GRAPHS

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GRAPHS
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Definition

• The Graph Data Structure
• set V of vertices
• collection E of edges (pairs of vertices in V)
• Drawing of a Graph
• vertex lt-gt circle/oval
• edge lt-gt line connecting the vertex pair

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Sample Uses

• Airports and flights
• Persons and acquaintance relationships
• Intersections and streets
• Computers and connections the Internet

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Graph Example

• Graph G(V,E)
• Va,b,c,d, E(a,b),(b,c),(b,d),(a,d)

a
b
c
d
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Adjacency

• Vertices connected by an edge are adjacent
• An edge e is incident on a vertex v (and
  vice-versa) if v is an endpoint of e
• Degree of a vertex v, deg(v)
• number of edges incident on v
• if directed graph, indeg(v) and outdeg(v)

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Graph Properties

• Graph G with n vertices and m edges
• ?v in G deg(v) 2m
• ?v in G indeg(v) ?v in G outdeg(v) m
• m is O(n2)
• m lt n(n-1)/2 if G is undirected
• m lt n(n-1) if G is directed

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Subgraphs

• Graph H a subgraph of graph G
• vertex set of H is a subset of vertex set of G
• edge set of H is a subset of edge set of G
• Spanning subgraph
• vertex set of H identical to vertex set of G

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Paths and Cycles

• (simple) Path
• sequence of distinct vertices such that
  consecutive vertices are connected through edges
• (simple) Cycle
• same as path except that first and last vertices
  are identical
• Directed paths and cycles
• edge directions matter

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Connectivity

• Connected graph
• there is a path between any two vertices

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Graph Methods

• Container methods
• General or Global methods
• numVertices(),numEdges(),vertices(),edges()
• Directed Edge methods
• indegree(v), inadjacentVertices(v),
  inincidentEdges(v)
• Update methods
• adding/removing vertices or edges

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General Methods

• numvertices()-returns the number of vertices in
  G
• numedges()- returns the number of edges in G
• vertices()- return an enumeration of the vertex
  positions in G
• edges()- returns an enumeration of the edge
  positions in G

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General Methods

• Degree (v) - returns the degree of v
• adjacentvertices(v)- returns an enumeration of
  the vertices adjacent to v
• incidentedges(v)- returns an enumeration of the
  edges incident upon v
• endvertices(e)- return an array of size 2
  storing the end vertices of e

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Directed Edges

• Directededges()- return an enumeration of all
  directed edges
• indegree(v)- return the indegree of v
• outdegree(v)- return the outdegree of v
• origin (v)- return the origin of directed edge e
• other functions- inadjacentvertices(),
  outadjacentvertices(), destination()

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Updating Graphs

• Insertedge(v,w,o)- insert and return an
  undirected edge between vertices v and w storing
  the object o at this position
• Insertvertex(o)- insert and return a new vertex
  storing the object o at this position
• removevertex (v)- remove the vertex v and all
  incident edges
• removeedge(e)- remove edge e

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• Implementation of a graph data structure

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Graph Implementations

• Edge List
• vertices and edges are nodes
• edge nodes refer to incident vertex nodes
• Adjacency List
• same as edge list but vertex nodes also refer to
  incident edge nodes
• Adjacency Matrix
• 2D matrix of vertices entries are edges

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Implementation Examples

• Illustrate the following graph using the
  different implementations

4
a
b
5
1
3
c
d
2
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Edge List
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Adjacency List
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Adjacency Matrix
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Implementation Examples-Query

• Illustrate the following graph using the
  different implementations

4
a
b
8
5
e
1
3
6
c
7
d
2
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Comparing Implementations

• Edge List
• simplest but most inefficient
• Adjacency List
• many vertex operations are O(d) time where d is
  the degree of the vertex
• Adjacency Matrix
• adjacency test is O(1)
• addition/removal of a vertex resize 2D array

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Graph Traversals

• Traversal- A systematic procedure for exploring
  a graph by examining all of its vertices and
  edges
• Consistency- rules to determine the order of
  visits (top-down, alphabetical,etc.)- make the
  path consistent with the actual graph (dont
  change conventions)

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Graph Traversals

• Depth First Search (DFS)
• visit a starting vertex s
• recursively visit unvisited adjacent vertices
• Breadth First Search (BFS)
• visit starting vertex s
• visit unvisited vertices adjacent to s, then
  visit unvisited vertices adjacent to them, and so
  on (visit by levels)

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DFS Traversal

• Depth First Search- recursively visit (traverse)
  unvisited vertices
• Dead End- reaching a vertex where all the
  incident edges go to a previously visited vertex
• Back Track- If a dead-end vertex is reached, go
  to the previous vertex

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Traversal Examples

• DFS a,b,c,d,h,e,f,g,l,k,j,i

c
b
d
a
e
g
f
h
i
j
l
k
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DFS Traversal

• Path of the DFS traversal- patha,b,c,d,h- h is
  a dead end vertex- back track to d,c,b,a and go
  to another adjacent (unvisited) vertex -gt e-
  patha,e,f,g,k,j,i- i is a dead end vertex -
  back track to all the way back to a (also a dead
  end vertex)
• finished

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BFS Traversal

• Visit adjacent vertices by levels
• discovery edges- edges that lead to an unvisited
  vertex
• cross edge- edges that lead to a previously
  visited vertex

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Traversal Examples

• BFS a,b,e,i,c,f,j,d,g,k,h,l

c
b
d
a
e
g
f
h
i
j
l
k
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DFS Algorithm

• Algorithm DFS(v)
• Input Vertex v
• Output Depends on the application
• mark v as visited perform processing
• for each vertex w adjacent to v do
• if w is unmarked then
• DFS(w)

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BFS Algorithm

• Algorithm BFS(s)
• Input Starting vertex s
• Output Depends on the application
• // traversal algorithm uses a queue Q
• // loop terminates when Q is empty

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BFS Algorithm continued

• Q ? new Queue()
• mark s as visited and Q.enqueue(s)
• while not Q.isEmpty()
• v ? Q.dequeue()
• perform processing on v
• for each vertex w adjacent to v do
• if w is unmarked then
• mark w as visited and Q.enqueue(w)

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Query

• Let G be a graph whose vertices are the integers
  1 through 8 and let the adjacent vertices of each
  vertex be given by the table below
  Vertex Adjacent Vertices1 (2,3,4)2 (1,3,4)3
  (1,2,4)4 (1,2,3,6)5 (6,7,8)6 (4,5,7) 7 (
  5,6,8)8 (5,7)

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Query

• Assume that in the traversal of G, the adjacent
  vertices of a given vertex are returned in the
  same order as they are listed in the above table
• a) Draw G
• b) Give the sequence of vertices of G visited
  using a DFS traversal order starting at vertex 1
• c) Give the sequence of vertices visited using a
  BFS traversal starting at vertex 1

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• Weighted GraphsOverview

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Weighted Graphs

• Graph G (V,E) such that there are weights/costs
  associated with each edge
• w((a,b)) cost of edge (a,b)
• representation matrix entries lt-gt costs

a
20
e
35
b
12
32
c
65
d
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Problems on Weighted Graphs

• Minimum-cost Spanning Tree
• Given a weighted graph G, determine a spanning
  tree with minimum total edge cost
• Single-source shortest paths
• Given a weighted graph G and a source vertex v in
  G, determine the shortest paths from v to all
  other vertices in G
• Path length sum of all edges in path

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Weighted Graphs-Concerns

• Minimum Cost
• Shortest Path

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Weighted Graphs

• Shortest Path- Dijkstras Algorithm
• Minimum Spanning Trees- Kruskals Algorithm