1. CS1022 Computer Programming & Principles
Lecture 7.2
Graphs (2)

2. Plan of lecture
Hamiltonian graphs
Trees
Sorting and searching
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3. Hamiltonian graphs (1)
Euler looked into the problem of using all edges once (visiting vertices as many times as needed)
Interesting related problem:
Find a cycle which passes through all vertices once
A Hamiltonian cycle includes all vertices in a graph
Hamiltonian graphs have Hamiltonian cycles
Useful for planning train timetables
Useful for studying telecommunications
W. R. Hamilton
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4. Hamiltonian graphs (2)
Unlike the Eulerian problem, there is no simple rule for detecting Hamiltonian cycles
One of the major unsolved problems in graph theory
Many graphs are Hamiltonian
If each vertex is adjacent (has an edge) to every other vertex, then there is always a Hamiltonian cycle
These are called complete graphs
A complete graph with n vertices is denoted Kn
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5. Hamiltonian graphs (3) a b e c d Example: complete graph K5
A Hamiltonian cycle is a b c d e a
There are several others
Since each vertex is adjacent to every other vertex
We have 4 options to move to the 2nd vertex, then
We have 3 options to move to the 3rd vertex, then
We have 2 options to move to the 4th vertex, then
We have 1 option to move to the 5th vertex
That is, 4  3  2  1  4!  24 cycles
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6. Hamiltonian graphs (4)
Finding Hamiltonian cycles in an arbitrary graph is not straightforward
Deciding if a graph is Hamiltonian is can be quite demanding
Problem:
Input a graph G  (V, E)
Analyse G
If it is Hamiltonian output YES, otherwise output NO
The test “if it is Hamiltonian” is not straightforward
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7. Travelling salesperson problem (1)
Hamiltonian graphs model many practical problems
Classic problem: travelling salesperson
A salesperson wishes to visit a number of towns connected by roads
Find a route visiting each town exactly once, and keeping travelling costs to a minimum
The graph modelling the problem is Hamiltonian
Vertices are town and edges are roads
Additionally, edges have a weight to represent cost of travel along road (e.g., petrol, time/distance it takes)
Search a Hamiltonian cycle of minimal total weight
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8. Travelling salesperson problem (2)
No efficient algorithm to solve problem
Complex graphs have too many Hamiltonian cycles
They all have to be considered, in order to find the one with minimal total weight
There are algorithms for sub-optimal solutions
Sub-optimal: not minimal, but considerably better than an arbitrary choice of Hamiltonian cycle
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9. Travelling salesperson problem (3)
Nearest neighbour (sub-optimal) algorithm
begin
choose v  V;
route := v; w := 0; v:= v; % initialise variables
mark v;
while unmarked vertices remain do
choose an unmarked vertex u closest to v;
route := route u; % append u to end of route
w := w  (weight of edge vu); % update weight of route so far
v:= u; % update current vertex
mark v; % mark current vertex
end
route := route v; % append origin to close cycle
w := w  (weight of edge vv);
output (route, w)
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10. Travelling salesperson problem (4)
Trace nearest neighbour (sub-optimal) algorithm
begin
choose v  V;
route := v; w := 0; v:= v;
mark v;
while unmarked vertices remain do
choose an unmarked vertex u closest to v;
route := route u;
w := w  (weight of edge vu);
v:= u;
end
route := route v;
w := w  (weight of edge vv);
output (route, w) A B D C 5 7 8 3 6 10 u
route w v
Initially – D C DC 3 A
DCA 9 B
DCAB 14 u
route w v
Initially – D C DC 3 A
DCA 9 B
DCAB 14
Exit loop
DCABD 24 u
route w v
Initially – D C DC 3 A
DCA 9 u
route w v
Initially – D C DC 3 u
route w v
Initially – D
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11. Travelling salesperson problem (5)
“Nearest”  “with lower weight on edge”
Exhaustive search finds 2 other solutions:
ABCDA (total weight 23)
ACBDA (total weight 31)
It is not the best solution, but it’s better than 31
A complete graph with 20 vertices has 6  1016 Hamiltonian cycles
Enumerating all would take too
much time and memory A B D C 5 7 8 3 6 10
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12. Trees (1) Special type/class of graphs called trees
Very popular in computing applications/solutions
A tree is a connected and acyclic graph G  (V, E)
In the literature, trees are drawn upside down A B D C E F
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13. Trees (2) Let G  (V, E) be a tree, |V|  n, |E|  m
We can state (all are equivalent)
There is exactly one path between any vertices of G
G is connected and m  n – 1
G is connected and the removal of one single edge disconnects it
G is acyclic and adding a new edge creates a cycle
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14. Spanning trees (1) Any connected graph G contains trees as sub-graphs
A sub-graph of G which is a tree and includes all vertices is a spanning tree
It is straightforward to build a spanning tree:
Select an edge of G
Add further edges of G without creating cycles
Do 2 until no more edges can be added (w/o creating cycle)
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15. Spanning trees (2) Find two spanning trees for the graph a b e c d f g
Solution 1
Solution 2 a b b d c e f g
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16. Minimal spanning tree Process adapted for minimum connector problem:
A railway network connecting many towns is to be built
Given the costs of linking 2 towns, find a network of minimal total cost
Spanning tree for a graph with weighted edges, with minimal total weight
This is called minimal spanning tree (MST)
Unlike the travelling salesperson, we have efficient algorithms to solve this problem
We can find the optimal solution!
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17. Minimal spanning tree algorithm (1)
G  (V, E) is a connected graph with weighted edges
Algorithm finds MST for G by successively selecting edges of least possible weight to build an MST
MST is stored as a set T of edges
begin
e := an edge of E of smallest weight;
T := e;
E:= E –e;
while E   do
e := an edge of E of smallest weight;
T := T  e;
E:= set of edges in (E – T) which do not create cycles if added to T;
end
output T;
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18. Rooted trees (1)
We often need to represent information which is naturally hierarchical
Example: family trees
We make use of rooted trees
A special vertex is called the root of the tree
The root of the tree has unique features
Oldest, youngest, smallest, highest, etc. A B D C E F
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19. ... Rooted trees (2) Rooted trees defined recursively:
A single vertex is a tree (with that vertex as root)
If T1, T2, , Tk are disjoint trees with roots v1, v2, , vk we can “attach” a new vertex v to each vi to form a new tree T with root v v
Each vertex in a rooted tree T forms the root of another rooted tree which we call a subtree of T T1 v1 T2 v2 Tk vk
...
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20. Rooted trees (3)
Top vertex is the root and vertices at bottom of tree (those with no children) are called leaves
Vertices other than root or leaves are called internal vertices
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21. Binary (rooted) trees Rooted trees used as models in many areas
Computer science, biology, management
Very important in computing: binary rooted trees
Each vertex has at most two children
Subtrees: left- and right subtrees of the vertex
A missing subtree is called a null tree v
Left
Right
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22. Sorting and searching (1)
Binary rooted trees are useful to support decisions, especially those requiring sorting/searching data
Ordered numbers, strings ordered lexicographically
Ordered data stored as vertices of binary tree
Data in left-subtree less than data item stored in v
Data in right-subtree greater than data item stored in v
These are called binary search trees v
<
Left
>
Right
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23. Sorting and searching (2)
Example of binary search tree with words
MY COMPUTER HAS A CHIP ON ITS SHOULDER MY
COMPUTER ON
SHOULDER A
HAS
ITS
CHIP
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24. Sorting and searching (3)
Binary search trees allow efficient algorithms for
Searching for data items
Inserting new data items
Printing all data in an ordered fashion
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25. Binary search (1) Algorithm to find (or not) item in binary tree
search(x, tree)
begin
if tree  null then
return false
else
Let tree be of form (left_subtree, root, right_subtree)
if x  root then
return true
if x  root then
return search(x, left_subtree)
return search(x, right_subtree)
end
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26. Binary search (2) Algorithm to find (or not) item in binary tree K C T
search(x, tree)
begin
if tree  null then
return false
else
Let tree be of form (left_subtree, root, right_subtree)
if x  root then
return true
if x  root then
return search(x, left_subtree)
return search(x, right_subtree)
End K C T V M K C T V M
search(R, )
left_sub= root= right_sub = C K T V M
search(R, ) T V M
left_sub= root= right_sub = M T V
search(R, ) V
left_sub= null root= right_sub = null V
search(R, null)
false
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27. Further reading
R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd (Chapter 7)
Wikipedia’s entry on graph theory
Wikibooks entry on graph theory
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