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Optimization Problems Minimum Spanning TreeBehavioral Abstraction
Optimization Problems
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?
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)?
Approximate Solutions
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
A Greedy Algorithm
Consider the weight and value of some foreign coins:foo $6.00 500 gramsbar $4.00 450 gramsbaz $3.00 410 gramsqux $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
Short-Sighted Decisions
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12
Start
End
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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?
A Greedy Algorithm
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14
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7
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76
Start
End
A Greedy Algorithm
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14
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76
Start
End
A Greedy Algorithm
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14
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76
Start
End
A Greedy Algorithm
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14
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76
Start
End
Path = 15
A Greedy Algorithm
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14
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76
Start
End
Shortest Path = 13
Dynamic Planning
Dynamic Planning
• Calculates all of the possible solution options, then chooses the best one.
• Implemented recursively.
• Produces an optimal solution.
• Spends more time.
Bellman’s 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.
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
Dynamic Planning
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Start
Enda
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d
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g
f
e b
Dynamic Planning
b = min(6+g, 5+e, 7+f)
End
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5
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g
f
e b
Notation: ‘x’ means “shortest path to x”
Dynamic Planning
g = min(6+d, 14)
e = min(3+c, 7+d, 7+g)
f = 2+c
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7
3
2
7
c
d
g
f
e
14
a
Dynamic Planning
c = min(5, 11+d)
d = 3
5
11
3
Start
a
c
d
b = min(6+g, 5+e, 7+f)
g = min(6+d, 14)
e = min(3+c, 7+d, 7+g)
f = 2+c
c = min(5, 11+d)
d = 3 via “a to d”
b = min(6+g, 5+e, 7+f)
g = min(6+d, 14)
e = min(3+c, 7+d, 7+g)
f = 2+c
c = min(5, 11+d)
d = 3 via “a to d”
b = min(6+g, 5+e, 7+f)
g = min(6+3, 14)
e = min(3+c, 7+3, 7+g)
f = 2+c
c = min(5, 11+3)
d = 3 via “a to d”
b = min(6+g, 5+e, 7+f)
g = min(9, 14)
e = min(3+c, 10, 7+g)
f = 2+c
c = min(5, 14)
d = 3 via “a to d”
b = min(6+g, 5+e, 7+f)
g = min(9, 14)
e = min(3+c, 10, 7+g)
f = 2+c
c = min(5, 14)
d = 3 via “a to d”
b = min(6+g, 5+e, 7+f)
g = 9 via “a to d to g”
e = min(3+c, 10, 7+g)
f = 2+c
c = 5 via “a to c”
d = 3 via “a to d”
b = min(6+g, 5+e, 7+f)
g = 9 via “a to d to g”
e = min(3+c, 10, 7+g)
f = 2+c
c = 5 via “a to c”
d = 3 via “a to d”
b = min(6+9, 5+e, 7+f)
g = 9 via “a to d to g”
e = min(3+5, 10, 7+9)
f = 2+5
c = 5 via “a to c”
d = 3 via “a to d”
b = min(15, 5+e, 7+f)
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”
b = min(15, 5+e, 7+f)
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”
b = min(15, 5+e, 7+f)
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”
b = min(15, 5+e, 7+f)
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”
b = min(15, 5+8, 7+7)
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”
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”
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”
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”
Dynamic Planning
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7
11
3
14
2
7
Start
Enda
c
d
7
6
g
f
e b
5
35
Shortest Path = 13
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
Questions?
Minimum Spanning Tree
Prim’s AlgorithmKruskal’s Algorithm
The Scenario
• Construct a telephone network…
• We’ve got to connect many cities together– Each city must be connected– We want to minimize the total cable used
The Scenario
• Construct a telephone network…
• We’ve got to connect many cities together– Each city must be connected– We want to minimize the total cable used
Minimum Spanning Tree
• Required: Connect all nodes at minimum cost.
Minimum Spanning Tree
• Required: Connect all nodes at minimum cost.
1 3
2
Minimum Spanning Tree
• Required: Connect all nodes at minimum cost.• Cost is sum of edge weights
1
2
3
2
1 3
Minimum Spanning Tree
• Required: Connect all nodes at minimum cost.• Cost is sum of edge weights
1
2
3
2
1 3
S.T. = 3 S.T. = 5 S.T. = 4
Minimum Spanning Tree
• Required: Connect all nodes at minimum cost.• Cost is sum of edge weights
1 3
2
1 3
2
1 3
2
M.S.T. = 3 S.T. = 5 S.T. = 4
Minimum Spanning Tree
• Required: Connect all nodes at minimum cost.• Cost is sum of edge weights• Can start at any node
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
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
4Pick any 2
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
– Prim’s– Kruskal’s
Prim’s 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”
Prim’s 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
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2124
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9 17
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Start
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2124
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9 3
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8 10
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9 17
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1320
2124
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2 7
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9 3
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8 10
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9 17
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1320
2124
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2 7
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9 3
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8 10
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9 17
11
1320
2124
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2 7
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9 3
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8 10
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9 17
11
1320
2124
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2 7
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9 3
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14
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8 10
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9 17
11
1320
2124
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2 7
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9 3
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14
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8 10
12
9 17
11
1320
2124
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2 7
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18
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9 3
15
14
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8 10
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9 17
11
1320
2124
NO!
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9 3
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8 10
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9 17
11
1320
2124
LOOP
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9 3
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8 10
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9 17
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1320
2124
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2 7
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18
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9 3
15
14
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8 10
12
9 17
11
1320
2124
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2 7
5
18
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9 3
15
14
19
8 10
12
9 17
11
1320
2124
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2 7
5
18
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9 3
15
14
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8 10
12
9 17
11
1320
2124
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2 7
5
6
9 3
10
9
11
Prim’s:MST = 66
An Alternate Algorithm:Kruskal’s Algorithm
Kruskal’s 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
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9 3
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8 10
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9 17
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1320
2124
Sort the Edges
2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 24
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2 7
5
18
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9 3
15
14
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8 10
12
9 17
11
1320
2124
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2 7
5
18
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9 3
15
14
19
8 10
12
9 17
11
1320
2124
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2 7
5
18
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9 3
15
14
19
8 10
12
9 17
11
1320
2124
4
2 7
5
18
6
9 3
15
14
19
8 10
12
9 17
11
1320
2124
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2 7
5
18
6
9 3
15
14
19
8 10
12
9 17
11
1320
2124
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2 7
5
18
6
9 3
15
14
19
8 10
12
9 17
11
1320
2124
4
2 7
5
18
6
9 3
15
14
19
8 10
12
9 17
11
1320
2124
StillNO!
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9 3
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14
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8 10
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9 17
11
1320
2124
Or
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9 3
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14
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8 10
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9 17
11
1320
2124
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2 7
5
18
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9 3
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14
19
8 10
12
9 17
11
1320
2124
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2 7
5
18
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9 3
15
14
19
8 10
12
9 17
11
1320
2124
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2 7
5
18
6
9 3
15
14
19
8 10
12
9 17
11
1320
2124
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2 7
5
6
9 3
10
9
11
Kruskal’s:MST = 66
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2 7
5
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9 3
10
9
11
Prim’s:MST = 66
Summary
• Minimum spanning trees– Connect all vertices– With no cycles– And minimize the total edge cost
• Prim’s and Kruskal’s algorithm– Generate minimum spanning trees– Give same total cost, but may give different
trees (if graph has edges with same weight)
Questions?
Introduction to theObject-Oriented Paradigm
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
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?
• We’d like some way of representing sucha structure in our programs.
Issues
• As is, there is no way to “protect” against violating the specified behaviors.
Items
Enqueue Dequeue
Sneak into Middle
Issues
• We’d like a way to put a “wrapper” around our structure to protect against this.
Items
Enqueue Dequeue
Sneak into Middle
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.
Procedural Abstraction
• Procedural Abstractions organize instructions.
Function PowerGive me two numbers (base & exponent)
I’ll return to you baseexponent
??? Implementation ???
Data Abstraction
• Data Abstractions organize data.
Name (string)
GPA (num)
Letter Grade (char)
Student Number (num)
StudentType
Behavioral Abstraction
• Behavioral Abstractions combine procedural and data abstractions.
Data State
EnqueueIs Full
Is Empty Dequeue
Initialize
Queue Object
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 doesn’t mean you’re doing OO programming.
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
Information Hiding
Information Hiding means that the user has enough information to use the interface, and no more
Example: stick shift• We don’t care about the inner workings…• We just want to get the car going!
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R
Encapsulation
Item 1 Item 2
Item3
Encapsulation is the grouping of related things together within some structure
Encapsulation via Algorithms
InstructionsProcedure
Function
Algorithm
Data
Algorithms encapsulate modules, data, and instructions.
Encapsulating Instructions
InstructionsInstructions
Module call
Procedure/Function
Instructions
Modules encapsulate instructions.
Encapsulating Data
Field 1 Field 2
Record
Record
Records allow us to do this with data
Revisiting the “Where is it?” Question
• Notice we still have no way of identifying the idea we’re discussing…– The Queue is still in the “ether.”
• We’d like to encapsulate the data (regardless of it’s actual representation) with the behavior (modules).
• Once we do this, we’ve got a logical entity to which we can point and say, “there it is!”
Encapsulating Data with Methods
EnqueueData
Dequeue
Queue
Abstract data types (ADTs) allow us to do this with logically related data and modules
• An idea, a concept, an abstraction of the “big picture”
• It encapsulates the abstract behaviors and data implied by the thing being modeled.
Abstract Data Types
Data State
EnqueueIs Full
Is Empty Dequeue
Initialize
Queue
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.
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 we’re representing as a thing regardless of it’s actual implementation.
Questions?
