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More Graph Algorithms

More Graph Algorithms. Applications of Graphs: Topological Sorting. Topological order A list of vertices in a directed graph without cycles such that vertex x precedes vertex y if there is a directed edge from x to y in the graph There may be several topological orders in a given graph

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More Graph Algorithms

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  1. More Graph Algorithms

  2. Applications of Graphs: Topological Sorting • Topological order • A list of vertices in a directed graph without cycles such that vertex x precedes vertex y if there is a directed edge from x to y in the graph • There may be several topological orders in a given graph • Topological sorting • Arranging the vertices into a topological order

  3. Topological Sort • Directed graph G. • Rule: if there is an edge u  v, then u must come before v. • Ex: A B G F C I H A B E D G F C I H E D

  4. Intuition • Cycles make topological sort impossible. • Select any node with no in-edges • print it • delete it • and delete all the edges leaving it • Repeat • What if there are some nodes left over? • Implementation? Efficiency?

  5. Implementation • Start with a list of nodes with in-degree = 0 • Select any edge from list • mark as deleted • mark all outgoing edges as deleted • update in-degree of the destinations of those edges • If any drops below zero, add to the list • Running time?

  6. Topological Sorting Figure 13.14 A directed graph without cycles Figure 13.15 The graph in Figure 13-14 arranged according to the topological orders a) a, g, d, b, e, c, f and b) a, b, g, d, e, f, c

  7. Topological Sorting • Simple algorithms for finding a topological order • topSort1 • Find a vertex that has no successor • Remove from the graph that vertex and all edges that lead to it, and add the vertex to the beginning of a list of vertices • Add each subsequent vertex that has no successor to the beginning of the list • When the graph is empty, the list of vertices will be in topological order

  8. Topological Sorting • Simple algorithms for finding a topological order (Continued) • topSort2 • A modification of the iterative DFS algorithm • Strategy • Push all vertices that have no predecessor onto a stack • Each time you pop a vertex from the stack, add it to the beginning of a list of vertices • When the traversal ends, the list of vertices will be in topological order

  9. Implementation • Start with a list of nodes with in-degree = 0 • Select any edge from list • mark as deleted • mark all outgoing edges as deleted • update in-degree of the destinations of those edges • If any drops below zero, add to the list • Running time? In all, algorithm does work: • O(|V|) to construct initial list • Every edge marked “deleted” at most once: O(|E|) total • Every node marked “deleted” at most once: O(|V|) total • So linear time overall (in |E| and |V|)

  10. Why should we care? • Shortest path problem in directed, acyclic graph • Called a DAG for short • General problem: • Given a DAG input G with weights on edges • Find shortest paths from source A to every other vertex

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