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Graphs

Learn about graphs and their definitions, terminology, and common traversal algorithms like Depth First Search (DFS) and Breadth First Search (BFS).

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Graphs

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  1. Graphs • Eugene Weinstein • New York University • Computer Science Department • Data Structures Fall 2008

  2. Graph Definitions • A graph G = (V, E) is a set of nodes/vertices (V) and edges (E) • Each edge connects a pair of vertices: (u,v) ∈ E • In an undirected graph, edges simply associate two vertices • In a directed graph, edges point from one vertex to another • e.g., (u,v) is an edge from u to v

  3. Graph Terminology • A path is a sequence of vertices connected by edges • A cycle is a path whose first and last vertices are the same and no vertex appears more than once • A graph is connected if there is a path from any vertex to any other vertex

  4. More Graph Terminology • A weighted graph is one with weights (a.k.a. costs or lengths) on edges 7 -2 3 5

  5. Graphs and Trees • What is the connection between graphs and trees? • A tree is an acyclic connected graph • So graphs can be viewed as a generalization of trees • Is this a tree? • Maybe it would help to redraw it?

  6. Graph Traversals • We can traverse/scan a graph just as we would a tree • But it is more tricky - what are the issues? • Cycles: we have to keep track of which vertices have been visited already • We will discuss two traversals • Depth first search (DFS): generalization of inorder, preorder, postorder traversal • Breadth first search (BFS): generalization of tree version

  7. Depth First Search • Assume all vertices are are colored white/black, initially white • DFSHelper(G,u) • u.color ← black • visit u // Print or do other work • for each vertex v such that (u,v) is an edge in G • if (v.color = white), DFSHelper(G, v) • DFS(G) • for each vertex v of G • if (v.color = white), DFSHelper(G, v) 7

  8. DFS Properties • Linear running time • Each node is colored once • Each edge is inspected once • O(|V|+|E|) runtime complexity • This is a tail-recursion implementation • i.e., we also use O(|V|) call stack space • There is also a non-recursive implementation, using a stack

  9. DFS Question • For a tree, this corresponds to a pre-order traversal • How do we change it to post-order? • DFSHelper(G,u) • u.color ← black • visit u // Print or do other work • for each vertex v such that (u,v) is an edge in G • if (v.color = white), DFSHelper(G, v) 9

  10. Breadth-First Search • Input to BFS: graph G and starting vertex s, all initially white • BFS(G,s) • s.color ← black • Q.enqueue(s) • while (!Q.empty) • u ← Q.dequeue • visit u • for each vertex v such that (u,v) is an edge in G • if (v.color = white) • v.color ← black • Q.enqueue(v) 10

  11. BFS Properties • Linear runtime complexity • Each vertex is visited at most once • Each edge is inspected at most once • Runtime: O(|V|+|E|) 11

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