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Questions and Answers
What does it mean for a graph to be connected?
What does it mean for a graph to be connected?
Which of the following correctly defines a directed graph?
Which of the following correctly defines a directed graph?
In graph terminology, what is the difference between a sparse graph and a dense graph?
In graph terminology, what is the difference between a sparse graph and a dense graph?
Which algorithm strategy is used to find the shortest path in a weighted graph?
Which algorithm strategy is used to find the shortest path in a weighted graph?
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What characterizes a vertex in a graph?
What characterizes a vertex in a graph?
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What is the key difference between BFS and DFS graph traversal methods?
What is the key difference between BFS and DFS graph traversal methods?
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Which of the following is NOT a property of directed graphs?
Which of the following is NOT a property of directed graphs?
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Which algorithm is primarily used for finding a minimum spanning tree in a graph?
Which algorithm is primarily used for finding a minimum spanning tree in a graph?
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What are edges in a directed graph characterized by?
What are edges in a directed graph characterized by?
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Which statement best defines a strongly connected graph?
Which statement best defines a strongly connected graph?
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How is the length of a path in a weighted graph determined?
How is the length of a path in a weighted graph determined?
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What represents the distance between two vertices in a graph?
What represents the distance between two vertices in a graph?
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In a directed graph, a graph is considered connected if:
In a directed graph, a graph is considered connected if:
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What is a key property of weighted graphs?
What is a key property of weighted graphs?
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Which of the following best describes a directed edge?
Which of the following best describes a directed edge?
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What implication does the presence of weights on edges have in a graph?
What implication does the presence of weights on edges have in a graph?
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What characterizes a minimum spanning tree in a weighted graph?
What characterizes a minimum spanning tree in a weighted graph?
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Which of the following is true about Prim's Algorithm's running time?
Which of the following is true about Prim's Algorithm's running time?
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What is the role of the for loop in Prim's Algorithm?
What is the role of the for loop in Prim's Algorithm?
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What is an essential property of a spanning tree?
What is an essential property of a spanning tree?
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Which process is NOT involved in Prim's Algorithm for finding a minimum spanning tree?
Which process is NOT involved in Prim's Algorithm for finding a minimum spanning tree?
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Which statement correctly describes the initialization step in Prim's Algorithm?
Which statement correctly describes the initialization step in Prim's Algorithm?
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What does the term 'Extract-Min' refer to in Prim's Algorithm?
What does the term 'Extract-Min' refer to in Prim's Algorithm?
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In what scenario would Prim's Algorithm NOT be applicable?
In what scenario would Prim's Algorithm NOT be applicable?
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What is a necessary condition for a graph to be amenable to topological sorting?
What is a necessary condition for a graph to be amenable to topological sorting?
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Which of the following statements about topological sorting is true?
Which of the following statements about topological sorting is true?
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In the second pass of Depth First Search for topological sorting, if vertex B has an unexplored child F, what should be done?
In the second pass of Depth First Search for topological sorting, if vertex B has an unexplored child F, what should be done?
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What is the result of adding vertex G to the stack during the third pass of the DFS?
What is the result of adding vertex G to the stack during the third pass of the DFS?
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Which element in the stack corresponds to the highest level of exploration after the entire DFS process?
Which element in the stack corresponds to the highest level of exploration after the entire DFS process?
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In a directed graph, which of the following pairs represents the correct structure of edges?
In a directed graph, which of the following pairs represents the correct structure of edges?
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What distinguishes a directed graph from other types of graphs?
What distinguishes a directed graph from other types of graphs?
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Which of the following is NOT a characteristic of topological sorting?
Which of the following is NOT a characteristic of topological sorting?
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When performing a topological sort, what happens to unexplored vertices once the DFS is complete?
When performing a topological sort, what happens to unexplored vertices once the DFS is complete?
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If vertex F is at level 4 in the stack during topological sorting, what does that signify?
If vertex F is at level 4 in the stack during topological sorting, what does that signify?
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Which of the following would invalidate a topological sorting attempt?
Which of the following would invalidate a topological sorting attempt?
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What role does backtracking play in the DFS process of topological sorting?
What role does backtracking play in the DFS process of topological sorting?
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What is the purpose of assigning numbers during the topological sort process?
What is the purpose of assigning numbers during the topological sort process?
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What is the overall running time of BFS on a graph?
What is the overall running time of BFS on a graph?
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What condition indicates that a directed graph is strongly connected?
What condition indicates that a directed graph is strongly connected?
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Which traversal method proceeds by exploring a vertex's neighbors before moving on to their neighbors?
Which traversal method proceeds by exploring a vertex's neighbors before moving on to their neighbors?
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In a directed graph, if BFS terminates before all vertices are visited, what can be inferred?
In a directed graph, if BFS terminates before all vertices are visited, what can be inferred?
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What is the concept behind Depth First Search (DFS)?
What is the concept behind Depth First Search (DFS)?
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What distinguishes a directed graph from an undirected graph?
What distinguishes a directed graph from an undirected graph?
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In the context of graph traversal, what is the role of a queue in BFS?
In the context of graph traversal, what is the role of a queue in BFS?
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What does a topological sort ensure in a directed acyclic graph?
What does a topological sort ensure in a directed acyclic graph?
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What can you say about the explored nodes in a stack during DFS?
What can you say about the explored nodes in a stack during DFS?
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When performing BFS, what does it indicate if multiple vertices are in the queue at the same time?
When performing BFS, what does it indicate if multiple vertices are in the queue at the same time?
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Which of the following is not a feature of directed graphs?
Which of the following is not a feature of directed graphs?
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Why is the adjacency list preferred for sparse graphs?
Why is the adjacency list preferred for sparse graphs?
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Which traversal method would you use to find the shortest path in an unweighted graph?
Which traversal method would you use to find the shortest path in an unweighted graph?
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What is a significant characteristic of a weighted graph?
What is a significant characteristic of a weighted graph?
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Study Notes
Topological Sort
- A topological ordering is a linear ordering of vertices in a directed acyclic graph (DAG) such that for every directed edge (u, v) from vertex u to vertex v, u comes before v in the ordering.
- Every DAG has at least one topological ordering.
- A topological sorting is a way to arrange the nodes of a graph in such a way that if there is a path from node A to node B, then A comes before B in the ordering.
- Topological sort is used in many applications, including:
- Project scheduling: Finding the order in which tasks must be completed.
- Compiler optimization: Finding the order in which instructions can be executed.
- Course scheduling: Finding the order in which courses must be taken.
Directed Acyclic Graphs
- Every directed acyclic graph has at least one source vertex.
- It's impossible to topologically order the vertices of a graph that contains a directed cycle.
Topological Sort
- A topological sort is an ordering of vertices in a directed acyclic graph (DAG) such that for every directed edge (u, v), u comes before v in the ordering.
Prim's Algorithm
- Goal: Find a minimum spanning tree (MST) in a weighted graph.
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Procedure:
- Initialize a set of vertices that are part of the MST.
- At each step, select the edge with the minimum weight connecting a vertex in the MST to a vertex outside the MST.
- Add the selected edge and vertex to the MST.
- Repeat until all vertices are in the MST.
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Example:
- Given a weighted graph, identify edges and vertices to include in the MST to obtain the minimum total weight.
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Running Time (Heap based):
- Initialization: Takes Θ(n) time.
- Constructing Heap: Takes Θ(n) time.
- Extract-Min: Takes O(log n) time, with n total calls, resulting in O(n log n) time.
- Loop: Executes Θ(m) times.
- Vertex Access: Takes Θ(1) time.
- Key Updating: Takes O(log n) time.
- Overall: O(n log n + m log n) = O((m+n) log n).
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Description
This quiz explores the concepts of topological sorting and directed acyclic graphs (DAGs). You'll learn about their applications, properties, and the significance of topological ordering in various fields like project scheduling and course sequencing. Test your understanding of how to correctly arrange nodes in a graph while considering directed edges.