prim's MST algorithm?
What method is used for Minimum Spanning Tree by adding edges with Union-Find?
What method is used for Minimum Spanning Tree by adding edges with Union-Find?
What is the runtime of Kruskal's MST algorithm?
What is the runtime of Kruskal's MST algorithm?
What method is used for Minimum Spanning Tree by removing edges?
What method is used for Minimum Spanning Tree by removing edges?
What is the runtime of Reverse-Delete MSTs algorithm?
What is the runtime of Reverse-Delete MSTs algorithm?
What principle is used to prove Kruskal's algorithm?
What principle is used to prove Kruskal's algorithm?
What principle is used to prove Prim's algorithm?
What principle is used to prove Prim's algorithm?
What principle is used to prove Reverse-Delete algorithm?
What principle is used to prove Reverse-Delete algorithm?
How many nodes does the root of rank k have?
How many nodes does the root of rank k have?
How many nodes are there at most of rank k?
How many nodes are there at most of rank k?
What is the runtime for Huffman's algorithm?
What is the runtime for Huffman's algorithm?
What is the runtime for Binary Heap Insert?
What is the runtime for Binary Heap Insert?
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Study Notes
Graph Algorithms Runtime Complexity
- Depth-First Search (DFS) runtime is O(n + m), where n is the number of vertices and m is the number of edges.
- Breadth-First Search (BFS) runtime is also O(n + m).
Graph Characteristics and Checks
- Removal of a vertex can be checked using DFS.
- To determine if a graph is bipartite, BFS is utilized.
Path and Cycle Determination
- The existence of a path can be verified using the HasPath function.
- The shortest cycle can be found by performing BFS from every vertex and checking the last edge.
- BFS is also used to count connected components and identify the longest connected component.
- BFS is optimal for finding the shortest path in unweighted graphs.
Topological Sorting Methods
- Tarjan's Algorithm provides a topological sort using a reverse DFS method.
- Kahn's Algorithm for topological sorting operates on vertices with an indegree of zero.
Algorithm Runtimes
- Both Tarjan's and Kahn's topological sort algorithms run in O(n + m).
- Dijkstra's Algorithm, used for finding the shortest path in weighted graphs, runs in O((V + E) log V) if the graph is connected, with V as vertices and E as edges.
Dijkstra's Specifics
- Proving Dijkstra's efficiency involves induction on the size of the path.
- Extracting the minimum once per vertex in Dijkstra's results in a time complexity of O(log V).
- Building the heap during Dijkstra's Algorithm takes O(V).
- Decreasing a key value E times in Dijkstra's runs in O(log V).
Directed Acyclic Graphs (DAG)
- To find the shortest path in a DAG, a topological sort is performed.
- The longest path in a DAG is computed as the shortest path multiplied by -1.
Minimum Spanning Tree (MST) Algorithms
- Prim's Algorithm, similar to Dijkstra's, has a runtime of O(m log n).
- Kruskal's MST algorithm, focused on adding edges, also runs in O(m log n).
- Reverse-Delete MST, which removes edges, operates with a runtime of O(m²).
Proving MST Algorithms
- Kruskal's and Prim's algorithms are proven using the Cut Property.
- The Reverse-Delete algorithm is demonstrated through the Cycle Property.
Union-Find Structures
- A root of rank k has at least 2^k nodes.
- There are at most n / 2^k nodes of rank k in a union-find structure.
Other Key Algorithms
- Huffman's Algorithm has a runtime of O(n log n) for constructing optimal prefix codes.
- Inserting into a binary heap has a runtime of O(log n).
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