18_Topological_sort.pptx

If we cannot find a vertex with in degree zero we

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If we cannot find a vertex with in-degree zero, we will show there must be a cycle Start with any vertex and define a list L = ( v ) Then iterate this loop |V | times: The first vertex 1 in the list L does not have in-degree zero So we can find a vertex w such that ( w , 1 ) is an edge Add w to the list: L = ( w , 1 , …, k ) By the pigeon-hole principle, at least one vertex must appear twice This forms a cycle; hence a contradiction, as this is a DAG

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Topological Sort First, we need a two lemmas: Any sub-graph of a DAG is a DAG Proof: If a sub-graph has a cycle, that same cycle must appear in the super- graph We assumed the super-graph was a DAG This is a contradiction the sub-graph must be a DAG
Topological Sort We will start with showing a b : If a graph is a DAG, it has a topological sort Proof by induction: A graph with one vertex is a DAG and it has a topological sort Assume a DAG with n vertices has a topological sort A DAG with n + 1 vertices must have at least one vertex v of in-degree zero Removing the vertex v and consider the vertex-induced sub-graph with the remaining n vertices If this sub-graph has a cycle, so would the original graph—contradiction Thus, the graph with n vertices is also a DAG, therefore it has a topological sort Add the vertex v to the start of the topological sort to get one for the graph of size n + 1

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Topological Sort Next, we will show that b a : If a graph has a topological ordering, it must be a DAG We will show this by showing the contrapositive: ¬ a ¬ b : If a graph is not a DAG, it does not have a topological sort By definition, it has a cycle: ( v 1 , v 2 , v 3 , …, v k , v 1 ) In any topological sort, v 1 must appear before v 2 , because ( v 1 , v 2 ) is a path However, there is also a path from v 2 to v 1 : ( v 2 , v 3 , …, v k , v 1 ) Therefore, v 2 must appear in the topological sort before v 1 This is a contradiction, therefore the graph cannot have a topological sort a b : A graph is a DAG if and only if it has a topological sorting
Outline Topological sorting Definitions Algorithm Finding the critical path

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Topological Sort Idea: Given a DAG V , iterate: Find a vertex v in V with in-degree zero Let v be the next vertex in the topological sort Continue iterating with the vertex-induced sub-graph V \ { v }
Example On this graph, iterate the following |V| = 12 times Choose a vertex v that has in-degree zero Let v be the next vertex in our topological sort Remove v and all edges connected to it

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Example Let’s step through this algorithm with this example Which task can we start with?
Example Of Tasks C or H, choose Task C

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Example Having completed Task C, which vertices have in-degree zero?
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