Chapter 16: Recursion

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C ProgrammingProgram Design IncludingData
Structures, Fifth Edition

• Chapter 16 Recursion

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Objectives

• In this chapter, you will
• Learn about recursive definitions
• Explore the base case and the general case of a
  recursive definition
• Discover what is a recursive algorithm
• Learn about recursive functions
• Explore how to use recursive functions to
  implement recursive algorithms

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Recursive Definitions

• Recursion solving a problem by reducing it to
  smaller versions of itself
• 0! 1 (1)
• n! n x (n-1)! if n gt 0 (2)
• The definition of factorial in equations (1) and
  (2) is called a recursive definition
• Equation (1) is called the base case
• Equation (2) is called the general case

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Recursive Definitions (cont'd.)

• Recursive definition defining a problem in terms
  of a smaller version of itself
• Every recursive definition must have one (or
  more) base cases
• The general case must eventually reduce to a base
  case
• The base case stops the recursion

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Recursive Definitions (cont'd.)

• Recursive algorithm finds a solution by reducing
  problem to smaller versions of itself
• Must have one (or more) base cases
• General solution must eventually reduce to a base
  case
• Recursive function a function that calls itself
• Recursive algorithms are implemented using
  recursive functions

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Recursive Definitions (cont'd.)

• Think of a recursive function as having
  infinitely many copies of itself
• After completing a particular recursive call
• Control goes back to the calling environment,
  which is the previous call
• Execution begins from the point immediately
  following the recursive call

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Direct and Indirect Recursion

• Directly recursive a function that calls itself
• Indirectly recursive a function that calls
  another function and eventually results in the
  original function call
• Tail recursive function function in which the
  last statement executed is the recursive call
• Example the function fact

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Infinite Recursion

• Infinite recursion every recursive call results
  in another recursive call
• In theory, infinite recursion executes forever
• Because computer memory is finite
• Function executes until the system runs out of
  memory
• Results in an abnormal program termination

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Infinite Recursion (cont'd.)

• To design a recursive function
• Understand problem requirements
• Determine limiting conditions
• Identify base cases and provide a direct solution
  to each base case
• Identify general cases and provide a solution to
  each general case in terms of smaller versions of
  itself

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Problem Solving Using Recursion
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Problem Solving Using Recursion (cont'd.)
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Problem Solving Using Recursion (cont'd.)
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Problem Solving Using Recursion (cont'd.)
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Problem Solving Using Recursion (cont'd.)
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Problem Solving Using Recursion (cont'd.)
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Problem Solving Using Recursion (cont'd.)
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Tower of Hanoi Analysis

• How long would it take to move 64 disks from
  needle 1 to needle 3?
• About 500 years (rate of 1 billion moves/sec)
• 264 ? 1 moves to move all 64 disks to needle 3
• If computer can generate 1 billion (109)
  moves/sec, the number of moves it can generate in
  one year is
• (3.2 107) 109 3.2 1016
• Computer time required for 264 moves
• 264 1.6 1019 1.6 1016 103 (3.2
  1016) 500

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Recursion or Iteration?

• There are usually two ways to solve a particular
  problem
• Iteration (looping)
• Recursion
• Which method is betteriteration or recursion?
• Factors
• Nature of the problem
• Efficiency

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Recursion or Iteration? (cont'd.)

• Whenever a function is called
• Memory space for its formal parameters and
  (automatic) local variables is allocated
• When the function terminates
• That memory space is then deallocated
• Every (recursive) call has its own set of
  parameters and (automatic) local variables

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Recursion or Iteration? (cont'd.)

• Overhead associated with executing a (recursive)
  function in terms of
• Memory space
• Computer time
• A recursive function executes more slowly than
  its iterative counterpart

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Recursion or Iteration? (cont'd.)

• On slower computers, especially those with
  limited memory space
• The slow execution of a recursive function would
  be visible
• Todays computers are fast and have inexpensive
  memory
• Execution of a recursion function is not
  noticeable

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Recursion or Iteration? (cont'd.)

• Choice between the two alternatives depends on
  the nature of the problem
• Mission control systems or similar
• Efficiency is critical and dictates the solution
  method
• Sometimes iterative solution is more obvious and
  easier to understand
• If the definition of a problem is inherently
  recursive, consider a recursive solution

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Programming Example Converting a Number from
Binary to Decimal

• Use recursion to convert a nonnegative integer in
  decimal format (base 10) into the equivalent
  binary number (base 2)
• Define some terms
• Let x be an integer
• Rightmost bit of x remainder of x after division
  by 2
• Rightmost bit of 33 is 1
• Rightmost bit of 28 is 0

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Programming Example (cont'd.)

• To find the binary representation of 35
• Divide 35 by 2
• The quotient is 17 and the remainder is 1
• Divide 17 by 2
• The quotient is 8 and the remainder is 1
• Divide 8 by 2
• The quotient is 4 and the remainder is 0
• Continue process until quotient becomes 0
• Rightmost bit of 35 cannot be printed until we
  have printed the rightmost bit of 17

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Programming Example (cont'd.)

• Binary representation of 35 is the binary
  representation of 17 (35/2) followed by the
  rightmost bit of 35

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Summary

• Recursion process of solving a problem by
  reducing it to smaller versions of itself
• Recursive definition defines a problem in terms
  of smaller versions of itself
• Has one or more base cases
• Recursive algorithm solves a problem by reducing
  it to smaller versions of itself
• Has one or more base cases

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Summary (cont'd.)

• The solution to the problem in a base case is
  obtained directly
• Recursive function function that calls itself
• Must have one or more base cases
• Recursive algorithms are implemented using
  recursive functions
• The general solution breaks the problem into
  smaller versions of itself

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Summary (cont'd.)

• The general case must eventually be reduced to a
  base case
• The base case stops the recursion
• Directly recursive a function calls itself
• Indirectly recursive a function calls another
  function and eventually calls the original
• Tail recursive the last statement executed is
  the recursive call