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Title: Optimization Problems Minimum Spanning Tree Behavioral Abstraction


1
Optimization Problems Minimum Spanning
TreeBehavioral Abstraction
Lecture 17
2
Optimization Problems
3
Optimization Algorithms
  • Many real-world problems involve maximizing or
    minimizing a value
  • How can a car manufacturer get the most parts out
    of a piece of sheet metal?
  • How can a moving company fit the most furniture
    into a truck of a certain size?
  • How can the phone company route calls to get the
    best use of its lines and connections?
  • How can a university schedule its classes to make
    the best use of classrooms without conflicts?

4
Optimal vs. Approximate Solutions
  • Often, we can make a choice
  • Do we want the guaranteed optimal solution to the
    problem, even though it might take a lot of
    work/time to find?
  • Do we want to spend less time/work and find an
    approximate solution (near optimal)?

5
Approximate Solutions
6
Greedy Algorithms
  • Approximates optimal solution
  • May or may not find optimal solution
  • Provides quick and dirty estimates
  • A greedy algorithm makes a series of
    short-sighted decisions and does not look
    ahead
  • Spends less time

7
A Greedy Algorithm
  • Consider the weight and value of some foreign
    coins
  • foo 6.00 500 grams
  • bar 4.00 450 grams
  • baz 3.00 410 grams
  • qux 0.50 300 grams
  • If we can only fit 860 grams in our pockets...
  • A greedy algorithm would choose
  • 1 foo 500 grams 6.00
  • 1 qux 300 grams 0.50
  • Optimal solution is
  • 1 bar 450 grams 4.00
  • 1 baz 410 grams 3.00

Total of 6.50
Total of 7.00
8
Short-Sighted Decisions
1
Start
2
End
12
1
9
The Shortest Path Problem
  • Given a directed, acyclic, weighted graph
  • Start at some vertex A
  • What is the shortest path from start vertex A to
    some end vertex B?

10
A Greedy Algorithm
2
7
5
Start
11
3
5
3
7
End
7
6
14
6
11
A Greedy Algorithm
2
7
5
Start
11
3
5
3
7
End
7
6
14
6
12
A Greedy Algorithm
2
7
5
Start
11
3
5
3
7
End
7
6
14
6
13
A Greedy Algorithm
2
7
5
Start
11
3
5
3
7
End
7
6
14
6
Path 15
14
A Greedy Algorithm
2
7
5
Start
11
3
5
3
7
End
7
6
14
6
Shortest Path 13
15
Dynamic Planning
16
Dynamic Planning
  • Calculates all of the possible solution options,
    then chooses the best one.
  • Implemented recursively.
  • Produces an optimal solution.
  • Spends more time.

17
Bellmans Principle of Optimality
  • Regardless of how you reach a particular state
    (graph node), the optimal strategy for reaching
    the goal state is always the same.
  • This greatly simplifies the strategy for
    searching for an optimal solution.

18
The Shortest Path Problem
  • Given a directed, acyclic, weighted graph
  • What is the shortest path from the start vertex
    to some end vertex?
  • Minimize the sum of the edge weights

19
Dynamic Planning
c
2
f
7
5
Start
11
b
3
e
5
3
7
d
a
End
7
6
14
6
g
20
Dynamic Planning
f
Notation x means shortest path to x
7
b
e
5
End
6
b min(6g, 5e, 7f)
g
21
Dynamic Planning
c
2
f
3
e
7
d
a
7
6
14
g min(6d, 14) e min(3c, 7d, 7g) f 2c
g
22
Dynamic Planning
c
5
Start
11
3
d
a
c min(5, 11d) d 3
23
b min(6g, 5e, 7f) g min(6d, 14) e
min(3c, 7d, 7g) f 2c c min(5, 11d) d 3
via a to d
24
b min(6g, 5e, 7f) g min(6d, 14) e
min(3c, 7d, 7g) f 2c c min(5, 11d) d 3
via a to d
25
b min(6g, 5e, 7f) g min(63, 14) e
min(3c, 73, 7g) f 2c c min(5, 113) d 3
via a to d
26
b min(6g, 5e, 7f) g min(9, 14) e
min(3c, 10, 7g) f 2c c min(5, 14) d 3
via a to d
27
b min(6g, 5e, 7f) g min(9, 14) e
min(3c, 10, 7g) f 2c c min(5, 14) d 3
via a to d
28
b min(6g, 5e, 7f) g 9 via a to d to g e
min(3c, 10, 7g) f 2c c 5 via a to c d
3 via a to d
29
b min(6g, 5e, 7f) g 9 via a to d to g e
min(3c, 10, 7g) f 2c c 5 via a to c d
3 via a to d
30
b min(69, 5e, 7f) g 9 via a to d to g e
min(35, 10, 79) f 25 c 5 via a to c d
3 via a to d
31
b min(15, 5e, 7f) g 9 via a to d to g e
min(8, 10, 16) f 7 via a to c to f c 5 via
a to c d 3 via a to d
32
b min(15, 5e, 7f) g 9 via a to d to g e
min(8, 10, 16) f 7 via a to c to f c 5 via
a to c d 3 via a to d
33
b min(15, 5e, 7f) g 9 via a to d to g e
8 via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
34
b min(15, 5e, 7f) g 9 via a to d to g e
8 via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
35
b min(15, 58, 77) g 9 via a to d to g e
8 via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
36
b min(15, 13, 14) g 9 via a to d to g e 8
via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
37
b min(15, 13, 14) g 9 via a to d to g e 8
via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
38
b 13 via a to c to e to b g 9 via a to d
to g e 8 via a to c to e f 7 via a to c
to f c 5 via a to c d 3 via a to d
39
Dynamic Planning
c
2
f
7
5
Start
11
b
3
e
5
3
7
d
a
End
7
6
14
6
Shortest Path 13
g
40
Summary
  • Greedy algorithms
  • Make short-sighted, best guess decisions
  • Required less time/work
  • Provide approximate solutions
  • Dynamic planning
  • Examines all possible solutions
  • Requires more time/work
  • Guarantees optimal solution

41
Questions?
42
Minimum Spanning Tree
  • Prims AlgorithmKruskals Algorithm

43
The Scenario
  • Construct a telephone network
  • Weve got to connect many cities together
  • Each city must be connected
  • We want to minimize the total cable used

44
The Scenario
  • Construct a telephone network
  • Weve got to connect many cities together
  • Each city must be connected
  • We want to minimize the total cable used

45
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.

46
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.

1
3
2
47
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights

1
1
3
3
2
2
48
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights

1
1
3
3
2
2
S.T. 3
S.T. 5
S.T. 4
49
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights

1
1
1
3
3
3
2
2
2
M.S.T. 3
S.T. 5
S.T. 4
50
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights
  • Can start at any node

51
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights
  • Can start at any node
  • Unique solution.
  • Can come from different sets of edges

4
4
4
52
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights
  • Can start at any node
  • Unique solution.
  • Can come from different sets of edges

4
4
Pick any 2
4
53
Minimum Spanning Tree
  • Required Connect all nodes at minimum cost.
  • Cost is sum of edge weights
  • Can start at any node
  • Unique solution.
  • Can come from different sets of edges
  • Two algorithms
  • Prims
  • Kruskals

54
Prims Algorithm
  • Start with any vertex.
  • All its edges are candidates.
  • Stop looking when all vertices are picked
  • Otherwise repeat
  • Pick minimum edge (no cycles) and connect the
    adjacent vertex
  • Add all edges from that new vertex to our
    candidates

55
Prims Algorithm
  • Pick any vertex and add it to vertices list
  • Loop
  • Exitif (vertices contains all the vertices
    in graph)
  • Select shortest, unmarked edge coming from
    vertices list
  • If (shortest edge does NOT create a cycle)
    then
  • Add that edge to your edges
  • Add the adjoining vertex to the vertices list
  • Endif
  • Mark that edge as having been considered
  • Endloop

56
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
57
Start
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
58
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
59
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
60
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
61
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
62
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
63
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
64
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
NO!
9
17
19
11
65
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
LOOP
9
17
19
11
66
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
67
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
68
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
69
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
70
4
5
Prims MST 66
3
9
2
7
6
10
9
11
71
An Alternate AlgorithmKruskals Algorithm
72
Kruskals Algorithm
  • Sort edges in graph in increasing order
  • Select the shortest edge
  • Loop
  • Exitif all edges examined
  • Select next shortest edge (if no cycle created)
  • Mark this edge as examined
  • Endloop
  • This guarantees an MST, but as it is built, edges
    do not have to be connected

73
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
74
Sort the Edges
  • 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14,
    15, 17, 18, 19, 20, 21, 24

75
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
76
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
77
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
78
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
79
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
80
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
81
4
5
3
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18
2
7
14
15
12
21
24
13
20
8
6
10
Still NO!
9
17
19
11
82
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18
2
7
14
15
12
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24
13
20
8
6
10
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17
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11
83
Or
84
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
85
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
86
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
87
4
5
3
9
18
2
7
14
15
12
21
24
13
20
8
6
10
9
17
19
11
88
4
5
3
9
2
7
Kruskals MST 66
6
10
9
11
89
4
5
Prims MST 66
3
9
2
7
6
10
9
11
90
Summary
  • Minimum spanning trees
  • Connect all vertices
  • With no cycles
  • And minimize the total edge cost
  • Prims and Kruskals algorithm
  • Generate minimum spanning trees
  • Give same total cost, but may give different
    trees (if graph has edges with same weight)

91
Questions?
92
Introduction to theObject-Oriented Paradigm
93
The Scenario
  • Recall the concept of a Queue
  • Defined by a set of behaviors
  • Enqueue to the end
  • Dequeue from the front
  • First in, first out

Items
94
Where is the Queue?
  • The logical idea of the queue consisted of
  • Data stored in some structure (list, array, etc.)
  • Modules to act on that data
  • But where is the queue?
  • Wed like some way of representing sucha
    structure in our programs.

95
Issues
  • As is, there is no way to protect against
    violating the specified behaviors.

Enqueue
Dequeue
Sneak into Middle
Items
96
Issues
  • Wed like a way to put a wrapper around our
    structure to protect against this.

Enqueue
Dequeue
Sneak into Middle
Items
97
Motivation
  • We need ways to manage the growing complexity and
    size of programs.
  • We can better model our algorithmic solutions to
    real world phenomenon.
  • Contracts of responsibility can establish clearer
    communication and more protected manipulation.

98
Procedural Abstraction
  • Procedural Abstractions organize instructions.

Function Power Give me two numbers (base
exponent) Ill return to you baseexponent ???
Implementation ???
99
Data Abstraction
  • Data Abstractions organize data.

StudentType
Name (string) GPA (num) Letter Grade
(char) Student Number (num)
100
Behavioral Abstraction
  • Behavioral Abstractions combine procedural and
    data abstractions.

Queue Object

Enqueue
Is Full
Data State
Is Empty
Dequeue
Initialize
101
The Object-Oriented Paradigm
  • Instances of behavioral abstractions are known as
    objects.
  • Objects have a clear interface by which they send
    and receive messages (communicate).
  • OO is a design and approach issue. Just because
    a language offers object-oriented features
    doesnt mean youre doing OO programming.

102
Benefits of Object-Oriented Paradigm
  • Encapsulation information hiding, control, and
    design
  • Reuse create libraries of tested, optimized
    classes
  • Adaptability controlled interactions with
    minimal coupling
  • Better model real world and larger problems
  • Break down tasks better

103
Information Hiding
  • Information Hiding means that the user has enough
    information to use the interface, and no more
  • Example stick shift
  • We dont care about the inner workings
  • We just want to get the car going!

1
3
5
2
4
R
104
Encapsulation
  • Encapsulation is the grouping of related things
    together within some structure

Item 1
Item 2
Item3
105
Encapsulation via Algorithms
  • Algorithms encapsulate modules, data, and
    instructions.


Algorithm
Procedure
Instructions
Function
Data
106
Encapsulating Instructions
  • Modules encapsulate instructions.


Procedure/Function
Instructions
Instructions
Module call
Instructions
107
Encapsulating Data
  • Records allow us to do this with data

Record
Field 1
Field 2
Record
108
Revisiting the Where is it? Question
  • Notice we still have no way of identifying the
    idea were discussing
  • The Queue is still in the ether.
  • Wed like to encapsulate the data (regardless of
    its actual representation) with the behavior
    (modules).
  • Once we do this, weve got a logical entity to
    which we can point and say, there it is!

109
Encapsulating Data with Methods
  • Abstract data types (ADTs) allow us to do this
    with logically related data and modules

Queue
Enqueue
Data
Dequeue
110
Abstract Data Types
  • An idea, a concept, an abstraction of the big
    picture
  • It encapsulates the abstract behaviors and data
    implied by the thing being modeled.

Queue

Enqueue
Is Full
Data State
Is Empty
Dequeue
Initialize
111
Achieving Behavioral Abstraction
  • Abstract data types (ADTs) are concepts.
  • We require some way to implement these common
    abstractions so we can write them once, verify
    that they are correct, and reuse them.
  • This would save us from having to re-do work.
    For example, every time we create a queue we did
  • List_Node definesa ...q_front isoftype
    ...q_tail isoftype ...
  • procedure Enqueue(...)
  • procedure Dequeue(...)
  • We need an algorithmic construct that will allow
    us to bundle these things together the class.

112
Summary
  • Behavioral abstraction combines data abstraction
    with procedural abstraction.
  • The object-oriented paradigm focuses on the
    interaction and manipulation of objects.
  • An Abstract Data Type (ADT) allows us to think of
    what were representing as a thing regardless of
    its actual implementation.

113
Questions?
114
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