Recursion

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Chapter 6

• Recursion

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Chapter Objectives

• Learn about recursive definitions
• Explore the base case and the general case of a
  recursive definition
• Discover recursive algorithms
• Learn about recursive functions
• Explore how to use recursive functions to
  implement recursive algorithms
• Learn about recursion and backtracking

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Recursive Definitions

• Recursion
• Process of solving a problem by reducing it to
  smaller versions of itself
• Recursive definition
• Definition in which a problem is expressed in
  terms of a smaller version of itself
• Has one or more base cases

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Recursive Definitions

• Recursive algorithm
• Algorithm that finds the solution to a given
  problem by reducing the problem to smaller
  versions of itself
• Has one or more base cases
• Implemented using recursive functions
• Recursive function
• function that calls itself
• Base case
• Case in recursive definition in which the
  solution is obtained directly
• Stops the recursion

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Recursive Definitions

• General solution
• Breaks problem into smaller versions of itself
• General case
• Case in recursive definition in which a smaller
  version of itself is called
• Must eventually be reduced to a base case

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Tracing a Recursive Function

• Recursive function
• Has unlimited copies of itself
• Every recursive call has
• its own code
• own set of parameters
• own set of local variables

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Tracing a Recursive Function

• After completing recursive call
• Control goes back to calling environment
• Recursive call must execute completely before
  control goes back to previous call
• Execution in previous call begins from point
  immediately following recursive call

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Recursive Definitions

• 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 recursive function in
  which the last statement executed is the
  recursive call
• Infinite recursion the case where every
  recursive call results in another recursive call

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Designing Recursive Functions

• Understand problem requirements
• Determine limiting conditions
• Identify base cases

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Designing Recursive Functions

• Provide direct solution to each base case
• Identify general case(s)
• Provide solutions to general cases in terms of
  smaller versions of itself

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Recursive Factorial Function

• int fact(int num)
• if(num 0)
• return 1
• else
• return num fact(num 1)

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Recursive Factorial Trace
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Recursive Implementation Largest Value in Array

• int largest(const int list, int lowerIndex, int
  upperIndex)
• int max
• if(lowerIndex upperIndex) //the size of
  the sublist is 1
• return listlowerIndex
• else
• max largest(list, lowerIndex 1,
  upperIndex)
• if(listlowerIndex gt max)
• return listlowerIndex
• else
• return max

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Execution of largest(list, 0, 3)
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Recursive Fibonacci

• int rFibNum(int a, int b, int n)
• if(n 1)
• return a
• else if(n 2)
• return b
• else
• return rFibNum(a, b, n - 1) rFibNum(a,
  b, n - 2)

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Execution of rFibonacci(2,3,5)
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Towers of Hanoi Problem with Three Disks
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Towers of Hanoi Three Disk Solution
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Towers of Hanoi Three Disk Solution
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Towers of Hanoi Recursive Algorithm

• void moveDisks(int count, int needle1, int
  needle3, int needle2)
• if(count gt 0)
• moveDisks(count - 1, needle1, needle2,
  needle3)
• coutltlt"Move disk "ltltcountltlt from
  "ltltneedle1
• ltlt to "ltltneedle3ltlt"."ltltendl
• moveDisks(count - 1, needle2, needle3,
  needle1)

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Decimal to Binary Recursive Algorithm

• void decToBin(int num, int base)
• if(num gt 0)
• decToBin(num/base, base)
• coutltltnum base

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Execution of decToBin(13,2)
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Recursion or Iteration?

• Two ways to solve particular problem
• Iteration
• Recursion
• Iterative control structures uses looping to
  repeat a set of statements
• Tradeoffs between two options
• Sometimes recursive solution is easier
• Recursive solution is often slower

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Backtracking Algorithm

• Attempts to find solutions to a problem by
  constructing partial solutions
• Makes sure that any partial solution does not
  violate the problem requirements
• Tries to extend partial solution towards
  completion

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Backtracking Algorithm

• If it is determined that partial solution would
  not lead to solution
• partial solution would end in dead end
• algorithm backs up by removing the most recently
  added part and then tries other possibilities

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Solution to 8-Queens Puzzle
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4-Queens Puzzle
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4-Queens Tree
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8 X 8 Square Board
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Chapter Summary

• Recursive definitions
• Recursive algorithms
• Recursive functions
• Base cases
• General cases

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Chapter Summary

• Tracing recursive functions
• Designing recursive functions
• Varieties of recursive functions
• Recursion vs. Iteration
• Backtracking
• N-Queens puzzle