Graphs Tessema M. Mengistu Department of Computer Science Southern - - PowerPoint PPT Presentation

Graphs

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Tessema M. Mengistu Department of Computer Science Southern Illinois University Carbondale tessema.mengistu@siu.edu Room - Faner 3131

Outline

• Introduction to Graphs
• Graph Traversals
• Finding a Path
• Graph ADT

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Introduction to Graph

• There are many applications that uses graphs

– Google Map

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Introduction to Graph

• Airline Reservation Systems

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Introduction to Graph

• Global Positioning System (GPS)

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Introduction to Graph

• Graph

– Consists of a finite set (distinct) of vertices or nodes or points that are connected together using lines called edges – A subgraph is a portion of a graph that is itself a graph

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Introduction to Graph

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Graph Terminology

• Nodes connected by edges
• Edges

– Undirected – Directed (digraph)

Graph Terminology

• Path between two vertices

– Sequence of edges

• A path in a directed graph must consider the

direction of the edges, and is called a directed path.

• The length of a path is the number of edges that

it comprises.

• If the path does not pass through any vertex

more than once, it is a simple path.

• Graph can be:

– Cyclic – has a path that begins and ends at the same vertex – Acyclic - A graph that has no cycles is

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Graph Terminology

• Weights

– Shortest, fastest, cheapest, costs

• A weighted graph

– has values on its edges – E.g distance in miles, cost, time of driving, etc

• A path in a weighted graph has a weight, or

cost, which is the sum of its edge weights

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Graph Terminology

• Connected graphs

– A graph that has a path between every pair of distinct vertices

• A complete graph

– Has an edge between every pair of distinct vertices.

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Graph Terminology

• Adjacent vertices

– Undirected graph

• Two vertices are adjacent if they are joined by an edge
• Also are called neighbors

– Directed graph

• Vertex i is adjacent to vertex j if a directed edge begins

at j and ends at i.

• Vertex A is adjacent to vertex B, but vertex B is not

adjacent to vertex A.

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Graph Terminology

• If a graph has n vertices, it can have at most

– n (n - 1) edges if the graph is directed – n (n - 1) / 2 edges if the graph is undirected

• A graph is sparse if it has relatively few edges

– Has O(n) edges

• A graph is dense if it has many edges

– Has O(n2) edges.

• Typical graphs are sparse.

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Graphs Vs Trees

• All trees are graphs, but not all graphs are

trees.

• A tree is a connected graph without cycles.

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Graph Traversals

• Graph Traversals

– Focus on the connections between vertices, rather than the contents of vertices

• In a graph, “visit a node” means simply to

“mark the node as visited.”

• Begins at any vertex—called the origin

vertex— and visits only the vertices that it can reach.

• Two types

– Breadth-first – Depth-first

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Breadth-first Traversal

• Breadth-first traversal

– Visits all neighbors of a node before visiting the neighbors’ neighbors.

• Traversal uses a queue to hold the visited

vertices.

• The traversal order is then the order in which

vertices are added to the queue.

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Breadth-first Traversal

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Breadth-first Traversal

• Example

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Depth-first Traversal

• Depth-first traversal

– Follows a path that goes as deeply into the graph as possible before following other paths. – After visiting a vertex, this traversal visits the vertex’s neighbor, the neighbor’s neighbor, and so

• n.

– Uses a stack in the iterative description of this traversal.

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Depth-first Traversal

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Depth-first Traversal

• Example

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Path

• Path examples

– Learning whether a particular airline flies between two given cities – Finding if there is a road connecting two cities

• A depth-first traversal stays on a path through the

graph as it visits as many vertices as possible.

– Each time we visit another vertex, we see whether that vertex is the desired destination. – If so, we are done and the resulting stack contains the path. – Otherwise, we continue the traversal until either we are successful or the traversal ends.

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Shortest Path – unweighted Graph

• A graph can have several different paths

between the same two vertices.

• We can find the path with the shortest length,

that is, the path that has the fewest edges.

– Example – a path between A and H

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Algorithm

• Every node keeps

– Its predecessor node – A length of the path to reach that specific node – Example

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Shortest Path – Weighted Graph

• The shortest path is not necessarily the one

with the fewest edges.

• The Shortest path is the one with the smallest

edge-weight sum.

• Example

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Figure 28-19 A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph

Figure 28-19 A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph

FIGURE 28-20 The graph in Figure 28-18a after finding the cheapest path from vertex A to vertex H

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Graph ADT

• No addition, removal, or retrieval

components.

• We use a graph to answer questions based on

the relationships among its vertices.

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• Example

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