Transcript Lecture 3

Lecture 3: Classifying Problems
• One of the main themes of this course will be to
classify problems in various ways
– By solvability
• Solvable, “half-solvable”, unsolvable
• We will focus our study on decision problems
– function (one correct answer for every input)
– finite range (yes or no is the correct output)
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Classification Process
• Take some set of problems and partition it
into two or more subsets of problems where
membership in a subset is based on some
shared problem characteristic
Subset 1
Set of Problems
Subset 2
Subset 3
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Classify by Solvability
• Criteria used is whether or not the problem
is solvable
– that is, does there exist a C++ program which
solves the problem?
Set of All Problems
Solvable Problems Unsolvable Problems
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Function Problems
• We will focus on problems where the
mapping from input to output is a function
Set of All Problems
Non-Function Problems Function Problems
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General (Relation) Problem
• the mapping is a relation
– that is, more than one output is possible for a
given input
Inputs
Outputs
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Criteria for Function Problems
• mapping is a function
– unique output for each input
Inputs
Outputs
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Example Non-Function Problem
• Divisor Problem
– Input: Positive integer n
– Output: A positive integral divisor of n
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Inputs
1
3
9
Outputs
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Example
• Sorting
• Multiplication Problem
– Input: 2 integers x and y
– Output: xy
2,5
Inputs
10
Outputs
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Another Example
• Maximum divisor problem
– Input: Positive integer n
– Output: size of maximum divisor of n smaller
than n
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Inputs
1
3
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Outputs
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Decision Problems
• We will focus on function problems where
the correct answer is always yes or no
Set of Function Problems
Non-Decision Problems Decision Problems
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Criteria for Decision Problems
• Output is yes or no
– range = {Yes, No}
• Note, problem must be a function problem
– only one of Yes/No is correct
Yes
No
Inputs
Outputs
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Example
• Decision sorting
– Input: list of integers
– Yes/No question: Is the list in nondecreasing
order?
(1,3,2,4)
(1,2,3,4)
Inputs
Yes
No
Outputs
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Another Example
• Decision multiplication
– Input: Three integers x, y, z
– Yes/No question: Is xy = z?
(3,5,14)
(3,5,15)
Inputs
Yes
No
Outputs
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A Third Example
• Decision Divisor Problem
– Input: Two integers x and y
– Yes/No question: Is y a divisor of x?
(14,5)
(14,7)
Inputs
Yes
No
Outputs
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Focus on Decision Problems
Set of All Problems
Solvable Problems Unsolvable Problems
Other
Probs
Decision
Problems
• When studying solvability, we are going to
focus specifically on decision problems
– There is no loss of generality, but we will not
explore that here
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Finite Domain Problems
• These problems have only a finite number of
inputs
Set of All Problems
Finite Domain Problems Infinite Domain Problems
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Lack of Generality
Set of All Problems
Solvable Problems Unsolvable Problems
Infinite
Domain
Finite
Domain
Empty
• All finite domain problems can be solved
using “table lookup” idea
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Table Lookup Program
int main(string x) {
switch x {
case “Bill”: return(3);
case “Judy”: return(25);
case “Tom”: return(30);
default: cerr << “Illegal input\n”;
}
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Key Concepts
• Classification Theme
• Decision Problems
– Important subset of problems
– We can focus our attention on decision
problems without loss of generality
– Same is not true for finite domain problems
• Table lookup
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