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Chapter 8, Part I

Chapter 8, Part I. Graph Algorithms. Chapter Outline. Graph background and terminology Data structures for graphs Graph traversal algorithms Minimum spanning tree algorithms Shortest-path algorithm Biconnected components Partitioning sets. Prerequisites.

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Chapter 8, Part I

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  1. Chapter 8, Part I Graph Algorithms

  2. Chapter Outline • Graph background and terminology • Data structures for graphs • Graph traversal algorithms • Minimum spanning tree algorithms • Shortest-path algorithm • Biconnected components • Partitioning sets

  3. Prerequisites • Before beginning this chapter, you should be able to: • Describe a set and set membership • Use two-dimensional arrays • Use stack and queue data structures • Use linked lists • Describe growth rates and order

  4. Goals • At the end of this chapter you should be able to: • Describe and define graph terms and concepts • Create data structures for graphs • Do breadth-first and depth-first traversals and searches • Find the minimum spanning tree for a connected graph • Find the shortest path between two nodes of a connected graph • Find the biconnected components of a connected graph

  5. Graph Types • A directed graph edges’ allow travel in one direction • An undirected graph edges’ allow travel in either direction

  6. Graph Terminology • A graph is an ordered pair G=(V,E) with a set of vertices or nodes and the edges that connect them • A subgraph of a graph has a subset of the vertices and edges • The edges indicate how we can move through the graph

  7. Graph Terminology • A path is a subset of E that is a series of edges between two nodes • A graph is connected if there is at least one path between every pair of nodes • The length of a path in a graph is the number of edges in the path

  8. Graph Terminology • A complete graph is one that has an edge between every pair of nodes • A weighted graph is one where each edge has a cost for traveling between the nodes

  9. Graph Terminology • A cycle is a path that begins and ends at the same node • An acyclic graph is one that has no cycles • An acyclic, connected graph is also called an unrooted tree

  10. Data Structures for Graphsan Adjacency Matrix • A two-dimensional matrix or array that has one row and one column for each node in the graph • For each edge of the graph (Vi, Vj), the location of the matrix at row i and column j is 1 • All other locations are 0

  11. Data Structures for GraphsAn Adjacency Matrix • For an undirected graph, the matrix will be symmetric along the diagonal • For a weighted graph, the adjacency matrix would have the weight for edges in the graph, zeros along the diagonal, and infinity (∞) every place else

  12. Adjacency Matrix Example 1

  13. Adjacency Matrix Example 2

  14. Data Structures for GraphsAn Adjacency List • A list of pointers, one for each node of the graph • These pointers are the start of a linked list of nodes that can be reached by one edge of the graph • For a weighted graph, this list would also include the weight for each edge

  15. Adjacency List Example 1

  16. Adjacency List Example 2

  17. Graph Traversals • We want to travel to every node in the graph • Traversals guarantee that we will get to each node exactly once • This can be used if we want to search for information held in the nodes or if we want to distribute information to each node

  18. Depth-First Traversal • We follow a path through the graph until we reach a dead end • We then back up until we reach a node with an edge to an unvisited node • We take this edge and again follow it until we reach a dead end • This process continues until we back up to the starting node and it has no edges to unvisited nodes

  19. Depth-First Traversal Example • Consider the following graph: • The order of the depth-first traversal of this graph starting at node 1 would be:1, 2, 3, 4, 7, 5, 6, 8, 9

  20. Depth-First Traversal Algorithm Visit( v ) Mark( v ) for every edge vw in G do if w is not marked then DepthFirstTraversal(G, w) end if end for

  21. Breadth-First Traversal • From the starting node, we follow all paths of length one • Then we follow paths of length two that go to unvisited nodes • We continue increasing the length of the paths until there are no unvisited nodes along any of the paths

  22. Breadth-First Traversal Example • Consider the following graph: • The order of the breadth-first traversal of this graph starting at node 1 would be: 1, 2, 8, 3, 7, 4, 5, 9, 6

  23. Breadth-First Traversal Algorithm Visit( v ) Mark( v ) Enqueue( v ) while the queue is not empty do Dequeue( x ) for every edge xw in G do if w is not marked then Visit( w ) Mark( w ) Enqueue( w ) end if end for end while

  24. Traversal Analysis • If the graph is connected, these methods will visit each node exactly once • Because the most complex work is done when the node is visited, we can consider these to be O(N) • If we count the number of edges considered, we will find that is also linear with respect to the number of graph edges

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