Graphs

GraphsGraphs

Chapter 13Chapter 13

CS 302 2Chapter 13 -- Graphs

PreviewPreview Graphs are an important mathematical concept that Graphs are an important mathematical concept that

have significant applications not only in computer have significant applications not only in computer science, but also in many other fields.science, but also in many other fields.

You can view a graph as a mathematical construct, You can view a graph as a mathematical construct, a data structure, or an abstract data type.a data structure, or an abstract data type.

This chapter provides an introduction to graphs This chapter provides an introduction to graphs and it presents the major operations and and it presents the major operations and applications of graphs that are relevant to the applications of graphs that are relevant to the computer scientistcomputer scientist


TerminologyTerminology

Def: A graph G consists of two sets Def: A graph G consists of two sets VerticesVertices EdgesEdges

Def: AdjacentDef: Adjacent Two vertices are said to be adjacent if they are Two vertices are said to be adjacent if they are

joined by an edge.joined by an edge.


Def: SubgraphDef: Subgraph A subgraph consists of a set of a graph’s vertices A subgraph consists of a set of a graph’s vertices

and a subset of its edgesand a subset of its edges


Def: PathDef: Path A path between two vertices is a sequence of edges A path between two vertices is a sequence of edges

that begins at one vertex and ends at another.that begins at one vertex and ends at another. Simple path: does not pass through the same Simple path: does not pass through the same

vertex more than once.vertex more than once. Def: CycleDef: Cycle

A cycle is a path that begins and ends at the same A cycle is a path that begins and ends at the same vertex.vertex.

Def: ConnectedDef: Connected A graph is connected if each pair of distinct A graph is connected if each pair of distinct

vertices has a path between themvertices has a path between them


A graph is complete if each pair of distinct A graph is complete if each pair of distinct vertices has an edge between themvertices has an edge between them


Since a graph has a set of edges it cannot have Since a graph has a set of edges it cannot have duplicate edges. duplicate edges. However, a multigraph does allow multiple edgesHowever, a multigraph does allow multiple edges A graph also does not allow self-edges (loops)A graph also does not allow self-edges (loops)


You can label the edges of a graphYou can label the edges of a graph When these labels represent numeric values, the When these labels represent numeric values, the

graph is called a weighted graphgraph is called a weighted graph


All of the previous examples were of All of the previous examples were of undirected graphs, because the edges did not undirected graphs, because the edges did not indicate a direction. If there is direction, then indicate a direction. If there is direction, then the graph is called a directed graph (or the graph is called a directed graph (or digraph)digraph)


Graphs as ADTsGraphs as ADTs

Basic FunctionsBasic Functions Create, Destroy, Determine emptyCreate, Destroy, Determine empty Determine number of vertices and edgesDetermine number of vertices and edges Is there an edge between I and J?Is there an edge between I and J? Insert a vertex, or Insert an edgeInsert a vertex, or Insert an edge Delete a vertex, or Delete an EdgeDelete a vertex, or Delete an Edge ……


Implementing GraphsImplementing Graphs The two most common implementations of a graph The two most common implementations of a graph

are the Adjacency Matrix and the Adjacency Listare the Adjacency Matrix and the Adjacency List

Adjacency MatrixAdjacency Matrix Given a graph G with n verticesGiven a graph G with n vertices Matix is n x n of BooleansMatix is n x n of Booleans Entries – True if an edge exists between I and J (false Entries – True if an edge exists between I and J (false

otherwise).otherwise).


Example: a directed graph.Example: a directed graph.


Example: a weighted graphExample: a weighted graph


An Adjacency ListAn Adjacency List Given a graph G with n verticesGiven a graph G with n vertices N linked listsN linked lists The iThe ithth list has node for vertex j iff there is an edge list has node for vertex j iff there is an edge

between i and jbetween i and j


Example: a directed graphExample: a directed graph


Example: a weighted graphExample: a weighted graph


Which implementation is better?Which implementation is better? Let’s look at the two most common operationsLet’s look at the two most common operations

Determine if there is an edge (i, j)Determine if there is an edge (i, j) Find all vertices adjacent to a given vertex i.Find all vertices adjacent to a given vertex i.

The adjacency matrix supports the first one The adjacency matrix supports the first one somewhat more efficiently than the listsomewhat more efficiently than the list

The second operation is supported more efficiently The second operation is supported more efficiently by the adjacency list than the matrix.by the adjacency list than the matrix.


Now consider the space requirementsNow consider the space requirements


Graph TraversalsGraph Traversals

Visit all the vertices in the graph.Visit all the vertices in the graph. There are two major ways to do this:There are two major ways to do this:

Depth first searchDepth first search Breadth first searchBreadth first search


Example: the order in which the nodes of a tree are Example: the order in which the nodes of a tree are visited in a depth-first search and a breadth-first visited in a depth-first search and a breadth-first search.search.


How do you implement them?How do you implement them? Depth-First SearchDepth-First Search

Typically done with a stack.Typically done with a stack. Breadth-First SearchBreadth-First Search

Typically done with a QueueTypically done with a Queue


Example: DFSExample: DFS Start with Node aStart with Node a

Trace the SearchTrace the Search


Example: DFSExample: DFS


Example: BFSExample: BFS Start with Node aStart with Node a

Trace the SearchTrace the Search


Example: BFSExample: BFS


Applications of GraphsApplications of Graphs

Topological SortingTopological Sorting Given the following graph, place the vertices in Given the following graph, place the vertices in

order so that the edges all point in one directionorder so that the edges all point in one direction


Multiple solutions:Multiple solutions:

Applications: Prerequisites ( 201,202,236,336,308,…)Applications: Prerequisites ( 201,202,236,336,308,…)


Spanning TreesSpanning Trees Given a graph G, construct a tree that has all the Given a graph G, construct a tree that has all the

vertices of Gvertices of G


A few observations about undirected graphsA few observations about undirected graphs A connected undirected graph that has n vertices must A connected undirected graph that has n vertices must

have at least n-1 edgeshave at least n-1 edges A connected undirected graph that has n vertices and A connected undirected graph that has n vertices and

exactly n-1 edges cannot contain a cycle.exactly n-1 edges cannot contain a cycle. A connected undirected graph that has n vertices and A connected undirected graph that has n vertices and

more than n-1 edges must contain at least 1 cycle.more than n-1 edges must contain at least 1 cycle.


The DFS Spanning TreeThe DFS Spanning Tree


The BFS spanning TreeThe BFS spanning Tree


Minimum Spanning TreesMinimum Spanning Trees Given a weighted undirected graph, compute the Given a weighted undirected graph, compute the

spanning tree with the minimum costspanning tree with the minimum cost


Prim’s AlgorithmPrim’s Algorithm Start with two setsStart with two sets

Vertices not visited (not part of the tree)Vertices not visited (not part of the tree) Vertices visited (part of the tree – initialized to some start Vertices visited (part of the tree – initialized to some start

vertex)vertex)

Find the shortest edge from the visited set to the not Find the shortest edge from the visited set to the not visited set.visited set.

Add the edge to the MSTAdd the edge to the MST Move the vertex on the other end from not visited to visited.Move the vertex on the other end from not visited to visited.

Continue until all vertices are visited.Continue until all vertices are visited.


Trace of Prim’sTrace of Prim’s


Shortest PathsShortest Paths Consider once again a map of airline routes.Consider once again a map of airline routes. The shortest path from 0 to 1 in the following The shortest path from 0 to 1 in the following

graph is not the edge 0-1 (weight 8) but the path graph is not the edge 0-1 (weight 8) but the path (0-4-2-1)(0-4-2-1)


The algorithm to calculate the answer is attributed The algorithm to calculate the answer is attributed to Dijkstrato Dijkstra

Need two arraysNeed two arrays Weight – Weight – Weight[v]Weight[v] = weight of the shortest path from 0 to v = weight of the shortest path from 0 to v Matrix – Matrix – Matrix[v][u]Matrix[v][u] = weight of edge from v to u = weight of edge from v to u

Then compare the Then compare the Weight[u]Weight[u] with with Weight[v]+Matrix[v][u].Weight[v]+Matrix[v][u]. Put the smallest in Put the smallest in Weight[u]Weight[u]

Then we need a set of vertices we have visitedThen we need a set of vertices we have visited


We begin with the vertex set initialized to 0We begin with the vertex set initialized to 0


CircuitsCircuits Eulerian CircuitsEulerian Circuits

Probably the first application of graphs occurred in the Probably the first application of graphs occurred in the early 1700s when Euler proposed the Königsberg bridge early 1700s when Euler proposed the Königsberg bridge problem. problem.

Can you start and end at the same point and only cross Can you start and end at the same point and only cross each bridge once? each bridge once?


Some Difficult ProblemsSome Difficult Problems The Traveling Salesperson ProblemThe Traveling Salesperson Problem

Def: A Hamiltonian circuit – pass through every vertex Def: A Hamiltonian circuit – pass through every vertex once.once.

Given Given a weighted graph a weighted graph an origin cityan origin city

Problem:Problem: the salesperson must begin at an origin city and visit all others the salesperson must begin at an origin city and visit all others

(exactly once) and return to the origin (exactly once) and return to the origin and do it with the least cost.and do it with the least cost.


The three utilities problemThe three utilities problem GivenGiven

A graph (3 houses, 3 utilities)A graph (3 houses, 3 utilities) ProblemProblem

Connect each house with each utility so the edges do not crossConnect each house with each utility so the edges do not cross

This is a planarity question (minimum crossing number = 0) This is a planarity question (minimum crossing number = 0) for Kfor K3,33,3


The four color problemThe four color problem GivenGiven

A planar graphA planar graph

ProblemProblem Can you color the vertices so no two adjacent vertices have the Can you color the vertices so no two adjacent vertices have the

same color?same color?

The question was posed more than a century ago (1852) The question was posed more than a century ago (1852) and was only solved in the 1976 (with the help of a and was only solved in the 1976 (with the help of a computer)computer)

Trivia: Nature July 17, 1879 carried a proof of this problem Trivia: Nature July 17, 1879 carried a proof of this problem (which was proved false in 1890)(which was proved false in 1890)


SummarySummary

Implementations:Implementations: Adjacency Matrix and Adjacency ListAdjacency Matrix and Adjacency List

Graph searching Graph searching uses stacks and Queuesuses stacks and Queues

TreesTrees Connected, undirected, acyclic graphsConnected, undirected, acyclic graphs

Problems looked atProblems looked at Shortest Paths, Eulerian circuits, Hamiltonian Shortest Paths, Eulerian circuits, Hamiltonian

CircuitsCircuits

Graphs

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