Module 3: Classifying Problems

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Module 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

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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?

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Function Problems

• We will focus on problems where the mapping from
  input to output is a function

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General (Relation) Problem

• the mapping is a relation
• that is, more than one output is possible for a
  given input

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Criteria for Function Problems

• mapping is a function
• unique output for each input

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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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Example Function Problems

• Sorting
• Multiplication Problem
• Input 2 integers x and y
• Output xy

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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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Decision Problems

• We will focus on function problems where the
  correct answer is always yes or no

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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

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Example

• Decision sorting
• Input list of integers
• Yes/No question Is the list in nondecreasing
  order?

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Another Example

• Decision multiplication
• Input Three integers x, y, z
• Yes/No question Is xy z?

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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?

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Focus on 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

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Lack of Generality

• 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 ltlt 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