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Chapter 8, Part I

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# Chapter 8, Part I - PowerPoint PPT Presentation

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

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
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
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

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