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CSE 780: Design and Analysis of Algorithms

CSE 780: Design and Analysis of Algorithms. Lecture 14: Directed Graph BFS DFS Topological sort. a. b. c. d. e. f. Un-directed graph Directed graph Each edge is directed from u to v. a. b. c. d. e. f. Vertex Degree. indeg(v) = # edges of the form

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CSE 780: Design and Analysis of Algorithms

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  1. CSE 780: Design and Analysis of Algorithms Lecture 14: Directed Graph BFS DFS Topological sort CSE 2331/5331

  2. a b c d e f • Un-directed graph • Directed graph • Each edge is directed from u to v CSE 2331/5331

  3. a b c d e f Vertex Degree • indeg(v) = # edges of the form • outdeg(v) = # edges of the form • Lemma: • Lemma: CSE 2331/5331

  4. Representations of Graphs • Adjacency lists • Each vertex u has a list, recording its neighbors • i.e., all v’s such that (u, v)  E • An array of V lists • V[i].degree = size of adj list for node • V[i].AdjList= adjacency list for node CSE 2331/5331

  5. Adjacency Lists • For vertex v  V, its adjacency list • has size: outdeg(v) • decide whether (v, u)  E or not in time O(outdeg(v)) • Size of data structure (space complexity): • O(V+E) CSE 2331/5331

  6. Adjacency Matrix • matrix • if is an edge • Otherwise, CSE 2331/5331

  7. Adjacency Matrix • Size of data structure: • O ( V  V) • Time to determine if (v, u)  E : • O(1) • Though larger, it is simpler compared to adjacency list. CSE 2331/5331

  8. Sample Graph Algorithm • Input: Directed graph G represented by adjacency list Running time: O(V + E) CSE 2331/5331

  9. Connectivity • A path in a graph is a sequence of vertices such that there is an edge between any two adjacent vertices in the sequence • Given two vertices we say that wis reachable from u if there is a path in G from u to w. • Note: w is reachable from uDOES NOT necessarily mean that u is reachable from w. CSE 2331/5331

  10. Reachability Test • How many (or which) vertices are reachable from a source node, say ? CSE 2331/5331

  11. BFS and DFS • The algorithms for BFS and DFS remain the same • Each edge is now understood as a directed edge • BFS(V,E, s) : • visits all nodes reachable from s in non-decreasing order CSE 2331/5331

  12. BFS • Starting from source node s, • Spread a wavefront to visit other nodes • First visit all nodes one edge away from s • Then all nodes two edges away from s • … CSE 2331/5331

  13. Pseudo-code Use adjacency list representation Time complexity: O(V+E) CSE 2331/5331

  14. Number of Reachable Nodes Compute # nodes reachable from Time complexity: O(V+E) CSE 2331/5331

  15. DFS: Depth-First Search • Similarly, DFS remains the same • Each edge is now a directed edge • If we start with all nodes unvisited, • Then DFS(G, visits all nodes reachable to node • BFS from previous NumReachable() procedure can be replaced with DFS. CSE 2331/5331

  16. More Example • Is reachable from ? • Is reachable from ? CSE 2331/5331

  17. DFS above can be replaced with BFS DFS(G, k); CSE 2331/5331

  18. Topological Sort CSE 2331/5331

  19. Directed Acyclic Graph • A directed cycle is a sequence such that there is a directed edge between any two consecutive nodes. • DAG: directed acyclic graph • Is a directed graph with no directed cycles. CSE 2331/5331

  20. undershorts socks watch pants shoes shirt belt tie jacket Topological Sort • A topological sort of a DAG G = (V, E) • A linear ordering A of all vertices from V • If edge (u,v)  E => A[u] < A[v] CSE 2331/5331

  21. Another Example Why requires DAG? Is the sorting order unique? CSE 2331/5331

  22. A topological sorted order of graph G exists if and only if G is a directed acyclic graph (DAG). CSE 2331/5331

  23. Question • How to topologically sort a given DAG? • Intuition: • Which node can be the first node in the topological sort order? • A node with in-degree 0 ! • After we remove this, the process can be repeated. CSE 2331/5331

  24. undershorts socks watch pants shoes shirt belt tie jacket Example CSE 2331/5331

  25. CSE 2331/5331

  26. Topological Sort – Simplified Implementation CSE 2331/5331

  27. Time complexity • O(V+E) • Correctness: • What if the algorithm terminates before we finish visiting all nodes? • Procedure TopologicalSort(G) outputs a sorted list of all nodes if and only if the input graph G is a DAG • If G is not DAG, the algorithm outputs only a partial list of vertices. CSE 2331/5331

  28. Remarks • Other topological sort algorithm by using properties of DFS CSE 2331/5331

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