1. Discrete Mathematics and its Applications
9/17/2018
University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter
Slides adapted from Michael P. Frank’s Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen
A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. The instructor can bring several modules to each lecture with him, to make sure he has enough material to fill the lecture, or in case he wants to preview or review slides from upcoming or recent past lectures.
9/17/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

2. Rosen 5th ed., chs. 8-9 ~44 slides (more later), ~3 lectures
Module #22: Graph Theory
Rosen 5th ed., chs. 8-9
~44 slides (more later), ~3 lectures
9/17/2018
(c) , Michael P. Frank

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 mathematics: A particular class of discrete structures (to be defined) that is useful for representing relations and has a convenient graphical representation.
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(c) , Michael P. Frank

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
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(c) , Michael P. Frank

5. We shall introduce a number of different types of graphs, starting with undirected graphs:
“simple” graphs
multigraphs
pseudographs
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(c) , Michael P. Frank

6. Visual Representation of a Simple Graph
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 Representation of a Simple Graph
9/17/2018
(c) , Michael P. Frank

7. Example of a Simple Graph
Let V be the set of states in the far-southeastern U.S.:
I.e., V={FL, GA, AL, MS, LA, SC, TN, NC}
Let E={{u,v}|u 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 MS AL SC GA LA FL
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(c) , Michael P. Frank

8. Extensions
All the main types of graphs can be extended to make them more expressive
For example, edges may be labelled: Labelled graphs
E.g., the edges in the previous example may be labelled with the type of border (e.g., patrolled/non-patrolled)
9/17/2018
(c) , Michael P. Frank

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 function f:E{{u,v}|u,vV  uv}.
E.g., nodes are cities, edges are segments of major highways.
Parallel edges
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(c) , Michael P. Frank

10. Multigraphs extended
Extension: edges may be labelled with numbers of miles. How about this example?
110 A B 90 20 80 C
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(c) , Michael P. Frank

11. Discrete Mathematics and its Applications
9/17/2018
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 A B
Exercise: formulate a constraint on labels that prevents this kind of mishap. 90 20 80 C
9/17/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

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 ) where f:E{{u,v}|u,vV}. Edge eE is a loop if f(e)={u,u}={u}.
E.g., nodes are campsites in a state park, edges are hiking trails through the woods.
9/17/2018
(c) , Michael P. Frank

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}
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(c) , Michael P. Frank

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 f:EVV.
E.g., V=web pages, E=hyperlinks. The WWW as a directed multigraph...
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(c) , Michael P. Frank

15. Types of Graphs: Summary
Summary of the book’s definitions.
This terminology is not fully standardized across different authors
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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.
9/17/2018
(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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
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(c) , Michael P. Frank

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.
9/17/2018
(c) , Michael P. Frank

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?
9/17/2018
(c) , Michael P. Frank

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.
9/17/2018
(c) , Michael P. Frank

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
9/17/2018
(c) , Michael P. Frank

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 v’s in-degree and out-degree.
9/17/2018
(c) , Michael P. Frank

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.
9/17/2018
(c) , Michael P. Frank

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
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(c) , Michael P. Frank

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 K2 K3 K5 K6
Note that Kn has edges.
9/17/2018
(c) , Michael P. Frank

29. Consider a complete graph G=(V,E)
Consider a complete graph G=(V,E). Can E contain any edges connecting a node in V to itself?
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(c) , Michael P. Frank

30. No: this would mean {u,v}E, where u=v hence u,vV: uv{u,v}E.
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 u=v
hence u,vV: uv{u,v}E.
9/17/2018
(c) , Michael P. Frank

31. Can you think of a natural example where complete graphs might model the facts?
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(c) , Michael P. Frank

32. 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
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(c) , Michael P. Frank

33. Discrete Mathematics and its Applications
9/17/2018
Cycles
For any n3, a cycle on n vertices, Cn, is a simple graph where V={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}.
Number of edges equals n. C3 C4 C5 C6 C8 C7
How many edges are there in Cn?
9/17/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

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)
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(c) , Michael P. Frank

35. Can a cycle be a complete graph?
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(c) , Michael P. Frank

36. Discrete Mathematics and its Applications
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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.
Can be proven by proving with induction that |Edges(C_n)|<|Edges(K_n)| for n>2.
9/17/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

37. Discrete Mathematics and its Applications
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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}}.
The number is 2n W3 W4 W5 W6 W8 W7
How many edges are there in Wn?
9/17/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

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?
9/17/2018
(c) , Michael P. Frank

39. 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
9/17/2018
(c) , Michael P. Frank

40. Number of vertices: 2n. Number of edges:Exercise to try!
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 edges:Exercise to try!
9/17/2018
(c) , Michael P. Frank

41. n-cubes (hypercubes)
For any nN, the hypercube Qn can be defined recursively as follows:
Q0={{v0},} (one node and no edges)
For any nN, if Qn=(V,E), where V={v1,…,va} and E={e1,…,eb}, then Qn+1 = …
9/17/2018
(c) , Michael P. Frank

42. n-cubes (hypercubes)
For any nN, the hypercube Qn can be defined recursively as follows:
Q0={{v0},} (one node and no edges)
For any nN, if Qn=(V,E), where V={v1,…,va} and E={e1,…,eb}, then Qn+1= (V{v1´,…,va´}, E{e1´,…,eb´}{{v1,v1´},{v2,v2´},…, {va,va´}}) where v1´,…,va´ are new vertices, and where if ei={vj,vk} then ei´={vj´,vk´}.
9/17/2018
(c) , Michael P. Frank

43. Bipartite Graphs
Def’n.: A graph G=(V,E) is bipartite (two-part) iff V = V1  V2 where V1∩V2= and eE: v1V1,v2V2: e={v1,v2}.
In English: The graph can be divided into two parts in such a way that all edges go between the two parts. V1 V2
This definition can easily be adapted for the case of multigraphs and directed graphs as well.
Can represent with zero-one matrices.
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(c) , Michael P. Frank

44. Bipartite graphs
… are extremely common, for example when you’re 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
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(c) , Michael P. Frank

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?
9/17/2018
(c) , Michael P. Frank

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 it’s not complete
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(c) , Michael P. Frank

47. Given a (bipartite) graph, can there be more than 1 way of partitioning V into V1 and V2 ?
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(c) , Michael P. Frank

48. Bipartite Graphs
Given a (bipartite) graph, can there be more than 1 way of partitioning V into V1 and V2 ?
Yes: isolated vertices can be put in either part: V1 V2
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(c) , Michael P. Frank

49. Complete Bipartite Graphs
Discrete Mathematics and its Applications
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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,v2}|v1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
Note: not every CBG is complete! (So “CBG” is a strange name)
K_{m,n} has m+n nodes and n*m edges.
Km,n has _____ nodes and _____ edges.
9/17/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

50. Subgraphs
A subgraph of a graph G=(V,E) is a graph H=(W,F) where WV and FE. G H
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

52. §8.3: Graph Representations & Isomorphism
Adjacency lists.
Adjacency matrices. [Not covered this year]
Incidence matrices. [Not covered this year]
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(c) , Michael P. Frank

53. Adjacency Lists
A table with 1 row per vertex, listing its adjacent vertices. b a d c e f
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(c) , Michael P. Frank

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 V<>V’ and E<>E’) may in some sense be equivalent:
Graph isomorphism (informal):
Two graphs are isomorphic iff they are identical except for their node names.
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(c) , Michael P. Frank

55. How would you define graph isomorphism formally?
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
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(c) , Michael P. Frank

56. Graph Isomorphism Formal definition:
Simple graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic iff  a bijection f:V1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.
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(c) , Michael P. Frank

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 …
9/17/2018
(c) , Michael P. Frank

58. Graph Invariants under Isomorphism
Necessary but not sufficient conditions for G1=(V1, E1) to be isomorphic to G2=(V2, E2):
|V1|=|V2| and |E1|=|E2|.
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.
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(c) , Michael P. Frank

59. Isomorphism Example
If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. d b a b a d c e e c f f
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(c) , Michael P. Frank

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
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(c) , Michael P. Frank

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?”
“What’s the quickest route from a to b?”
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(c) , Michael P. Frank

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 u=v.
A path traverses the vertices along it.
A path is simple if it contains no edge more than once.
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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 )
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

66. §8.5: Euler & Hamilton Paths
We’ll show you the problem that prompted Euler to invent the theory of graphs: the bridges of Koenigsberg (town later called Kaliningrad)
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(c) , Michael P. Frank

67. Bridges of Königsberg Problem
Can we walk through town, crossing each bridge exactly once, and return to start?
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(c) , Michael P. Frank

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
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(c) , Michael P. Frank

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
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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 …
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(c) , Michael P. Frank

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
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.
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(c) , Michael P. Frank

73. Not all edges have even degree
… so there is no Euler circuit. A D B C
Equivalent multigraph
The original problem
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(c) , Michael P. Frank

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: it’s a constructive proof of an existential proposition
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(c) , Michael P. Frank

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. )
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(c) , Michael P. Frank

76. ( Hamiltonian Path Theorems
Dirac’s 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.
Ore’s corollary: If G is connected, simple, has n≥3 nodes, and deg(u)+deg(v)≥n for every pair u,v of non-adjacent nodes, then G has a Hamilton circuit. )
9/17/2018
(c) , Michael P. Frank

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. )
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(c) , Michael P. Frank

78. §8.6: Shortest-Path Problems
Not covering this year.
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79. §8.7: Planar Graphs Not covering this year. 9/17/2018
(c) , Michael P. Frank

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 ≥ ___ ).
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(c) , Michael P. Frank

81. Tree and Forest Examples
Leaves in green, internal nodes in brown.
A Tree:
A Forest:
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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.
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(c) , Michael P. Frank

83. Same tree except for choice of root
Rooted Tree Examples
Note that a given unrooted tree with n nodes yields n different rooted trees.
Same tree except for choice of root
root
root
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(c) , Michael P. Frank

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 p j k
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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 ≤ ___.
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(c) , Michael P. Frank

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 n>2, can use terms like “leftmost”, “rightmost,” etc.
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(c) , Michael P. Frank

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?
9/17/2018
(c) , Michael P. Frank

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 n=mi+1 nodes, and =(m−1)i+1 leaves.
Proof: There are mi children of internal nodes, plus the root. And,  = n−i = (m−1)i+1. □
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.
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(c) , Michael P. Frank

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.
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(c) , Michael P. Frank

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

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 x’s left subtree is less than x.
Every node in x’s right subtree is greater than x.
Example: 7 3 12 1 5 9 15 2 8 11
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(c) , Michael P. Frank

94. Recursive Binary Tree Insert
procedure insert(T: binary tree, x: item) v := root[T] if v = null then begin root[T] := x; return “Done” end else if v = x return “Already present” else if x < v then return insert(leftSubtree[T], x) else {must be x > v} return insert(rightSubtree[T], x)
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(c) , Michael P. Frank

95. Decision Trees (pp )
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.
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(c) , Michael P. Frank

96. Coin-Weighing Problem
Imagine you have 8 coins, one of 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? ?
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(c) , Michael P. Frank

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 of weighings thus yields a decision tree with branching factor 3.
The balance then “decides” whether to tip left, tip right, or stay balanced.
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(c) , Michael P. Frank

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 height h ≥ log38 = 1.893… = 2.
Let’s see if we solve the problem with only 2 weighings…
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(c) , Michael P. Frank

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 what’s 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.
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(c) , Michael P. Frank

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.
You can prove that this strategy always leads to a balanced 3-ary tree.
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(c) , Michael P. Frank

101. Coin Balancing Decision Tree
Here’s what the tree looks like in our case:
123 vs 456
left:
123
balanced: 78
right:
456
1 vs. 2
4 vs. 5
7 vs. 8
L:1
R:2
B:3
L:4
R:5
B:6
L:7
R:8
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(c) , Michael P. Frank

102. Prefix Codes & Huffman Codingpp
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103. Game Trees pp
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104. §9.3: Tree Traversal Universal address systems Traversal algorithms
Depth-first traversal:
Preorder traversal
Inorder traversal
Postorder traversal
Breadth-first traversal
Infix/prefix/postfix notation
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105. §9.4: Spanning Trees Not covering this semester. 9/17/2018
(c) , Michael P. Frank

106. §9.5: Minimum Spanning Trees
Not covering this semester.
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(c) , Michael P. Frank