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# Algorithm Complexity and Graph Algorithms

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@CourteousNewton

## Questions and Answers

cyclic

cyclic

acyclic

### A digraph is considered cyclic if it has a strongly connected component with more than one ______.

<p>node</p> Signup and view all the answers

### The worst-case scenario for deciding whether a digraph is cyclic involves computing the intersection of the forward and backward sets of nodes and checking if the intersection's size is larger than ______.

<p>1</p> Signup and view all the answers

### A cyclic digraph has a strongly connected component with more than one ______ because it forms a cycle within that component.

<p>node</p> Signup and view all the answers

### Bob's algorithm for computing strongly connected components focuses on identifying components with more than one ______.

<p>node</p> Signup and view all the answers

### The worst-case complexity of Bob's algorithm for determining if a digraph is cyclic is ______.

<p>not specified</p> Signup and view all the answers

### Bob's algorithm checks if the intersection's size is larger than 1 to determine if the digraph is ______.

<p>cyclic</p> Signup and view all the answers

### The worst-case complexity of Bob's algorithm for determining if a digraph is cyclic is ______.

<p>not specified</p> Signup and view all the answers

### A weakly connected component of a digraph is a largest possible subgraph of mutually reachable ______, if we ignore the direction of edges.These components can be found by computing the connected components of the undirected version of the digraph.The complexity is also Θ(n + e). 21.6.Summary 647 Algorithms, data structures and computability A strongly connected component of a digraph is a largest possible subgraph of mutually reachable ______.These components can be found with repeated traversals of the original graph and its reverse.The worst-case complexity is Θ(n×(n+e)).The reverse graph of a digraph has the same ______ and edges but with the directions reversed.The reverse graph represents the inverse relation of the original graph, e.g.‘A follows B’ becomes ‘B is followed by A’.The reverse graph is computed in Θ(n + e) time.A topological sort of a directed acyclic graph (DAG) is a permutation of its ______ so that for every edge A−→B, node A comes before node B in the permutation.A DAG has one or more topological sorts.Cyclic digraphs have no topological sort.Kahn’s algorithm computes a topological sort in a greedy fashion.In each iteration it (virtually) removes one node with in-degree zero from the digraph and appends it to an initially empty sequence.The complexity is Θ(n + e).If the digraph has a cycle, Kahn’s algorithm returns a sequence without all the graph’s ______, thus making it easy to detect cycles in digraphs. 648 Chapter 21.Graphs 2

<p>nodes</p> Signup and view all the answers

### A digraph is considered cyclic if it has a strongly connected component with more than one ______.

<p>cycle</p> Signup and view all the answers

### A cyclic digraph has a strongly connected component with more than one ______ because it forms a cycle within that component.

<p>cycle</p> Signup and view all the answers

### Bob's algorithm returns false if the digraph is ______.

<p>acyclic</p> Signup and view all the answers

### Bob's algorithm returns true if the digraph is ______.

<p>cyclic</p> Signup and view all the answers

### The worst-case scenario for deciding whether a digraph is cyclic involves computing the intersection of the forward and backward sets of nodes and checking if the intersection's size is larger than ______.

<p>one</p> Signup and view all the answers

### Bob's algorithm for computing strongly connected components focuses on identifying components with more than one ______.

<p>node</p> Signup and view all the answers

### Bob adapts the algorithm for computing the strongly connected components by checking if the intersection's size is larger than 1 to determine if the digraph is ______.

<p>cyclic</p> Signup and view all the answers

### Bob's algorithm checks if the intersection's size is larger than 1 to determine if the digraph is ______.

<p>cyclic</p> Signup and view all the answers

### The worst-case complexity of Bob's algorithm for determining if a digraph is cyclic is ______.

<p>Θ(n * (n+e))</p> Signup and view all the answers

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