1. Discrete Mathematics and its Applications
10/14/2018
University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank
Slides for a 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.
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(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
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(c) , Michael P. Frank

3. What are Graphs? Not Our Meaning
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 webby-looking graphical representation.
Not Our
Meaning
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4. Applications of Graphs
Potentially anything (graphs can represent relations, relations can describe the extension of any predicate).
Apps in networking, scheduling, flow optimization, circuit design, path planning.
More apps: Geneology analysis, computer game-playing, program compilation, object-oriented design, …
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(c) , Michael P. Frank

5. 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 / arcs / links: unordered pairs of [distinct] elements u,v  V, such that uRv.
Visual Representation of a Simple Graph
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(c) , Michael P. Frank

6. 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

7. 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

8. Pseudographs
Like a multigraph, but edges connecting a node to itself are allowed. (R may 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.
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(c) , Michael P. Frank

9. Directed Graphs
Correspond to arbitrary binary relations R, which need not be symmetric.
A directed 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

10. Directed Multigraphs
Like directed graphs, but there may be more than one arc 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 is a directed multigraph...
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11. Types of Graphs: Summary
Summary of the book’s definitions.
Keep in mind this terminology is not fully standardized across different authors...
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12. §8.2: Graph Terminology You need to learn 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

13. 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 / neighbors / 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

14. 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.
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(c) , Michael P. Frank

15. 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.
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(c) , Michael P. Frank

16. 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
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(c) , Michael P. Frank

17. 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.
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(c) , Michael P. Frank

18. 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.
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(c) , Michael P. Frank

19. 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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20. 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.
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(c) , Michael P. Frank

21. How many edges are there in Cn?
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}}. C3 C4 C5 C6 C8 C7
How many edges are there in Cn?
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(c) , Michael P. Frank

22. How many edges are there in Wn?
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?
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(c) , Michael P. Frank

23. 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!
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(c) , Michael P. Frank

24. 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´}.
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(c) , Michael P. Frank

25. 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

26. 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
Km,n has _____ nodes and _____ edges.
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(c) , Michael P. Frank

27. 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

28. Graph Unions
The union G1G2 of two simple graphs G1=(V1, E1) and G2=(V2,E2) is the simple graph (V1V2, E1E2). a b c d e  a b c d f
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(c) , Michael P. Frank

29. §8.3: Graph Representations & Isomorphism
Adjacency lists.
Adjacency matrices.
Incidence matrices.
Graph isomorphism:
Two graphs are isomorphic iff they are identical except for their node names.
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30. Adjacency Lists
A table with 1 row per vertex, listing its adjacent vertices. b a d c e f
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31. Directed Adjacency Lists
1 row per node, listing the terminal nodes of each edge incident from that node.
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32. Adjacency Matrices A way to represent simple graphs
possibly with self-loops.
Matrix A=[aij], where aij is 1 if {vi, vj} is an edge of G, and is 0 otherwise.
Can extend to pseudographs by letting each matrix elements be the number of links (possibly >1) between the nodes.
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(c) , Michael P. Frank

33. 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 easily be extended to other types of graphs.
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(c) , Michael P. Frank

34. Graph Invariants under Isomorphism
Necessary but not sufficient conditions for G1=(V1, E1) to be isomorphic to G2=(V2, E2):
We must have that |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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35. 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

36. 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

37. §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

38. Paths in Directed Graphs
Same as in undirected graphs, but the path must go in the direction of the arrows.
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(c) , Michael P. Frank

39. 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.
Connected component: connected subgraph
A cut vertex or cut edge separates 1 connected component into 2 if removed.
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(c) , Michael P. Frank

40. 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.
Note strongly implies weakly but not vice-versa.
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41. 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

42. Counting Paths w Adjacency Matrices
Let A be the adjacency matrix of graph G.
The number of paths of length k from vi to vj is equal to (Ak)i,j.
The notation (M)i,j denotes mi,j where [mi,j] = M.
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(c) , Michael P. Frank

43. §8.5: Euler & 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

44. 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

45. Euler Path Theorems
Theorem: A 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 (but not an Euler circuit) 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

46. Euler Circuit Algorithm
Begin with any arbitrary node.
Construct a simple path from it till you get back to start.
Repeat for each remaining subgraph, splicing results back into original cycle.
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(c) , Michael P. Frank

47. Round-the-World Puzzle
Can we traverse all the vertices of a dodecahedron, visiting each once?`
Equivalent graph
Dodecahedron puzzle
Pegboard version
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(c) , Michael P. Frank

48. 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.
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(c) , Michael P. Frank

49. 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

50. §8.6: Shortest-Path Problems
Not covering this semester.
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51. §8.7: Planar Graphs Not covering this semester. 10/14/2018
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52. §8.8: Graph Coloring Not covering this semester. 10/14/2018
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53. §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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54. Tree and Forest Examples
Leaves in green, internal nodes in brown.
A Tree:
A Forest:
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55. 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.
You should know the following terms about rooted trees:
Parent, child, siblings, ancestors, descendents, leaf, internal node, subtree.
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56. 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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57. 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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58. 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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59. 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

60. 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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61. 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?
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(c) , Michael P. Frank

62. 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

63. 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

64. §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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65. 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

66. 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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67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

75. Prefix Codes & Huffman Codingpp
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76. Game Trees pp
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77. §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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78. §9.4: Spanning Trees Not covering this semester. 10/14/2018
(c) , Michael P. Frank

79. §9.5: Minimum Spanning Trees
Not covering this semester.
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(c) , Michael P. Frank