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Lecture 9:. Graphs & Graph Models. Definition of a Graph. cycle. path. edge. vertex. A. D. F. C. H. B. E. G. Graph Representations. edge list. node list. A B A C A D A E A F B A B C B E B H C A C B C D C E C H D A D C D F D G D H. A - B C D E F B - A C E H - PowerPoint PPT Presentation
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Lecture 9:
Graphs & Graph Models
Definition of a Graph
edge
vertex
cycle
path
Graph Representations
adjacency matrix
node list edge listA D F
C H
B E G
A B C D E F G HA - 1 1 1 1 1 0 0B 1 - 1 0 1 0 0 1C 1 1 - 1 1 0 0 1D 1 0 1 - 0 1 1 1E 1 1 1 0 - 1 1 0F 1 0 0 1 1 - 1 1G 0 0 0 1 1 1 - 1H 0 1 1 1 0 1 1 -
A - B C D E FB - A C E HC - A B D E HD - A C F G HE - A B C F GF - A D E G HG - D E F HH - B C D F G
A BA CA DA EA FB AB CB EB HC AC BC DC EC HD AD CD FD GD H
E AE BE CE FE GF AF DF EF GF HG DG EG FG HH BH CH DH FH G
node list - lists the nodes connected to each node
edge list - lists each of the edges as a pair of nodes undirected edges may be listed twice XY and YX in order to simplify algorithm implementation
adjacency matrix - for an n-node graph we build an nxn array with 1's indicating edges and 0's no edge the main diagonal of the matrix is unused unless a node has an edge connected to itself. If graph is weighted, 1's are replaced with edge weight values
Example Applications
This graph could represent...
a computer network
an airline flight route
an electrical power grid
WalMart warehouse supply lines
...
Niche Overlap Graph
Animals are represented as nodes. We place an edge between two nodes if the corresponding animals compete for resources (food, habitat, ...).
Web Links
Vertices indicate we pages and arcs indicated links to other web pages.
Round-Robin Tour
undefeated 5-0
biggest loser 0-5
A directed graph represents a round-robin tour in which each team plays every other team exactly one time. Arc points from winning team to losing team.
An Acquaintanceship Graph
EduardoKamini
JanPaulaTodd
KamleshLilaAmy
ChingLiz
SteveJoelGailKokoKari
Shaquira
Eduardo
Kamini
Jan
Paula
Todd
Kamlesh
Lila
Amy
Ching
Liz
Steve
Joel
Gail
Koko
Kari
Shaquira
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 1 0 0 1 0 0 0 0 1 0 0 0 01 0 1 0 1 0 1 1 0 1 0 1 0 0 0 00 0 0 1 0 1 0 1 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 1 0 0 0 0 0 0 00 0 1 1 0 0 0 0 0 1 0 1 1 0 0 00 0 0 1 1 0 0 0 0 1 1 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 1 1 0 0 1 0 0 0 0 00 0 0 0 1 0 0 1 0 1 0 0 0 1 0 10 0 1 1 0 0 1 0 0 0 0 0 1 0 1 00 0 0 0 0 0 1 0 0 0 0 1 0 0 0 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0
0 3 2 1 2 3 2 2 4 2 3 2 3 4 3 43 0 1 2 3 4 2 3 5 3 4 2 3 5 3 42 1 0 1 2 3 1 2 4 2 3 1 2 4 2 31 2 1 0 1 2 1 1 3 1 2 1 2 3 2 32 3 2 1 0 1 2 1 2 2 1 2 3 2 3 23 4 3 2 1 0 3 2 1 3 2 3 4 3 4 32 2 1 1 2 3 0 2 4 1 2 1 1 3 2 22 3 2 1 1 2 2 0 3 1 1 2 3 2 3 24 5 4 3 2 1 4 3 0 4 3 4 5 4 5 42 3 2 1 2 3 1 1 4 0 1 2 2 2 3 23 4 3 2 1 2 2 1 3 1 0 3 2 1 4 12 2 1 1 2 3 1 2 4 2 3 0 1 4 1 23 3 2 2 3 4 1 3 5 2 2 1 0 3 2 14 5 4 3 2 3 3 2 4 2 1 4 3 0 5 23 3 2 2 3 4 2 3 5 3 4 1 2 5 0 34 4 3 3 2 3 2 2 4 2 1 2 1 2 3 0
static void floyd(){ for (int k = 0; k < n; k++) { for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if(i!=j) W[i, j]=min(W[i,j],W[i,k],W[k,j]); } } }}
In an acquaintanceship graph people are represented as nodes. Nodes are connected by an edge when two people know each other. We can compute the degree of separation between any two people using Floyd's All-Pairs Shortest-Path Algorithm.
Simple Graphs vs Multigraphs
A simple graph is one in which each edge connects two different vertices and where on two edges connect the same pair of vertices.
Graphs that have multiple edges connecting the same pairs of vertices are called multigraphs or pseudographs.
An edge that connects a node to itself is called a loop.
A D F
C H
B E G
ring complete graph bipartite graph
directed acyclic graph
Types of Graphs
A D F
C H
B E G
Graph Breadth-First Traversal
Given a graph G(V,E) and a starting vertex s, performa breadth-first traversal (BFT) of G. such that eachreachable vertex is entered exactly once.
If all vertices are reachable, the edges traversed andthe set of vertices will represent a spanning treeembedded in the graph G.
1) BFT suggests an iterative process (rather than a recursive one)
2) BFT vertices order of traversal can be maintained using a Queue data structure
3) The preferred representation for the graph is an adjacency matrix
4) We will need a way to keep up with which vertices have been "used" (e.g. a Boolean list)
5) Process begins by placing the starting vertex in the Queue
6) A vertex is taken from the Queue, every unused vertex adj to this vertex is added to the Queue This operation is repeated until the Queue is empty.
8) The output (answer) is returned in the form of a list of vertices in the order they entered the Queue
Graph Depth-First Traversal
A D F
C H
B E G
Given a graph G(V,E) and a starting vertex s, performa depth-first traversal (BFT) of G. such that eachreachable vertex is entered exactly once.
If all vertices are reachable, the edges traversed andthe set of vertices will represent a spanning treeembedded in the graph G.
1) DFT suggests a recursive process (rather than an iterative one)
2) DFT vertices order of traversal are maintained automatically by the recursion process (as a Stack)
3) The preferred representation for the graph is an adjacency matrix.
4) We will need a way to keep up with which vertices have been "used" (e.g. a Boolean list)
5) Process begins by passing the starting vertex to a recursive function DFT(s)
6) For the current vertex, s DFT(s) calls itself for each adjacent, unused vertex remaining. This operation is completed when all calls to DFT( ) are completed.
8) The output is returned as a of a list of vertices in the order they were passed to DFT( ).