University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter

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Title: University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter

1
University of Aberdeen, Computing
ScienceCS3511Discrete MethodsKees van Deemter

Slides adapted from Michael P. Franks Course
Based on the TextDiscrete Mathematics Its
Applications (5th Edition)by Kenneth H. Rosen

2
Module 22Graph Theory

Rosen 5th ed., chs. 8-9
44 slides (more later), 3 lectures

3
What are Graphs?
Not

General meaning in everyday math A plot or
chart of numerical data using a coordinate
system.
Technical meaning in discrete mathematicsA
particular class of discrete structures (to be
defined) that is useful for representing
relations and has a convenient graphical
representation.

4
Applications of Graphs

(At least) anything that can be modelled using 1-
or 2-place relations.
Apps in networking, scheduling, flow
optimization, circuit design, path planning,
search genealogy, any kind of taxonomy,
Challenge name a topic that graphs cannot model

We shall introduce a number of different types of
graphs, starting with undirected graphs
simple graphs
multigraphs
pseudographs

6
Simple Graphs

Correspond to symmetric,irreflexive binary
relations R.
A simple graph G(V,E)consists of
a set V of vertices or nodes (V corresponds to
the universe of the relation R),
a set E of edges unordered pairs of distinct
elements u,v ? V, such that uRv.

Visual Representationof a Simple Graph
7
Example of a Simple Graph

Let V be the set of states in the
far-southeastern U.S.
I.e., VFL, GA, AL, MS, LA, SC, TN, NC
Let Eu,vu adjoins v
FL,GA,FL,AL,FL,MS, FL,LA,GA,AL,AL,
MS, MS,LA,GA,SC,GA,TN,
SC,NC,NC,TN,MS,TN, MS,AL

NC
TN
SC
MS
AL
GA
LA
FL
8
Extensions

All the main types of graphs can be extended to
make them more expressive
For example, edges may be labelledLabelled
graphs
E.g., the edges in the previous example may be
labelled with the type of border (e.g.,
patrolled/non-patrolled)

9
Multigraphs

Like simple graphs, but there may be more than
one edge connecting two given nodes.
A multigraph G(V, E, f ) consists of a set V of
vertices, a set E of edges (as primitive
objects), and a functionfE?u,vu,v?V ? u?v.
E.g., nodes are cities, edgesare segments of
major highways.

Paralleledges
10
Multigraphs extended

Extension edges may be labelled with numbers of
miles. How about this example?

110
B
A
90
80
20
C
11
Multigraphs extended

Can go from A to B via C in less than 110.The
distances in the graph are not minimal,
or something
is wrong.

110
B
A
90
80
20
C
12
Pseudographs

Like a multigraph, but edges connecting a node to
itself are allowed. (R may even be reflexive.)
A pseudograph G(V, E, f ) wherefE?u,vu,v?V
. Edge e?E is a loop if f(e)u,uu.
E.g., nodes are campsitesin a state park, edges
arehiking trails through the woods.

13
Directed Graphs

Correspond to arbitrary binary relations R, which
need not be symmetric.
A di(rected) graph (V,E) consists of a set of
vertices V and a binary relation E on V.
E.g. V set of People,E(x,y) x loves y

14
Directed Multigraphs

Like directed graphs, but there may be more than
one edge from a node to another.
A directed multigraph G(V, E, f ) consists of a
set V of vertices, a set E of edges, and a
function fE?V?V.
E.g., Vweb pages,Ehyperlinks. The WWW asa
directed multigraph...

15
Types of Graphs Summary

Summary of the books definitions.
This terminology is not fully standardized across
different authors

16
8.2 Graph Terminology

Introducing the following terms
Adjacent, connects, endpoints, degree, initial,
terminal, in-degree, out-degree, complete,
cycles, wheels, n-cubes, bipartite, subgraph,
union.

17
Adjacency

Let G be an undirected graph with edge set E.
Let e?E be (or map to) the pair u,v. Then we
say
u, v are adjacent / connected.
Edge e is incident with vertices u and v.
Edge e connects u and v.
Vertices u and v are endpoints of edge e.

18
Degree of a Vertex

Let G be an undirected graph, v?V a vertex.
The degree of v, deg(v), is its number of
incident edges. (Except that any self-loops are
counted twice.)
A vertex with degree 0 is called isolated.
A vertex of degree 1 is called pendant.

19
Degree of a Vertex

The degree of v, deg(v), is its number of
incident edges. (Except that any self-loops are
counted twice.)
Exercise construct a pseudograph with three
vertices, all of which have different degrees.

20
Handshaking Theorem

Let G be an undirected (simple, multi-, or
pseudo-) graph with vertex set V and edge set E.
Then
Proof every edge causes degree degree 2

21
Handshaking Theorem

Let G be an undirected (simple, multi-, or
pseudo-) graph with vertex set V and edge set E.
Then
Corollary Any undirected graph has an even
number of vertices of odd degree.

22
Sociological example

(Like previous exercise, this is from Jonathan
L.Gross coursenotes to Rosen)
Suppose the students in this class are
represented as vertices. An edge between a and b
means that a and b were acquainted before the
course began. This is a simple graph.
How many students knew an odd number of other
students? Could there be 3?

23
Sociological example

(Like previous exercise, this is from Jonathan
L.Gross coursenotes to Rosen)
Suppose the students in this class are
represented as vertices. An edge between a and b
means that a and b were acquainted before the
course began. This is a simple graph.
By corollary to handshaking theorem The
number of students who knew an odd number of
other students is even.

24
Directed Adjacency

Let G be a directed (possibly multi-) graph, and
let e be an edge of G that is (or maps to) (u,v).
Then we say
u is adjacent to v, v is adjacent from u
e comes from u, e goes to v.
e connects u to v, e goes from u to v
the initial vertex of e is u
the terminal vertex of e is v

25
Directed Degree

Let G be a directed graph, v a vertex of G.
The in-degree of v, deg?(v), is the number of
edges going to v.
The out-degree of v, deg?(v), is the number of
edges coming from v.
The degree of v, deg(v)?deg?(v)deg?(v), is the
sum of vs in-degree and out-degree.

26
Directed Handshaking Theorem

Let G be a directed (possibly multi-) graph with
vertex set V and edge set E. Then
Note that the degree of a node is unchanged by
whether we consider its edges to be directed or
undirected.

27
Special Graph Structures

Special cases of undirected graph structures
Complete graphs Kn
Cycles Cn
Wheels Wn
n-Cubes Qn
Bipartite graphs
Complete bipartite graphs Km,n

28
Complete Graphs

For any n?N, a complete graph on n vertices, Kn,
is a simple graph with n nodes in which every
node is adjacent to every other node ?u,v?V
u?v?u,v?E.

K1
K4
K3
K2
K5
K6
Note that Kn has edges.
29

Consider a complete graph G(V,E). Can E
contain any edges connecting a node in V to
itself?

Consider a complete graph G(V,E). Can E
contain any edges connecting a node in V to
itself?
No this would mean u,v?E, where uv
hence ??u,v?V u?v?u,v?E.

Can you think of a natural example where complete
graphs might model the facts?

Can you think of a natural example where complete
graphs might model the facts?
E.g., Graphs that model, for groups of n people,
which people are seen by each member of the group

33
Cycles

For any n?3, a cycle on n vertices, Cn, is a
simple graph where Vv1,v2, ,vn and
Ev1,v2,v2,v3,,vn?1,vn,vn,v1.

C3
C4
C5
C6
C8
C7
How many edges are there in Cn?
34

Can you think of a natural example where cycles
might model the facts?
E.g., Graphs that model, for groups of n people,
which people are sitting next to each other (on a
round table)

Can a cycle be a complete graph?

Can a cycle be a complete graph?
Yes every cycle with exactly 3 elements is a
complete graph.
No other cycle can be a complete graph.

37
Wheels

For any n?3, a wheel Wn, is a simple graph
obtained by taking the cycle Cn and adding one
extra vertex vhub and n extra edges vhub,v1,
vhub,v2,,vhub,vn.

W3
W4
W5
W6
W8
W7
How many edges are there in Wn?
38

Definition A graph is regular iff every vertex
has the same degree
Which of these are regular? (What degree?)
Complete graphs?
Cycle graphs?
Wheel graphs?

A graph is regular iff every vertex has the same
degree
Which of these are regular? (What degree?)
Complete graphs? Yes degree n-1 (for n nodes)
Cycle graphs? Yes degree 2
Wheel graphs? No, except when they have 3 nodes

40
n-cubes (hypercubes)

For any n?N, the hypercube Qn is a simple graph
consisting of two copies of Qn-1 connected
together at corresponding nodes. Q0 has 1 node.

Q0
Q1
Q2
Q4
Q3
Number of vertices 2n. Number of edgesExercise
to try!
41
n-cubes (hypercubes)

For any n?N, the hypercube Qn can be defined
recursively as follows
Q0v0,? (one node and no edges)
For any n?N, if Qn(V,E), where Vv1,,va and
Ee1,,eb, then Qn1

42
n-cubes (hypercubes)

For any n?N, the hypercube Qn can be defined
recursively as follows
Q0v0,? (one node and no edges)
For any n?N, if Qn(V,E), where Vv1,,va and
Ee1,,eb, then Qn1 (V?v1,,va,
E?e1,,eb?v1,v1,v2,v2,,va,va)
where v1,,va are new vertices, and where if
eivj,vk then eivj,vk.

43
Bipartite Graphs

Defn. A graph G(V,E) is bipartite (two-part)
iff V V1 ? V2 where V1nV2? and ?e?E
?v1?V1,v2?V2 ev1,v2.
In English The graph canbe divided into two
partsin such a way that all edges go between
the two parts.

V2
V1
This definition can easily be adapted for the
case of multigraphs and directed graphs as well.
Can represent withzero-one matrices.
44
Bipartite graphs

are extremely common, for example when youre
modelling a domain that consists of two different
kinds of entities
Animals in a zoo, linked with their keepers
Words, linked with numbers of letters in them
Logical formulas, linked with English sentences
that express their meaning

45
Some questions

Can you think of a graph with two vertices that
is not bipartite?
Can you think of a simple graph with two vertices
that is not bipartite?
Can you think of a simple graph with three
vertices and a nonempty set of edges that is
bipartite?

46
Some questions

Can you think of a graph with two vertices that
is not bipartite? Yes if there are self-loops
Can you think of a simple graph with two vertices
that is not bipartite? No must be bipartite
Can you think of a simple graph with three
vertices and a nonempty set of edges that is
bipartite? Yes as long as its not complete

Given a (bipartite) graph, can there be more than
1 way of partitioning V into V1 and V2 ?

48
Bipartite Graphs

Given a (bipartite) graph, can there be more than
1 way of partitioning V into V1 and V2 ?
Yes isolated verticescan be put in either part

V2
V1
49
Complete Bipartite Graphs

For m,n?N, the complete bipartite graph Km,n is a
bipartite graph where V1 m, V2 n, and E
v1,v2v1?V1 ? v2?V2.
That is, there are m nodes in the left part, n
nodes in the right part, and every node in the
left part is connected to every node in the
right part.

K4,3
Km,n has _____ nodesand _____ edges.
50
Subgraphs

A subgraph of a graph G(V,E) is a graph H(W,F)
where W?V and F?E.

G
H
51

A subgraph of a graph G(V,E) is a graph H(W,F)
where W?V and F?E.
Note since H is a graph, F can only involve
nodes that are elements of W.

52
8.3 Graph Representations Isomorphism

Graph representations
Adjacency lists.
Adjacency matrices. Not covered this year
Incidence matrices. Not covered this year

53
Adjacency Lists

A table with 1 row per vertex, listing its
adjacent vertices.

b
a
d
c
e
f
54
An equivalence relation between graphs

Unlike ordinary pictures, we can say precisely
when two graphs are similar
Obviously, a given graph (V,E) may be drawn in
different ways.
But even (V,E) and (V,E) (where VltgtV and
EltgtE) may in some sense be equivalent
Graph isomorphism (informal)
Two graphs are isomorphic iff they are identical
except for their node names.

Graphs that are isomorphic share all their
important properties, e.g.,
The number of nodes and edges
The degrees of all their nodes
Whether they are bipartite or not, etc.
How would you define graph isomorphism formally?
For simplicity focus on simple graphs
Hint use the notion of a bijection

56
Graph Isomorphism

Formal definition
Simple graphs G1(V1, E1) and G2(V2, E2) are
isomorphic iff ? a bijection fV1?V2 such that ?
a,b?V1, a and b are adjacent in G1 iff f(a) and
f(b) are adjacent in G2.
f is the renaming function between the two node
sets that makes the two graphs identical.
This definition can be extended to other types of
graphs.

57
Graph Isomorphism

How can we tell whether two graphs are
isomorphic?
The best algorithms that are known to solve this
problem have exponential worst-case time
complexity. (Faster solutions may be possible.)
In practice, a few tests go a long way

58
Graph Invariants under Isomorphism

Necessary but not sufficient conditions for
G1(V1, E1) to be isomorphic to G2(V2, E2)
V1V2 and E1E2.
The number of vertices with degree n is the same
in both graphs.
For every proper subgraph g of one graph, there
is a proper subgraph of the other graph that is
isomorphic to g.

59
Isomorphism Example

If isomorphic, label the 2nd graph to show the
isomorphism, else identify difference.

d
b
b
a
a
d
c
e
f
e
c
f
60
Are These Isomorphic?

If isomorphic, label the 2nd graph to show the
isomorphism, else identify difference.

Same of vertices

a
b

Same of edges

Different of verts of degree 2! (1 vs 3)

d
e
c
61

We now jump to the area that gave raise to the
invention of (a formal theory of) graphs
Questions like
Can I travel from a to b?
Can I go travel a to b without going anywhere
twice?
Whats the quickest route from a to b?

62
8.4 Connectivity

In an undirected graph, a path of length n from u
to v is a sequence of adjacent edges going from
vertex u to vertex v.
A path is a circuit if uv.
A path traverses the vertices along it.
A path is simple if it contains no edge more than
once.

63
Connectedness

An undirected graph is connected iff there is a
path between every pair of distinct vertices in
the graph.
Theorem There is a simple path between any pair
of vertices in a connected undirected graph.

64
( Directed Connectedness

A directed graph is strongly connected iff there
is a directed path from a to b for any two verts
a and b.
It is weakly connected iff the underlying
undirected graph (i.e., with edge directions
removed) is connected.
)

65
Paths Isomorphism

Note that connectedness, and the existence of a
circuit or simple circuit of length k are graph
invariants with respect to isomorphism.

66
8.5 Euler Hamilton Paths

Well show you the problem that prompted Euler to
invent the theory of graphs the bridges of
Koenigsberg (town later called Kaliningrad)

67
Bridges of Königsberg Problem

Can we walk through town, crossing each bridge
exactly once, and return to start?

68
Bridges of Königsberg Problem

Can we walk through town, crossing each bridge
exactly once, and return to start?

A
Can you model the
situation using a graph?
D
B
C
The original problem
69
Bridges of Königsberg Problem

Can we walk through town, crossing each bridge
exactly once, and return to start?

A
D
B
C
Equivalent multigraph
The original problem
70
8.5 Euler Hamilton Paths

Terminology
An Euler circuit in a graph G is a simple circuit
containing every edge of G.
An Euler path in G is a simple path containing
every edge of G.

71
Bridges of Koenigsberg

Bridges are edges.So the answer to the problem
is YES iff its graph contains an Euler circuit.
In fact, it does not

72
Euler Path Theorems

Theorem A finite connected multigraph has an
Euler circuit iff each vertex has even degree.
Proof
(?) The circuit contributes 2 to degree of each
node.
(?) By construction using algorithm on p. 580-581
Theorem A connected multigraph has an Euler
path iff it has exactly 2 vertices of odd degree.
One is the start, the other is the end.

73
Not all edges have even degree

so there is no Euler circuit.

A
D
B
C
Equivalent multigraph
The original problem
74
Euler Circuit theorem

Sketch of proof that even degree implies
existence of Euler circuit
Start with any arbitrary node. Construct simple
path from it till you get back to start. (Graph
is connected every node has even degree, so you
can leave any node that you have entered)
Repeat for each remaining subgraph, splicing
results back into original cycle. Finiteness of
graph implies that this process must end.
Note that the complete version of this proof
provides an algorithm its a constructive proof
of an existential proposition

75
( Hamilton Paths

An Euler circuit in a graph G is a simple circuit
containing every edge of G.
An Euler path in G is a simple path containing
every edge of G.
A Hamilton circuit is a circuit that traverses
each vertex in G exactly once.
A Hamilton path is a path that traverses each
vertex in G exactly once. )

76
( Hamiltonian Path Theorems

Diracs theorem If (but not only if) G is
connected, simple, has n?3 vertices, and ?v
deg(v)?n/2, then G has a Hamilton circuit.
Ores corollary If G is connected, simple, has
n3 nodes, and deg(u)deg(v)n for every pair u,v
of non-adjacent nodes, then G has a Hamilton
circuit. )

77
( HAM-CIRCUIT is NP-complete

Let HAM-CIRCUIT be the problem
Given a simple graph G, does G contain a
Hamiltonian circuit?
This problem has been proven to be NP-complete!
This means, if an algorithm for solving it in
polynomial time were found, it could be used to
solve all NP problems in polynomial time. )

78
8.6 Shortest-Path Problems

Not covering this year.

79
8.7 Planar Graphs

Not covering this year.

80
9.1 Introduction to Trees

A tree is a connected undirected graph that
contains no circuits.
Theorem There is a unique simple path between
any two of its nodes.
A (not-necessarily-connected) undirected graph
without simple circuits is called a forest.
You can think of it as a set of trees having
disjoint sets of nodes.
A leaf node in a tree or forest is any pendant or
isolated vertex. An internal node is any
non-leaf vertex (thus it has degree ___ ).

81
Tree and Forest Examples
Leaves in green, internal nodes in brown.

A Tree

A Forest

82
Rooted Trees

A rooted tree is a tree in which one node has
been designated the root.
Every edge is (implicitly or explicitly) directed
away from the root.
Concepts related to rooted trees
Parent, child, siblings, ancestors, descendents,
leaf, internal node, subtree.

83
Rooted Tree Examples

Note that a given unrooted tree with n nodes
yields n different rooted trees.

Same tree exceptfor choiceof root
root
root
84
Rooted-Tree Terminology Exercise

Find the parent,children, siblings,ancestors,
descendants of node f.

o
n
h
r
d
m
b
root
a
c
g
e
q
i
f
l
j
k
p
85
n-ary trees

A rooted tree is called n-ary if every vertex has
no more than n children.
It is called full if every internal (non-leaf)
vertex has exactly n children.
A 2-ary tree is called a binary tree.
These are handy for describing sequences of
yes-no decisions.
Example Comparisons in binary search algorithm.

86
Which Tree is Binary?

Theorem A given rooted tree is a binary tree iff
every node other than the root has degree ___,
and the root has degree ___.

87
Ordered Rooted Tree

This is just a rooted tree in which the children
of each internal node are ordered.
In ordered binary trees, we can define
left child, right child
left subtree, right subtree
For n-ary trees with ngt2, can use terms like
leftmost, rightmost, etc.

88
Trees as Models

Can use trees to model the following
Saturated hydrocarbons
Organizational structures
Computer file systems
In each case, would you use a rooted or a
non-rooted tree?

89
Some Tree Theorems

Any tree with n nodes has e n-1 edges.
Proof Consider removing leaves.
A full m-ary tree with i internal nodes has
nmi1 nodes, and ?(m-1)i1 leaves.
Proof There are mi children of internal nodes,
plus the root. And, ? n-i (m-1)i1. ?
Thus, when m is known and the tree is full, we
can compute all four of the values e, i, n, and
?, given any one of them.

90
Some More Tree Theorems

Definition The level of a node is the length of
the simple path from the root to the node.
The height of a tree is maximum node level.
A rooted m-ary tree with height h is called
balanced if all leaves are at levels h or h-1.
Theorem There are at most mh leaves in an m-ary
tree of height h.
Corollary An m-ary tree with ? leaves has
height h?logm?? . If m is full and balanced
then h?logm??.

91
9.2 Applications of Trees

Binary search trees
A simple data structure for sorted lists
Decision trees
Minimum comparisons in sorting algorithms
Prefix codes
Huffman coding
Game trees

92
Binary Search Trees

A representation for sorted sets of items.
Supports the following operations in T(log n)
average-case time
Searching for an existing item.
Inserting a new item, if not already present.
Supports printing out all items in T(n) time.
Note that inserting into a plain sequence ai
would instead take T(n) worst-case time.

93
Binary Search Tree Format

Items are stored at individual tree nodes.
We arrange for the tree to always obey this
invariant
For every item x,
Every node in xs left subtree is less than x.
Every node in xs right subtree is greater than x.

Example
7
3
12
1
5
9
15
0
2
8
11
94
Recursive Binary Tree Insert

procedure insert(T binary tree, x item)v
rootTif v null then begin rootT x
return Done endelse if v x return Already
presentelse if x lt v then return
insert(leftSubtreeT, x)else must be x gt
v return insert(rightSubtreeT, x)

95
Decision Trees (pp. 646-649)

A decision tree represents a decision-making
process.
Each possible decision point or situation is
represented by a node.
Each possible choice that could be made at that
decision point is represented by an edge to a
child node.
In the extended decision trees used in decision
analysis, we also include nodes that represent
random events and their outcomes.

96
Coin-Weighing Problem

Imagine you have 8 coins, oneof which is a
lighter counterfeit, and a free-beam balance.
No scale of weight markings is required for this
problem!
How many weighings are needed to guarantee that
the counterfeit coin will be found?

?
97
As a Decision-Tree Problem

In each situation, we pick two disjoint and
equal-size subsets of coins to put on the scale.

A given sequence ofweighings thus yieldsa
decision tree withbranching factor 3.
The balance thendecides whether to tip left,
tip right, or stay balanced.
98
Applying the Tree Height Theorem

The decision tree must have at least 8 leaf
nodes, since there are 8 possible outcomes.
In terms of which coin is the counterfeit one.
Recall the tree-height theorem, h?logm??.
Thus the decision tree must have heighth
?log38? ?1.893? 2.
Lets see if we solve the problem with only 2
weighings

99
General Solution Strategy

The problem is an example of searching for 1
unique particular item, from among a list of n
otherwise identical items.
Somewhat analogous to the adage of searching for
a needle in haystack.
Armed with our balance, we can attack the problem
using a divide-and-conquer strategy, like whats
done in binary search.
We want to narrow down the set of possible
locations where the desired item (coin) could be
found down from n to just 1, in a logarithmic
fashion.
Each weighing has 3 possible outcomes.
Thus, we should use it to partition the search
space into 3 pieces that are as close to
equal-sized as possible.
This strategy will lead to the minimum possible
worst-case number of weighings required.

100
General Balance Strategy

On each step, put ?n/3? of the n coins to be
searched on each side of the scale.
If the scale tips to the left, then
The lightweight fake is in the right set of ?n/3?
n/3 coins.
If the scale tips to the right, then
The lightweight fake is in the left set of ?n/3?
n/3 coins.
If the scale stays balanced, then
The fake is in the remaining set of n - 2?n/3?
n/3 coins that were not weighed!
Except if n mod 3 1 then we can do a little
better by weighing ?n/3? of the coins on each
side.